"SPACE AND FUNCTION ANALYSIS"
"A Computer System for the generation of
functional layouts in the S.A.R. Methodology".
by
Alfonso Govela
B. Arch. Universidad Iberoamericana,
Mexico D.F.,
1974
submitted in partial fulfillment
of the requirements for the degree of
Master of Architecture
in
Advanced Studies.
at the
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
February 1977.
Signature of Author
Departlent of Architecture, February 1977
Certified by
Prof. Nicholas Negroponte, As ociat
Thesis Supervisor.
-
Accepted by
Chairman, Departmental Committee for
Graduate Students.
Rotch
APR
1 1977
rofessor
2
ACKNOWLEDGEMENTS
I thank my Thesis Advisor, Associate Professor Nicholas
Negroponte, for presenting, in a Su'mmer Session back in 1973, the
possibilities of computer applications in Architecture, and for
providing later on the opportunity and the support to work with him
project.
and to develop this
I thank John N. Habraken, Head of the Department, for the
patient sessions where he explained in simple terms the basic concepts
of the S.A.R. Methodology that appear here in a convoluted way.
I thank Mike Gerzso for the discussions we have had on
Architecture and Computers, and for starting, through his work, my
interest in Production Systems.
issues,
I thank Seth Steinberg for his constant help in conceptual
matters.
programming techniques and practical
The work reported herein has been sponsored by CONACYT,
Consejo Nacional de Ciencia y Tecnologia, Mexico, D.F., Mexico.
Alfonso Govela
February 1977.
3
TABLE OF CONTENTS
Abstract .. ...................................................
5
0.
Introduction .................................................
6
1.
Problem Definition and Assumptions ...........................
1.1. Background ..............................................
1.2. Problem Definition ......................................
1.3. General Assumptions .....................................
9
9
12
13
2.
Methodological Basis and Enumeration Techniques ..............
2.1. S.A.R. Formulation of Design Problems
...........
2.1.1. Parts and Relations .........
...........
2.1.2. Levels of Definition ........
...........
2.1.3. Operations of Analysis ......
2.2. Enumeration Techniques .............
...........
on
2.2.1. Problem-Solving, Representati
2.2.2. Representation ..............
...........
2.2.2.1. Basic Model ........
and.Search.
2.2.2.2. Graph Notation.....
...........
2.2.3. Main Types of Representations
...........
2.2.3.1. State-Space........
...........
2.2.3.2. Problem-Reduction
...........
2.2.3.3. Theorem-Proving
2.2.4. Search ......................
18
19
19
22
24
31
33
35
36
38
44
44
50
54
56
56
63
2.2.4.1. Basic Techniques ...
2.2.4.2. Backtracking .......
3. Space and Function Analysis .............
...........
3.1. Formulation ........................
...........
3.1.1. Site ........................
. .. .. .. .. . .
3.1.1.1. Actual Site ........
...........
3.1.1.2. Formal Site ........
...........
3.1.2. Elements ....................
...........
3.1.2.1. Physical Units.....
...........
3.1.2.2. Use Space ..........
...........
3.1.2.3. Restrictions .......
3.1.3. Relations between Elements ..
3.1.3.1. Simple and Compound Relations ...........
3.1.4. Position of Elements .............................
3.1.4.1. Levels of Definition,
Expansion of Position Rules.............
3.1.4.2. Position Rules in overlapping zones.....
3.2. Standards ................................ ..............
79
79
79
80
82
87
88
88
89
90
92
97
99
105
109
4
3.3. Enumeration ............................................
111
3.3.1. Overview ........................................ 112
3.3.2. Description
..................................... 114
3.3.2.1. Truth Table ............................ 120
3.3.2.2. Evaluation of Position Rules ...........
123
3.3.2.3. Preclusion --.............................
126
3.3.2.4. Zone Dimensions
........................
124
3.3.2.5. Sector Dimensions ...................... 130
3.3.2.6. Permutation of Elements ........ ........
133
3.3.2.7. Margin Dimensions ........................ 146
3.3.2.8. Evaluation of Element Constraints ......
147
3.3.2.9. Circulation ............................... 154
4. Computer System ---.............................................
4.1. Relational Data Base
....................- --- -- --- -----4.2. Spatial Representation ...................................
4.3. Query Language .........................................
4.4. Search .................................................
5. Conclusions -.--. ---.----.-.-.--.--.-.----.-Appendix
158
159
168
174
177
.......-.--.--. 181
....................................................
185
185
5
Abstract
"SPACE AND FUNCTION ANALYSIS"
"A Computer System for the generation of
functional layouts in the S.A.R. Methodology".
Alfonso Govela
Submitted to the Department of Architecture on February 1977
in partial fulfillment of the requirements for the degree of Master
of Architecture in Advanced Studies.
As part of the S.A.R. Methodology, a set of computer
programs has been implemented to carry on the systematic generation
the possible functional layouts for a given design criteria.
of all
They are intended to help the designer analyze and evaluate
function, and display the
the relationships between a space and its
consequences of different design standards on different sizes and
layouts of spaces.
The main assumption is that a space can be analyzed
functionally by looking at characteristic arrangements of furniture
or equipment, that correspond to a certain function.
A function can be defined in terms of the location,
dimensions and relations between furniture elements and spaces.
A set of design standards describe a spatial system and
constraint a solution space where particular layouts can be
effectively, and if necessary, exhaustively explored by a procedure
that generates as many arrangements as desired.
This generative capability is aimed to help in the
development and evaluation of standards for spatial performance. By
studying the different layouts that each set of standards permits,
different evaluation techniques can be defined to compare, select
and agree on the most adequate criteria for an actual situation or
an hypothetical case.
TSa
Thesis Supervisor:
Nicholas Negroponte
Title: Associate Professor of Architecture.
6
0.
Introduction:
This T hesis
architectural
fu nctions
it reflects this
comes
it
hand
design
from
an
on
their
other
the
an
from
generat ive
and
tools within
design and
grown
it has
one
On
sections.
in spatial
Interest
soatial
Interests
through several
apolicatlons
as a nalytical
techniques,
formulation of
of
different
the
computer
in
with the
the analysis
in t he
methodolo gles,
Interest
and
I t has origirated
consequences.
and
concerned
is
of
framework
design.
The main proble m that
logical
in
S.A.R.
Methodol ogy.
characteristics,
all
the
conceptual
a participatory design
alternative
models
that
field
of
solution
have
become
Artificial
structured view of the
which
basis
functional
to
mass
spatial
enumerates
function an d
come
from
two
a space.
different
discipline origin ated as an
housing
standard practice
Intelligence.
problems in
standards
provide a
an d
present a procedure that
a
first
" supports"
to
functions
possible relations between
Its
fields,
and
attempts
spatial
way to define
systematic
It
the
is
designing
orocess of
operation in the
the
it approaches
are
probl em s,
in the
The
first
design on the
described,
and
the
computer
provided a
basis
while
of
the
7
"Problem-So lving"
through
second,
A I thou gh
around
of
first
'Tree-Searching'
different
generating the
permit.
represents
new
Ideas
sometime. However,
its application
an
In
different
in
Idea that
we might say, is an
second,
for
is
case
the
t he
architecture,
this
way
layouts that these standards
functional
has been
a
provided
techniques,
and
in
sense.
Important
Pr cb Iem-So lv Ing
has been oriented to the representation of
procedures
find
of
range
that
solutions
po ssi bilitles
procedure tha t
"
sense, beside s
b eing far
design, would
by
searching
according to some
given rules.
designs* or finds design solutions
mI ss the
away in terms of
issue of
the
prob lem
and
A
in this
our knowledge of
in design.
values implicit
What matters is not only the solution, but the
of
a
through
definition
the process to reach this result,
as
values are de fin ed constantly In each decision taken.
In design, it is more relevant the problem of
Not
how
we
ge t
solutions
but how do we evaluate
consequences. To analyze decisions, we
their
consequences and
through
the
we have
enumeratior
of
have
to enumerate
results
decision against another and select
more
evaluation.
to
their
their
evaluate
results,
we can com pare one
the
adequate. Searching and enumeration
one
we
consider
are in p rincip le
8
equivalents,
and
it
fun ction
At the scale of
small
a
extend
of
par t
oroblem of values,
analysis,
computer
use the
approach to
as
enumeration
used In
Prob lem-Solving are
techniques of
the
in the context of
Is
is
it
and
a room,
method t hat can be used as its
generative
Thesis.
no t
another
attempt to
that reco gnizes the
a methodology
t o the scale of
this
but an
designer,
that
provide
a
main analytical
operation.
In
chapter
first
the
assumptions.
main
presentation of
the
problem area
Following
it,
is defined with
there
Is
an
the S.A.R. principles and
a
our
general
o utline
of
the Problem-Solvirng representations and search t echniaues.
In
the
Artificial
SPACE
of
third
Inte
chapter
of
the principles
I ligence are applied and
and FUNC T ION
ANALYSIS,
the computer system
and in
Is presented.
both S.A.R. and
extended
the fourth an
to
the
out Iine
9
DEFINITION AND ASSUMPTIONS.
PROBLEM
I.-
Background:
1.1.-
a
have
Spaces
in human
purpose,
they
Within them,
activities.
continuously
identifying
refer
its
to
in
in
actions
or
call,
think as, the
"function* that has
or carrying
kinds
are
ways.
The
fact ,
our
in
names that
language, with
that can be reali zed within
activities
on a series
beer
proce ss
created,
or simply
result
later
point
from
In time,
can
a
Usi ng a space
from
a
th e space
is
adaptation,
spontaneous
that
to a function
was
neither
considered in its creation, nor planed to be contained
it.
In
depends
of
either
case,
the success
of
or fallure
In the relations between different
the space,
such
as shaoe,
proportions
we
space, or
resul t
the time when
at
what
is made of a
assigned to it.
of activities
planning
a
constitute
*use* that
careful
at
humzn
limits.
This series of
the
of
in
shows
everyday
the action or
different
multitude
purpose,
their
them,
a
for
containers
actions of
performed
of
Importance
as
exist
frequently
must
environments,
this
by
"use"
characteristics
or dimensions,
to
10
name a
to
few,
it.
the kind of
Spatial
sometimes
certain
and
characteristics
in a definite strong
the process
to
spatial
types
initial,
if
spatial
solutions.
set
at
up
not
often
form
the
standards
These
global,
implications
to
a
of
the
the
of
of
the
the
range
seldom
understand what
solutions might
in
the
of
possible
a
in
form of
spaces
for
framework
existence,
or
a
for
the
definition is made. By
we
sometimes
the two,
the impact
to
norms or personal
characteristics
rationale,
be on different
exist
or
provide
comprehend the relations between
their
and
in the design.
These
their
of
always roughly
very end, according
acceptable
their
one
assumptions already
when a change in
of
Is
process
circumstances
function.
the
very
unaware
to
assignment
taken in the generation
understanding the reasons behind their
fall
prevent,
of
the
uses,
preferences
delimit
oreferences, set
being
or
performance
types are almost
the time,
of
that
but
or
until
cultural
corresponding
design,
beginning
more
of
of
activities
fluctuates
Must
space,
way,
the first step
the
definition
the
permit
assigned
functions.
In
other,
function that can be
of
uses.
fall
to
and consequently
different
spatial
11
might
be
considered a tri vial
increasin g
*performa nce*
anal ysis of buildings
cost of
actual
more
"sof ter"
life
per ioc.
las
interest
spaces
out
years,
last
the
being
shifted
fcr
The economic
from
to tte
the building construction,
in the builoing use over
more
the
Its
whole
costs between the Initial
(
the maintenance bills
X)
more substantial,
understanding how
the importance of
are used.(1)
At
other
there has been an
where sav ings can become
pointed
has
exploration is needed.
however,
The propor tion of
indicates areas
and
of design experience
different kinds.
construct ion expenses ( X ) and
for
any need
matter
level,
of
studies
functions
without
d uring
interest
to
spaces
problem
further systemat ization or
At a practical
"sol id"
a
simply
or
explanati ons,
where no
rel ating
of
The problem
a deeper,
hand,
values and
becomes
there
more
has
assumptions
aparent
that
significant
level,
been an increasing questioning
behind
design
desigr problems
with
solutions,
are
clearcut
not
situations
difficult
problems which solutions represent
of values,
and
which values must
technical
be
agreed
implementation.
as
the
of
it
we Il defined
solutions,
technical
attempting any
on
but
implicit set
upon
before
12
(2),
As Rittel
critique
increasing
the
the
challenging
their
with
preocupation
lob,
professional's
pointed out
consequ ences
seen
fact that "...the
by
eroded
seems
to be
support
renewed
equity.
The
as
solving
an
to be definable,
is being confronted now
of
Is beirg
....
"...There
values...".
a growing realization
professional's
by
seeming consensus
differentiation
...
stems from
for
ap peared
that
understandab le and consensual...",
with the
work
ef ficiency,
for
correctly,
quite
on prof essional
"....once
problems
of
assortment
tests
has
that a weak strut
in the
where
li es at the juncture
system
goal-formulation, problem-definition,
Issues
equi ty
and
meet...".
How to provide a basis for
spatial
and
evaluation of
level,
-of
analysis,
are
course-
the
and
how
to coordinate
standa rds
main two
they were not
the
or
values
first
kind
of
the formulation
at
this
second
issues which motivated,
although
solved,
In
the work
done
this
thesis.
1.2.-
Problem definition:
The idea behind SPACE
to develop, along
the
lines of
analysis
was
S.A.R. methodology
(3),
and FUNCTION
the
13
a
systematic
way of -figuring out the relations between a
space and its function or its set of possible functions.
Its main
a
makes
space
objectives
for
adequate
to
were
understand
a certain use, when can we
realistically assign activities to a given soace,
the
are
what
or
what
consequences of changing spatial characteristics
or functional requirements.
How to
formulate functions, describe spaces, and
analyze the relations between
of
parts
the
thesis
function, one or all
in
two,
these
are
the
main
work. How to find out, for a given
the spaces where it can be
contained
satisfactory way; or for a given soace, how to find
a
out, one or all
the functiors ti-at can be contained by it,
are the particular questions
that
we
attempt
to
development
of
would
answer.
1.3.-
Assumotions:
A design problem In itself,
this
the
SPACE and FUNCTION analysis has been approached with
certain assumptions In mind:
-
Independently of defending,
explaining where personal
it
was
considered
more
rationalizing
or
values for a function come from,
relevant
to
find
out how can
14
in a way that
functions be expressed
in
terms
a function can be
It was assumed that
of
pieces of
the
fur niture or
a space,
of
that shou ld
positions
we can define
be
in
contained
and
space
To talk about
then the series of
by
together
it,
the
poss
defined
equipment that are
needed to carry on a certair a ctivit Y.
use
understand
implicatlors.
their spatial
-
us
hel Ps
the
el ements
with
their
ible
relations beween
these
terms,
different pieces.
-
a set of
define
An
Functions
or
standaras
a range of
arrangement
"acceptable'
of
for one of
elements In
-
To
arrangements
that are
by
what
of
their
elements
the
have
to
a space
can have.
correspond
a *functional
to
layout*,
distributions
variations
implications,
Functional
c onseouences
lo ok
implicitly
we can analyze,
-
that
that
I mplicitly
which
possibl e valid
the
evaluate
we
possible,
u ses
represent
this
and
of
space.
standards,
knowing
in
systems
value
definition is called
functional
it stanas
formulated
at
of
all
the
permited by
In
functiona I
understand or
design values in
definitions
different
possible
them.
Only
layouts are
refutate
,
in terms
design s olutions.
are,
there fore,
not
15
attempted
as
definite
statements
open
to
evaluated
through
or
laws
refutation,
the
patterns, but rather
as
enumeration
are
implications
of
their
as
possible
consequences.
The formulatior of
enumeration
assumptions
of
behind
methodological
the next
their
basis
chapter,
Implementation of
four.
its
functional standards and
possible
SPACE
and
layouts
FUNCTION
are
the
the
main
analysis.
The
for these two parts are explored
final
form In chapter
in
three, and the
a generative computer system in
chapter
16
One,
Chapter
No test
-----(1) "Life-Cycle costing in the Public BulIdings Servi ce",
G.S.A. General Services Administration,
Public Building Service,
for
Inc.
& Hamilton
Study made by Booz-Allen
G.S.A. ,P.B.S.
Theory
General
a
in
"0ilemmas
(2)
RITTEL
HORST,
PlannIng",
Working Paper 194, University of California,
Berkeley, Cal. Nov. 1972.
BOEKHOLT
(3)
HABRAKEN N.J.
J.TH.,
THIJSSEN
A.P.,
0INJENS
"Variations: The Systematic Design of
forthcomming
M.I.T.
Press.
of
P.J.M.,
Supports",
Cambrldge Mass. 1976.
-7
18
2.-
METHODOLOGICAL BASIS AND ENUME RATION TECHNIQUES
or norms
Standaros
that
the
permits
particular
so obvious,
they
at
least,
layouts;
should
permits the systematic
be
be
permit
if
understanding
transparent
Architecten
be
problem
scale
analysis;
reducedand
Problem-Solving
as a
two
Group
basis that,
at the
to
of
the
second,
and
and not
a
way
to
point and can
formulation must
the
but
that
defined
possible
from
ma in
of
principles
Research>
allohs
layouts
standpoint
is
it
not
of
so
implemen ted.
this chapter
general
way
layouts.
these
Tha t their
clear
methodological
rather,
in
is a clear
implications ,
their
a
that
be s ufficiently
layouts,
how it can be
In
the
might
in
important,
exaustive enumeration o f all
necessary,
First
way
but, more
in different ways.
the
a
stru ctured
must
the evaluation of
solved
in
enumeration of
That standards
permit
formulated
their conseauences. They
evaluation of
should be described,
testing
be
must
(1),
the S.A.R.
having solve d
of
scale
the
can
be
aoplled
in
the
as
a
principles
actual
-or
and functlon
Enumeration techni cues are
framework that can be used
<St1tching
this formulation
s pace
general
revised.
presen ted
are
housing,
are
Id eas
of
presented
generation
19
of
layouts.
2.1.- S.A.R. formulation of Design Problems$
2.1.1.- Parts and Relations:
In
the S.A.R.
formulated
be
in
terms
of "...an
that have to be placed in that
is
a site,
always
problems
methooology, design
environment
environment
and
elements
environment..."(2).
or context where
in
terms of
that
space,
fro m
define
in design solution can always be expressed
site,
Depending on
vary
There
different
elements can be positioned, and the stanoards that
a set of values
can
its
and their
the scale we
the spaces in
an d accordingly,
elements
city block,
the elements
work,
positions.
the
site
to one area in
that
are
can
room
positioned
20
them
In
can range
from
perform a function to
dwell
in an
already and
they
pieces
as
dimensions,
well
needed
can appear
is
that
in
design
or
contexts
that are external
problems
an
oeople
constrains
given
that
exist
to the designer actions,
urban
blcck,
as might
where
be
situations, and
or
they
stand
instances
The elements
or
They can
the case
of
infrastructure, size,
and surrounding buildings are established
defined;
for several
to
as
be defined during the design process.
can
site
furniture
the support structures where
represent one specific situation,
a
of
urban environment.
Sites
situations,
the
can be used to
then as general
and
describe multiple
or
schemes
models
of similar sites.
that
are
positioned
In
a
site
can
21
be
defined by the designer
possible options.
that
relate
In either
elements
and
or
selected
case, to formulate
elements
site,
described with sufficient exactitude.
elements,
from some range
shape and dimensions,
of
standards
have
to
be
Tyoe and number
of
are the basic information
-. A I
IzzIIzzzzz~
t
71
1
J-
777f
t
0
7t
22
that
them.
description of
for any
be required
aoula
Among these elements there are certain relations
have
that should
in
other
each
being
Elements
different ways,
that
that
requirements
solution,
as
ard
In
present
to
their
from,
seDarated
result
relations
Their
etc..
by,
relate
can
is in terms of
other,
each
to
near
adjacent,
contained
also be
to be defined.
from
program
the
design
should
be
part of
these requirements they should
aescribed precisely.
A
is
methodology,
designed
defined"
"well
then
formed
by the
problem
the
in
following parts and
relations:
environment.
1.
A descriotion of
the
2.
A defined set of
elements
that could
be
used
in the environment.
3.
Data about
the
Iocation of
the
Iocation ot
elements
relative
to one another.
4.
the
Data about
the
elements
In
envir6nment.
2.1.2.-
Levels of d ef inItion:
The formulation of
Positions,
can be
applied
different
scales
of
Site,
at
Elements,
different
the design problem.
Relations and
Ievel s
and
at
As said before,
23
L--J
the Site can be
the
elements
an urban
can
be
block, or a s pace
in a room,
struc tures,
housing
or
and
furniture
pieces.
It
the
Site
at
is characteristic
one
scale,
in
this
formulatIon,
becomes an Element at
that
the next
24
level up,
and its elements
f or
next
the
con sequently,
formulation one
level
stand
can
positions
used
be
in
their
s ite,
the defin itio n of
in
layout
with one possible functional
site.
an element
elements
A dwelling w ith all
In the
2.1.3.-
Operations of
The evaluation
layouts
of
is performed by
f or a given site
We
structure ,
start
explore
next
up.
and together
a
for
element
and
become
buildings
level
up.
the
selected
to pop up as elements
we reach
To continue
among
the
the
possible room
level
the
of
possible
la yer
an d
in
t1e
if satisfled,
ns
as
elements
genera te aga in all
wh1ch,
in
t ur n, -
layouts of
in
the fi na
the
the
are
some
to the nex t el em ent
example
this
level.
smaller
In there we
a ny
al I
layout variatio
variations
u ntil
Analysis:
given
possible
the
level
one
its rooms becomes then
alternatives
its
for
above
be an
generating
with
select some basic
levels,
standards
standards at
at a
agree to
the
site,
building
that
elements
the urban block.
in
hierarchy ,
the
together with
furnitur e layouts,
for
A room can be the Site
dwelling
one of
Sites
The variations
down.
of el ement layouts at one scal e become then,
their
as
in the
I layer.
IlIustrations
layouts, or the uses that can be
25
1en
I
-
1
26
e;2'C7R A4AAs:15
L4
/A45,C
J4ZI4771N5 :
4::,
27
All
00;/B0g
<//E
gq/Bl N/
A/ E
401/C VA/477V6
grAr/<!
V
M/4
o,*,y<!
some
would
of
m would
of 3.60 x 4.10
And the variations
floor,
X/
</'
CoA%';
assigned to a room
apartment
AV/
a// 0-3
of
room
be: (
arrangements
be genericallyt4fiSc
which could result
in
an
V4A%9/74WS>
in the oossible
floor
p I a n s ,(i!-V)e45W.
in
an
This formulation structures a design problem
In
ano which could be Positioned, similarly
urban block
a
way
that
as: 6/-,A
r7ststrE}
permits the systematic enumeration of
In chapter 3 these general
the
scale
principles will
of SPACE and FUNCTION ANALYSIS,
shown through an.example how is
the
be
layo uts.
applied
to
and it wil I be
enumeration
car -red
28
$3
9Z as
X/
'V) '/
&/9-Y477Nw
-
29
4
1
"'1o
ii j~J
/w
A4P14
iii
iii~i
ate
111 1[~
'
UAwAV
A
I
gfW4*
7/55r
30
~~LT3
on.
In
from
techniques
drawn from
the
fol Iowing
other
then the
enumeration.
sections,
fields
basic
will
some
relevant
be described as we
aporoach
for
our
method
have
of
31
Enumeration Techniques:
2.2.-
the beginning of
At
Combinatorial
furniture
Theory were explored as it
arrangements
layouts
enumeration of
are both problems
These
whole
booy
particular
represented
problem,
of
among
of
where
elements
furniture
as
set,
A furniture
a
layout
SYSTEM
of
for
for
of
repeated)
a
in
on
In
a
our
could
the site
it,
and we
of
be
a
set
see each
pieces,
set,
positioned
one
in
for all
our
factors,
of our problem,
piece
a
and considering the
of
the sets
case,
like size,
this
distinct
we
have.
assignment
fitting in site,
relations to other elements, that cannot be
formulation
of
REPRESENTATIVES
pieces as a 'system
representatives'
several
to use
in
selection
the spatial
Unfortunately,
depends
nature.
are positioned respectively in each
collection of positioned
(not
therefore the
examole,
each space
can be positioned
that
assigning
atractive
DISTINCT
the spaces. Considering then each
*representative'
of
that could be brought
configuration as
each
that
and the existence of configurations
we define
that
was thought
of a strong combinatorial
theory
case.
from
the site, and
techniques presented the
of
techniques
constitute a problem
to parts in
pieces
furniture
the Thesis,
defined in the
as our problem consist
In
fact
32
In
finding out
at
aone
first,
and
all,
correspond
what of
Artificial
and
and
as
of
were
techniques
functional
of
evolved
explore,
other
among
for
frameworks
studies of
the development
of
these
heated
an
for
our
use
for
the
expanding
the
in
Artificial
computers,
during
the
the
last
20
formulation
From
years
of
of
psychological
techniques that per.mit
approaches,
(3),
it
has
a
Whether the existance of
a
tasks,
procedurzi
produced,
series of
to our
besides
principles and
problem.
these
techniques
show
Intel ligence in the person or system
any
degree
uses
them, or whether there can
be
to
general
proceed in solving particular
are relevant
of
of
can
interest
things,
polemics
techniques that
models
standards.
*problem-solving".
how people
definition
we
from
methods
explored as
that
application
Intelligence has
general
be
that
layouts
'tree-searching'
Generated from
quite
can
"problem-solving"
alternative,
Intelligence,
evaluation
to
the
enumerating
then
an
representations
areas
assignments
to them.
As
problem
these
such
a
thing
that
as
a
problem solver, are questions not only beyond the
Interests of
this
thesis,
but
questions
that
tend
to
33
the relevance that these methods
from
us
distract
have
In
themselves.
The
problem-solving
of
outline
basic
follows
tree-searching methods
representations and
general,
in
Problem-Solving
2.2.1.-
on.
Representation and Sea rch:
We can say t tat
Is and
situation
a
the
this
actio n,
and
bring the actual
our
in
A problem,
recogniti on
and
from
we think
we ought
case,
terms,
it
that we
to t ave.
desired
in
the
situations
can
take
us
to the conditions that
have,
for
discussion on
go about producing plans of
be
of
(4)
is enough room
might
the
consists
that
that we actually
our
these
a plan of
conditions
reduce
situatlon.
actions
There
for
to
different
of
find
to
that can
of discre pancles between
the p roposal
the
ideal
forced
it
be.
should
characteristics
the prese nt situation as near as possible
character istics of
it
we are
or the col lection of actions,
dif ference
as
a situation
that
discrecance,
this
with
Confronte d
think
we
as
solved
is a problem to be
perceive a d iscrepancy between
we
when
there
sufficient
how
action,
to say that
it
is
but
effective
34
of
must
actions,
the time, result from previous thought
anO evaluation, and such thought and evaluation arise from
an
understanding,
through
an
internal
model,
of
the
in
an
Problem structure that we are confronted with.
A
model
of
internalization of the
structure
this
consists
main characteristics of the problem
the set of possible operations that we can perform to
and
bridge the gap between present and desired conditions.
To build models or representations of a problem,
a process of urderstanding,
we engage in
have to demand certain conditions in
we
build.
reality,
produce,
the
with
we
situation,
that
results
or otherwise
of
the
processes
our
formulation
in
of
problem
that
to reach its solutions.
structure of the problem
practical
the descriptions that
the
want
to
*real'
solutions
can
Descriptions should lend themselves to
expression
expression of
attempts
the actual
results
our
not be of any use.
practical
we
we wcrk with a representation instead of
since if
situation might
so
should not contradict aspects of
Descriptions
dealing directly with
correlate
and to do
a
can
and
be
permit
used
in
the
our
We want to describe the
consistent
way,
with
a
its Information and a relevant
representation of the processes involved in
changing
old
35
conditions into
new,
more
desirable ones.(5)
and
how
to manipulate
the
two
main
then
conceptual
methods, corresponding
in
problem
for solutions, are
looking
in
issues
the
for
to build such descriptions
How
*problem-solving*
to REPRESENTATION and
SEARCH.
2.2.2.- Representation:
problem
that specify how
initial
actions
that
defined by
the
a set of
As
problem
is
structures
symbol
problem
be
used
in
among
laws,
we are
Newell
which
actual
or
situati on,
legal
of
set
the oroblem are
and the combination of
specify
there
the
might
extent
of
the
exist
looking for.
and Simon present
to designate
(1)
a TEST
generate
it
a
"'...To
for
the problem),
structures (potential
to
The
solving
and transformations,
(solutions of
is
transformation
the
describe
situation.
transformation
situations
solution that
laws of
an intermedlate or partial
or *goal'
can
both conditions
of
conditions
situation,
desired
with
to change one condition into another.
Problem
and the
together
conditions
describe
representations
"Problem-solving*
a
a class of sym bol
and
(2)
solutions).
structure,
state
a
To
using
GENERA TOR
solve
(2)
a
t hat
36
satisfies the
test of
(6)
(i)..."
Basic Model,
2.2.2.1.-
Post Production
Systems:
The basic principle behind these representations
computation mechanism presented
goes back
to a general
Emil Post
in 1943. Post
systems
logical
finite alphabet,
that
rules
transformed
logical
some
string
how
of
other string
systems as
of
symbols".
words
in
backwards,
that
in
in some
"*sets
symbols
may
of
be
(7)
for example, the
that represents,
"palindromes*,
and
forwards
symbols written
and analyze
A simple model
structure of
of
strings
as
tell
into
proposed to analyze expressions
by
read
identically
a Post's Production System,
would bet
Alphabet.
a,b,c
Axioms or
initial
situationst
a,b,c,aabbcc
Productions:
--
>
a$a
(P1)
$-->
b$b
(P2)
$
c$c
(P3)
$
-- >
where,
can
use
Initial
in
constructing rew strings.
situations,
the
the alphabet represents
or
the strings
that
symbols
The axioms
we
take as
we
are our
given,
37
not
derived
that
string,
any
string,
from
other
that
is either
from
has been generated
to
productions
ordered
*$'
any
and
symbol
a right
side such as
on the right.
That
As can
will
system
is *$*
into
be seen
from
only
on the
given
*c*,
already,
and
elements
the fact
each
that
the
of
2;-
c c
A-
aca
I-
b |acb
an initial
problem of
that
will
left
the string
Into
r ules,
the
and
alI
tte
the
letters
rule
mirrors
the
same
a previous word.
*bacacab* would bet
peda1ron (p;) a - Cac
(P/) cac
.
_
dt'4e
(
and the *Problem* of
out of
as
e,4'g/ron
.Mi/A
-
such
production
The generation of the word
/,
an
the axioms are " oa lindromes'
production
in both sides
rep resent
'a$a*.
possible *palindromes* compose d from
or
appilcation of
left side,
"palindromes*
generate
string
which Indicat e possible
" a$*,
the string
of
transformations
with a
a
or
productions
strings
pair of
for a
stands
an ax lom
the success ive
The
axioms.
"*
source.
situation
finding out
take us from
'a'
ca4'
Into
-4CQCW
generating
a goal
would be
the sequence
*a'
c4
-+a
of
*bacacab*
equivalent to the
production
*bacacab".
rules
38
the
both
then,
solutions.
The structure of palindromes
expressed
as
a
yield resul ts external
that are ex ternal
"f...
short)
of
of
" ... In
as
of
transformations
these
to the system nor
by
enriched
I aws,
transformation
its
of
Interp lay
and
collection
mere
a
the structure is preserved or
Involve law s!
the
properties,
their
and
elements
not
and
system
test for
is understood
of transformati ons
'system
much as It is a
and the
for generation
capability
i tse If
has built within
problem descr iption
Our
which never
elements
employ
to it..."
(Its)
three key
transformation,
notion of structure is
the
Ideas:
and
the
comprised
(In
Idea of
wholeness,
idea of
self-regu lation..."
the
idea
(8).
2.2.2.2.-
of
this
Graph Notation:
Before
going
basic
model,
principles
notational
problem-solving
methods
we
should
used
in
look
the
that are relevant
Problem-solving representations
being
systems
mathematical
of
notion of
variations
Into the description of
transformations,
first
at
some
of
description
to this
share,
the
use
thesis.
besides
of
a GRAPH as a common notation.
the
(9)
39
G =
A "GRAPH":
(NE),
of
'nodes'
used to represent
a set of
nonempty
exist
set
between
elements we
want
corresponds
to
these
elements
by
represented
lines that
and a set
"N',
elements
The set
them.
consists
of
and
nodes "'N
to talk about, and
of
of
a
finite,
"edges'
relations
stands for
the set of
edges
"E",
that
the
"E"
the relations that exist between pairs of
in
the
dots
Graphically,
set.
circles, and
or
link related nodes.
A
edges
"GRAPH"
nodes
are
are shown as
would
be,
for
instance:
10
G =
(NE)
N =
(1,m,n)
E =
((I,.m),(I,n),(m,n))
If
we
reoresenta tions
as
think,
being
before,
of
pro blem
descriptions
of
pro blem
did
we
general
situations related by transformations, we can begin to see
the corres pondance between the notion of a graoh
and
the
notion of problem representation.
At
elements,
a
first
conditions
level,
or
nodes
can
characteristics
edges can correspond to desired relations
correspond
in
a
to
prob Iem,
among them,
and
40
a
final
problem situation.
the s et
graph standing then for
we
descriptio n of
their
can
it
which
set
a
of
above in
as
in
used to reoresen t bot h the
p rob lem
the
of
generation
nodes were not
listed one by one,
as
simply
or
edges were defined as
between
or it can be
f igure(2.1),
situations
edges,
general
addi t ion
of
e xamp Ie
valid
of
words
palindrome"
but
combi nati ons;
or
as
the
letters
,
where
Initial
as
defined
produc tion rules,
schemes
I mpl i cit ly,
described
and where
relations
strings, denoted b y * $',
new conf i gurations that contain the previous
an
the
and
situations,
described expli citly, as In the
be
can
th e
and
p arts
According on how w e defi ne the nodes and
graph
"a",
"b*
or *c*.
scheme
It
situations
and
production
and
plus
is said
that the graph Is defined implicitly, because by means
initial
the
general problem transformations.
structure of
a
or
Being the graph a n abs tract
be
oarticular
of
s ituaticns am 0ng
a structure, tha t is
rel ations,
structure
and edges
rules,
production
of
solution.
a
for
search
can
one situation in t o a new diffe rent one;
from
changes
in themselv es,
la ws 9
transformatlor
represent
nodes
level ,
second
situati on s
problem
now
become
a
At
or a
a nartial
an initial,
for
then
stand
would
graph
of
rules, we can always
41
have
a way of
the
set
of
set
of
links,
each and
exo I icitty
everyone of
of
of
is a
member
to
define
both sets.
of one portion
of
the
would be:
graph,
"palindrome*
having
the members
representation
A graphical
Impl icit
Instead
of
a member
a string Is
a transformation
or whether
nodes,
of
the
finding out whether
4040
"Ccc"
AaCa94=iv
where
we
we can select each of
can
continue
each
apply
so
doing
"palindromes"
further
too we
at the
where
of
as
everytime
the
long
the
have an empty
string,
the possible axioms,
possible
as
to which
productions,
we want,
graph
from
grows
and
generating new
further
and
down.
If
this
graph
notation is going to be
used as
a
42
besides
purpose,
we
what
is
for
concepts that are relevant
important to define several
this
then it
for problem-solving representations,
convention
about
have already said
nodes and eages.
have a seouence of
When we
edges of
the form:
(nign2),(n2,n3),(n3,n4),....,(nnnn-1)
corresponds to
this
sequence
at
node
the
where
the
end
each
of
the node at the beginning of the
Is
called a
"path".
A
linearly
it can also
be represented
next edge,
*oath' goes along a
connected nodes and
sequence of
edge
for
this
reason
ast
n1,n2,n3,......,nn
P1'z
and be said
to have a "path
length*
of
n-1,
that
43
the
in
edges included
to the number of
equal
length
Is a
sequence.
with
s
ay that
and
las t
nodes
another,
a
sev e ral
oaths.
the
graph, t hen
graph
"Trees*
are
is
is a
Initial
node,
bottom
of
the root
the
set
has
n o cycl e at
each of
from
of
has
nodes
first
node,
In
to some
because
the
in
sequences of nodes from an
final
nodes, down
at
tte
Initia I situation p lus the rules
a 'tree'
that might
whose branches
situation,
transformations
to
constitute a solution;
finding a solution becomes then equivalent to
sequen ce
all,
the other
t he
the possible paths from tha t given
situations
m * I*
fro
called a *tree*.
the graoh. The
of
t o one
path' to ever y node
transfor mation, can then ' grow'
are all
equal
one cycle,
has
Important graphs to us,
,
first
"cycle'.
and
p aths they show distinct
their
the
th ere
the *root',
called
edge,
a re
the
like the second one ,
a graph,
When
When
path*
figure(2.1)
one entering
have onl
and
last node,
n ode which no edge enters,
first
of
first and the
a nd the graph In figure(2.2)
*i*,
but
distinct,
*simple'.
Is
this path a
in
graph
the
'simple
a
of
then we call
The
to
the 'path*
are
'path'
a
in
of
possible
exception
then we
nodes
all the
When
finding
through a certain path
44
that can
node down
desired
to a
root
the tree
take us from
its branches.
Everytime
in
the
down.
The set of
at
in turn becomes
the
of
say we expand the node one
the
end of
these expanded
a
level
edges
the expanded node, and
of
'successors*
the
called
these
nodes
out
edges
grow a series of
we
tree,
node
are
we
to
the
newly
paths,
are
basic
*predecessors*
created nodes.
Nodes and edges,
of
concepts
expressive
methods
theoretical
on
chapter(3)).
2.2.3.There
representations
not
but
possibilities,
to other
(ref.
and
Theory which are used as notation for
Graph
problem-solving
trees
only
of
because
because they give
notions that will
Main types of
have been three
be
general
kinds
of
(10)
STATE-SPACE REPRESENTATIONS.
2.-
PROBLEM-REDUCTION REPRESENTATIONS.
3.-
THEOREM
State-Space:
later
Representations:
1.-
2.2.3.1.-
us access
explained
in problem-solving methods:
PROVING.
their
problem
45
In
conditions or
problem
of
of
characteristics
STATE-SPACE
representations,
problem situations
which
DESCRIPTORS"
"STATE
in
case
the
are
described by
certain
represent
the problem
solution at
a certain poirt
and
situations
are
time.
Initial
final
expressed
respectively as existing and desired characteristics which
not
necessarely
have
but that
strings,
approplate
problem
"palindromes*,
for
a form
words *a'
and the
for the problem
.
appear
Legal
as
the same
fashion
description
back
combinations
what
In
of
more
In the case of tte
In hand.
and
of
transformations
in
In Post
DESCRIPTOR"
all
is
this
Production
'bacacab".
representation
our
the
very much
how
can
the
be reached by
initial
state
"'STATE-SPACE", that
example,
contain the
to
DESCRIPTOR".
into a new "STATE
'pallndrome'
words that
in
Systems,
situations that
called
the
state descriptors
final
generating the expression
as
such
'bacacab* would have been
aoplication of these operators to
constitute
format of
instance, strings would have been
The set of
still
the
"'OPERATORS" or rules that soecify,
change a "STATE
the
of
form
descriptor and the
state
Initial
be restricted to
can take any
the
for
to
letter
all
*a'
Is,
the
in the
46
middle and
or
ec"
whose
letters repeat
on each side of
where
Nilsson(1971),
a
is
strings,
a
model
for
a
8-puzzle.
sliding-block
In
agains
from
representation has neither state-descriptors,
nor operators described as
located
"b"
it.
An example, taken
STATE-SPACE
or
alternatively "a'
in
this
a
9,
puzzle
there
3 by 3,
cell
the empty cell
are
space,
8
numbered
block
which can be slide
to form certain configurations
such
as:
1,5
1
71
187
Operators
in this
and possible movements of
to
another,
as
case correspond to
the empty cell
from
blocks are slided to occupy
olace, and an example of the operators
the
one
valid
location
its previous
'rules"
wculd
be
figure(2.3).
Supposing that the initial
and the final,
situation Is:
desired configuration is:
then one sequence of
transformations
that
can
47
RYl,
R,-
417,
gA4
2~3:
48
g4eWOT=
final
produced the
configuration would bet
STATE-SPACE
practical
expressions
have a sequential
be explored
from
zi-
representations
of
problems
characteristic.
previous situations.
lend
themselves
with structures
Different
At
to
that
situations
can
any point
in time
49
we can analyze the state we are
existing
difference to our
in,
final
to
goal,
find
what
is
and the process
reaching a solution can be composea of a concatenation
the
of
of
50
one
operations,
after
the other,
Prcblem-Reduction:
2.2.3.2.-
representations,
In PROBLEM-REDUCTIOIN
to
taken
problem
can
be
larger
or
reduced
the
into
decomposed
methods
simpler
concerned
are
with
that can be pursued to reach a solution. These
strategies are oriented to decompose an
a set
of smaller
components
original
problem
which solutions might
be
to obtain.
The elements
are
representation
called
steps
one.
strategies
easier
different
which solutions imply the solution of
Problem reduction
into
Rather
explore how an original
we
problem,
problems"
"primitive
the
a
solve
of
descriptions
with
than working
deal
we
the problem itself.
the structure of
with
a
state.
state into a new
instead
continuously modifying
"PROBLEM
situations,
goal
situations
as
was
mode
of
of
problems,
consisting
of
Initial
and operators that
transform
presented
representation.
this representation as being
with in this
descriotions
therefore
DESCRIPTORS",
one into the other,
"state-space'
that we deal
We might,
one
level
in
the
in fact,
higher
previous
think
from
cf
the
51
previous one, and contain
Droblem
in
a
transformations
in
this
situations.
legal
The
decompositions
descriptor
specify how one problem
these operato rs,
of
application
subproblems and among
we
those descriptors that
If
the "state-space'
applications
of
problem-reduction
more
easily satisf y.
of a
for
operators
generates
r eached
be
can
that
can chose
of
problem repr esents the
o perators,
we
Iook
combinations
state-space
strategies
by
the appl ication of
se t
the
before
from,
of
all
exp Ior ing
state-space des criptor.
any particular
One example
of this
*decision
tree",
following
Protection
Association
protectionsystems
general
can
generat e a set
we can
its
possible situations
all
possible
transformed
be
might
of possible subproblem descriptors. Through the
into a set
related
possible
its
into
problem
one
of
are
case,
They are accomplished through "OPERATORS" that
components.
set of
of
hierarchy
components
of
descriptors
problem
*state-space' representations as
statement
protection",
but
for
reduction method c an be
p roposed by the Nat iona I Fire
for
the
analysis
buildi ngs.(11)
of the problem
the
the
design
of
might
fire
In
be
here
of
fire
the most
simply
'fire
protective measures
52
desired cost
taken
implies a
fiqure(2.7).
or performance.
The elements
are cifferent
protective
measures,
as
subset
Subproblems
of
in
of
get
for
"primitives"
be
to
have
represent
problems
solved.
different
and so on until
every ki nd of
imp l ementation.
different
altern a tives
of
For
different
in
Tne
measures.
our
problem descriptor nodes. Either
itself
our
problem,
OR alternative
avallable,
second
two,
can
be
alternatives
descriptor
nodes.
solve the problem,
first
graph rotation, as
of
alternatives
OR
w hathever
'AND*
are
All
of
tnem
can
th em
we say then tha t
as
with
are
"OR' successors of
and
selec ted
its
There might
be used only i n combination
which can
or alternatives
described,
themselves can
a
at ternatives
of
combinations co nstitute a sol ution.
be alternatives which by
other
collection
eac h
into
solve
can
that
we
action.
The operators b r eak a protecti on measure
col l ection
be
can
and their Implementation
turn,
their
acti ons with smaller subset
some basic
actions that
that
problems
which,
strategies,
protection and the
desired
the
provides
that
one
the
valid
different
the selection among
implies
othe r
solve
by
one,
alternative
alternative
is
a win ning strategy. The
succes sors
have
of
to be satisfied
problem
in
Ln
Decision Tree
jF5ZPO
7
54
to have our
order to
al ternative
that
alternati ve
proceed
problem solved, and therefore we
say
four,
AND
five,
in t he picture
then
consists
strategy
AND*/"OR' nodes,
through
in
has
structure
a
this
and when the
stated as a
when
described
can be
soluticn
its
its
different
reaching a solution can
process of
be
sol utions.
Theorem-Proving:
using
predicate
final
be
can
we want to model
order among
hierarchi cal
In THEOREM-PROVING
and
that
synthesis of related "primitive"
2.2.3.3.-
"first-order
found.
f ashion,
similar
a
'primitive
representations
the problems
when
decompose d
initial
to
paths,
of
tha t reaches a set of
Problem-reduction
attempted
(figure(2.7)),
combination
the
of
problems" which solution can be
are
satisfied
i n our solution.
As can be seen
parts,
and
selected
be
must
alternative
AND
three,
a
representations,
situations
formalism,
f or example
logical
calculus",
conditions of
as a
language
a problem can be
in
which
expressed
55
as
valid
sentences,
and
performed in order to
find
The elements
the
case
alohabet.
strings
which
with
or
in
this
Post
symbols
relevance,
be
"well
their
besides
from
in particular,
by
statements.
*syntax'
or
the
formulas"
can
formulas"
or out
as
assertions
be
Its
that
or
symbols
the
formulas"
to the domain
'true'
'false*
Although
two
of
on
'rules
or
of
Inference'
to
regulates
how
out of
"we
first,
Il
other "well
in the alphabet;
and
that relates "well
of Interest,
the
deduced
be
include
part
all
can be made about an
main parts
constructed
of
set
as new statements can
part
'semantics"
or
formulas";
formulation, can
the
statements that
the application of
previous
the
legitimate
formed
legal
formalism represents
and meaningful
manipulated
into
that
of Interest.
The
area
as
to a given
operations
assembled
called
belong,
Systems,
always be decided by interpreting them
valid
or
proofs
representation,
Production
can
expressions,
some domain
analysis can be
implications,
out
Their combination result
how
dictate
logical
statements of our problem.
deductions about
in
where
by assigning
the
formed
formed
second,
formed
them
a
value.
this
representation
offers
the
56
advantages of
the
power of
logical
always
generality, uniformity of representation and
remains
express our
difficult
is
that
formalization
for making deductions,
techniques
to
reach
and
demanded,
knowledge
of
specific
the
level
difficult
problems
in
it
of
also
to
logical
formalisms as the predicate calculus.
2.2. 4.-
Search:
For all
the previous
representations,
our
in
their
ved
is
have
formulated
Issue
that remains
problem
be
to
sequence
of
operators
transform
a
state
descriptor,
so
or
descrip t or
break a problem
a new statement
out
nference
of
an old
control
paths
transformations
the
bounds, are
still
find
the
rules,
that
can
anoth er
that
state
or deduce
ne.(12)
to
growth of branches
that
0
the second
to
how
in t o its components,
Which alternatives
several
into
terms,
we
once
select
when
can be apolied,
there
are
and how
in our tree to a number
can be explored within reasonable
to
of
tiffe
the main problems of search.
2.2.4.1.For
the
Basic Techniquest
first
problem,
that
is
which
57
to
alternatives
next,
select
conventions
two
be
can
the paths
established on how to explore systematically all
in the solution space of a given problem. Depending on how
generate
at each
alternatives
to
or in what orded we decide
we proceed exploring nodes,
of the graph, we can
level
move along the breadth or along the
the
paths
the successor
nodes
depth
of
that extend out of the tree root.
If we decide to explore all
at
a
point, before continuing to expand them into
given
other levels further down, we say we conduct the search in
a BREADTH-FIRST manner as shown
in
figure(2.8). If we decide to explore only one
node at each level of
the tree, and proceed doing so,
each successive nodes until we reach a terminal
until
have explored all
we
for
branch, or
of the possible paths, we say
that we conduct the search then in a DEPTH-FIRST way,
see
figure(2.9).
BREADTH-FIRST
on
conventions
how
to
DEPTH-FIRST
or
solution space, and depending
problem,
each
disadvantages.
through
the
of
them
has
When solutions
levels
of
the
of
each
visit
on
the
tree,
the
are
nodes in a
structure
particular
are
searches
of
the
advantages
unevenly
or
distributed
as in figure(2.8),
a
58
Sl=( 1,2,)
S2*( 2,3,4,)
S3x( 4,)
S4-( 4,5,6,)
S5=( 5,)
SS=( 6,1,2,)
S7s( 1,8,)
SO-(
1,2,9,)
SOa( 1)
S1=C(1112)
SIe (13)
Sl=(
1,2,3,)
S2-( 2,3,4,)
53(
3,4,1,)
54u( 4,1,2,)
PAWM&57 Z-
59
depth-first
beyond
alternatives
have
on
But,
Instead.
unnecessary
we
had
the other
would
Independent
oprators
in
of
Even small
quite rapidly.
the
breadth-fir st
use
solution muc h mo re
tend
incre ase
to
genera t ors
alternative
8-puzzle presented in
the
of
positionfour
-at
three
that produces
the
and paths. For
operators,
four
sides-
four -at
possible paths,
into
extending
out
that
a
their size
,
like the
example,
-at
when
state-space
of our
initial
of
other into
In
the center
applied,
produce
each of the cornerscan combine
only two alternatives;
that contain 14 moves,
one
alternatives
different
alternatives each, and
generate
of
kind
for
d escr iption
large number
possibilities
faster.
trees
convention s,
these
representations, combine with e ach
case,
at
of
lot
a
2.9)
state-space
this
search
hand, when solu t ion s exist
find the
situations
problem-solving
path might
would be done by breadth- firs t searches
work
when depth-first
a shorter
where
tree,(figure
the
of
levels
similar
level
if
found
been
the
exploring
time
longer
spend
may
search
three
that
in sequences
of
1,497,792
configuration
60
in
figure(2.4).
and
with
Problems with
operators
oossibilities
can
that
these terms,
numb er
larger
Iarg er
expand
quite
reduce a
easy
exponential
of
this
*ever
explosion of search"
of the problem structure.
understanding
r epre sentation
of
a
(6),
We can,
problem
in
have to
the
less
we know about
of
demands a knowledge
fact,
measure
our
ope ration,
and
a problem, the more we
A policy for
is
next
successive nodes
suppress
expansion
against their
altogether
rules
of
of
the
a pr oblem space,
the
process
of
alternative s.
branch selection can ta ke advantage
characteristics of
the
in
knowledge in
branch selection and generation of
of particular
or a reduct ion
that have to be explored
can only come from embedded
certain
In
terms of rep resentations
In
*pruning" of branches,
combinations
or
of
search for its solution.
A
which
threat
present
which processes reduce search to a minimal
that
number
to a non-operative alternative.
The control
say
o f operators,
the
the problem,
and
to
by
proceed,
*promise*
(10)
to
decide
ranking
succeed;
exploration of branches, by
thumb ,
'heuristic
called
Information* (10).
Simple
heuri stic
information
can
reduce
H
to
62
substan tially
all
figure( 2.6),
state,
problem
problem
a
the 8-puzzle tree of
In
space.
for
recognizing that
without
the
state we had
the
to
situation
every
partial
tree of
n ode
path
a nd
can be evaluated
the
'EVALUA TION
of
number
FUNCTION
in
the
figure(2.10).
If besides ttese reductions,
each
that
before its
applica tion. Preventing their application results
smaller
each
type of
movement
a
operati on -centersidecorner- there is
reverse s
to
blindly
operators were applied
the
in terms of
misolaced
the
*promise*
of
length of
the
blocks,
then
the
(10)2
f (n)=g(n)+W(n)
where
number
nodes in
the
g(n)
is the path
misplaced
blocks,
a "EST-FIRST"
manner
of
tre e in
figure
(2.11).
length, and
can
W(n)
is
the
help us to select the
and reduce
the
search
to
63
AV4t47T
)1
1 eV10
1iaer F,//
2.2.4.2.-
Backtrackingl
One of
the exhaustive
techniques
for
searching
44
the
set
of
all
a given
problem
explores systematically the solution space of
It
at
problem, by partially expanding solutions an element
or retracing its steps
all
It
reaches a
where no
further elements can be aaded, or
whenever
the components
have beer, added to form a valid result.
a combinatorial
technique
have
sequential
expansion of
several
several
the.
parts,
clearly
In
established
restrictions
or
together
-
from
the
definition
set
their
that
taking one
of
of
of
parts
is
values and the
stipulate
what
structure consists oft
X2,
each of
values
formed
result.
A set of
( X1,
general
with all
"constraints'
constitutes a valid
Their
them capable of
definition
this
that permits tte
their solutions. These are
each of
this
by
solved
structure
depending on some
values,
problem.
being
to
amenable
Problems
by
a previous decision
to the state of
try another possibility, whenever
in order to
point
fashion, and by "backtracking"
in a depth-first
time,
the
Is
(13)
'BACKTRACKING*.
a
possible solutions to
parts, or
.. **.**
which
.,,
"selection spaces"
Xn
)
represents
a
set
of
possible
where a particular decision or selection can
65
be made according to a
of constraints
Criteria
-
(
00000
xi, x2,
09
)
xn
In order t o expand a
-
Solution represented as a "vector* of
(
x1,
x2,.
....
elemert *x1"
where every
)
*x2*
*xn* correspond
to
the set of parts * X1' or
to a vaid selection f rom
'X2*
or
respectively.
*Xn*
An
exhaustive
search
values
compared
with
'selection
spaces*
until
product of
the
all
the elements
in
alI
and,
possibilities
that
'Xr*
out the set of
all
with all
the set 'X2'
for
vectors
this
for
*Xi
have
again st
i ts
of
all
the
ele ments
has
all t hese sets,
and so o n until
to be explored
that
of
t he set
represents
satisfy
al I the
simi larly
result
val ues
the elements in
have
In
their
possible
........ x Xr,
X1 x X2 x
so I utions
or another ,
The 'Cartesian Product*
or
restrictions
the
all
been explored.
i.e.:
*X2'
for
possible
way
one
considered,
be
for
possible val ues
the
structure means that all
to
xn
000.0
length n
the
"XI'
times
set
"Xn*,
numbe r
of
in order to find
the
constraint
valid solutions.
In a BRUTE FORCE approach, what
we would do,
Is
66
proceed to construct each of
test
combinations and once constructed,
these
from
resulting
the possible complete vectors
we
If
order to find out
them against our criteria in
have a valid solution.
in the
spaces' ,
*selection
proceeding
By
point
whatever
the vector being
succeeding in
Looking at
always
an
whether
tell
expanded.
of
the criteria
At
solution,
being
valid
by
whether
or
solution can not
be expanded
any point
can
not
provide
explicitly,
and
a valid solution
is
partially
be extended anymore.
We can not
without
without
a
a
know
a stopping rule that
our solution space,
more.
generation of
during the
always
formed, but we can
solution
time
our
these,
that direction any
in
not guarantee that
can
we
in
of
none
having
guarantee continuous advance towards a
of
can select
from where we
the next set,
can
we
constraints
element, contains a candidate for a valid extension of
the vector,
can
for
chances
the
are
what
are,
we
in
sequentially
we can always test
the selection of values for a solution,
at
the values
explicit consideration of all
the
unnecessary
makes
however,
works
backtracking
way
The
having
when
solution,
excludes
having
to
to
but
we
large sectiors
explore
them
wait for a complete
67
vector in order to test for its
validity.
Backtracking can be better understood using
our
Stop(11)
Se( 2122)
S3nC 3132)
S4(
4142)
S5u( 51)
Sea( 61)
F/6e4rME
graph notation:
-
fZI
figure(2.12).
At the beginring of the tree of possibilities,
solution, or a node that represents our
we have an initial
have
initial vector of length zero, as no decisions
been
made yet.
- From this starting point, we can
tree
by
representing
construct
a
each of the possible selections as
branches that grow out of this root. Each of the values in
*Xi'
would appear as a node at the end of
and
stand
for
a
possible
these
branches
selection to be added to our
vector.
-
From each of
these nodes, we
can
select
now
68
one of
into
the
a
values In "X2'
of
level
second
in its turn,
which,
branches;
nodes we continue branching on until
from the
and
expand
would
resulting
have included
we
all
the possible selections of "Xn'.
-
our path towards
pushing
until
Constructing
we reach a
terminal
a
consists
solution,
further
then
in
levels down in the tree,
branch, or until
hit
we
dead
a
end.
By convention, we can select
node,
out
the
always
in a
left-to-right
retrace our
we
left branch when we
and
down
directions,
visiting the
importance
however,
branch selection
that
'stopping
nodes that have
criteria,
and
node,
nodes
in
the
following
the
2,I3)
The
or
a
can move systematically across all
we
paths in the tree,
tests
available
two
these
left-to-right,
mechanism,
a
steos to return to a previous node.
With
way: {{/,4/A
left to exit
the next
to
return
of
manner, such that we always pick
in the
first branch
branches out
to
everytime
of
relies
backtracking
In
a
be
we
a
search
stronger criteria for
incorporates, as
rules'
as
described
before,
to help reduce the number of
explicitly
explored.
advance from
With
such
a given node to its
69
SI-t 1,2,)
S2--( 6,5,)
it
S3- 3,4,
S4( 5,4,)
we
successors,
that
not
do
a%-
look
10%
for a subset
first
lef tmost member.
we find
we advance there.
subset of *X2*
are
both
At this
as a Possible
point
the
we
extension
and
took now for the vaid
and find out that the nodes *x21* and *x22*
selections
Invaid
constitutes a vatid
doing
*xt1'
shows
Starting from the root,
this procedure.
the node
first
figure(2.14)
In
A graphics example
of
choices
vaid
the problem constraInts, and
violate any of
from this subset we pick out Its
consequences
of
%~
and
only
the
node *x23*
possiblity of extending our path
By
so, we can see now, how a whole region of the tree,
extending
below
consideration,
the
since
invalid
nodes,
is
ruled
out
of
all the paths that go through these
70
SIS( 11)
S2=C212223)
S3*( 313233)
S4=( 414244)
h
IWNALID MEES
FA*M4
u
.
2-N
nodes would by definition be wrong.
This
process is
at
formulate
the
search
By discovering a dead end
the tree, we *preclude* or exclude
from further considerations,
points.
during
regions
called *preclusion*.
certain level of
a
out
cutting
paths
the
all
below
such
To preclude large regions of the tree we have to
our
constraints
analysis
sequential
in
a
way
that
makes
such
possible, and structure our solution
space in a way that brings
these
forward
violations
as
soon as possible.
One way of doing this is to sort the
spaces*
by
increasing
different levels of
with
the
smaller
number
of
the tree, so that
choices
we
have
*selection
along
the
the
sets
number of elements at the beginning or
71
and the
larger
violations
occur
near the root,
As
tree.
the
of
certain combinations
in
having the fewest choices at the beginning
elements,
produce
to
tend
bottom
sets at the
will
If
than
preclusions of paths
larger
of
It
were done otherwise.
of
techniques
as
such
to help
are used
'branch merging',
bound' and
amount
other
some
ordering',
*Branch and Bound' incorporates to
for branch selection, considerations
Is
against
and
preference,
or
maximize
solution.
modified
as
encounter new
pro ceed
minimize
Bou nded
our criteria
for
we
out
by
to
by
the
criteria
the
lower of
along
we can rank It now
upper
scale
of
trying
to
predefined
selection
in
preferences
solutions
a
respectively,
limits
is
the branches of
choices that can be made.
increasing or
branches
overall
acc eptable
move
a
out
in
the
we can decide whether or not
and
reduce
for preference values
option,
Invalid
an
ot hers acc ording
the
and
successors. Besides knowing if a successor
or
valid
a
*branch
for solutions.
work spent searching
among different
"branch
and
*preclusion'
with
Together
continuously
the
Looking
tree and
at
them
we can Improve our situation,
decreasing our previous bounds, drop
that extend beyond
out
limits,
effectively
72
reducing the regions of nodes that have to be considered.
the
that
fact
solution space,
of elements, but
a
spend
might
that
in many cases what
redunaancy
As
solution.
solutions
diff erent
only
might
t hat
of them transfor mations
of
out
properties
are
said
or
transformations,
merging
al I ow
of
regiors
same
pr oblems
whose
to
us
c onstruct
*isomo rphi c*
as
tries
paths
merge
to
of
into seque nces
reduce
our
sol ution
expand the results
or collapse
to
under
it
loks
is also called,
of
*clusters'
space, but allow us
the
such
to one an other. Branch
non-isomorphic
all
new
tha t share these
for these equiva Ience s either before or during the
and
We
a
e quivalen t
be
to
In
So lutions
*isomorph r ejetion*,
or
paths.
rot a ti ons or t ran slations, all
one S *
old
versio n
a
combinatins
In
definition of
case
the
be
share symmetr i es,
solutions
in the
size of
time considering diff erent
lot of
constitute
increases the
only the explotion
is not
recognizes
hand,
the other
'Branch merging", on
search
equivalent
solutions
which
nevertheless
to
variations
If
possib le
desired.
2.3. Conclusions:
As
backtracking
a
combination
of
all
these
provides an organized approach to
techniques,
exhaustive
73
even with
more
some cases,
of
that
implications
a
that
importance
in
the same grounds.
find out
Having to construct solutions in orded to
exist at
theoretical
it
bad,
might not be a graceful
all,
but
sense,
Is the only choice
lacks
structure stlil
we
a more
have
powerful
In our particular case,
to
Indication
a
larger
problems
explanatory
not
but rather
using tree-searching methods,
for good or
theory.
as
It
need that demands
for
whose
from
us
stop
take
a
of design
the generation
configurations, this criticism should
an
for
they
if
way in
or elegant
the time,
must of
generating
in
has
it
frequently critiziced
is
solutions,
enumeration
be solved otherwise.
which could not
The
for
method
valid
stli]4
resources,
memory
on
has
the
with
expansion,
sequertial
this
use
a clever
But
in
take
vs. explicit enumeration of
Implicit
backtracking
makes of
problems
than anyone could wait.
the
and
solutions
solution
its
and
computers, can
the use of digital
the
preclusion,
spaces* can still
"selection
explosions,
combinatorial
bring back
time,
size of
Increase in
searches.
it is,
future and
related development.
And
these
realize
processes
that
is only of
the
"...computerization
secondary importance.
of
The main
74
Issues are still
spatial
the better understanding of
configurations
manipulating
them.
investigations..."
Here
(14).
and
seem
of
to
our
the
theory of
reasoning
In
lie the significance of
75
Chapter Two, Notest
A.P.,
DINJENS
P.J.M.,
J.T.,
THIJSSEN
(1)(2)
BOEKHOLT
HABRAKEN N.J.
"Variations:
The Systematic Design of Supports",
forthcomming M.I.T. Press. Cambridge Mass. 1976.
(3)
WEIZENBAUM J.,
"Computer Power and Human Reason".
W.H. Freeman, San Francisco 1976.
---
DREYFUS H.L.,
"What computers can't do".
Harper & Row, N.Y. 1972.
---
PAPERT S.,
"The
(4)
ARCHER,
Artificial
B.L.,
"An Overview of
Intelligence
the
of
Structure
H.Dreyfus"...
the
of
(esign
Process".
in "Emerging Methods In Ervi ronmental Design and
M.I.T. Press,
Planning", (Ed.
G.T.
Moore),
Cambridge 1970, pp.285-307.
---
RITTEL, H.,
for
the
design
of
an
principles
"Some
educational
system for design".
part I, DMG-Newsletter Vol. 4 No. 12, Dec. 1970
pp.3-10.
part II, DMG-Newsletter Vol.
5 No. 1, Jan. 1971
pp.2-11.
(5)
KAPLAN A.,
"The Conduct of Inquiry",
Chandler Pub.Co., San Francisco 1964.
---
McCARTHY J.,
"Some
Philosophical
Problems
from the Standpoint
of
Artificial
Inteligence".
Machine Intelligence 4/1969,
(Ed.
Meltzer
Michle).
American Elsevier, N.Y.,
1969 pp.463-502.
and
76
A., & SIMON H.,
"Comouter Science as
Symbols and Search".
Vol.
A.C.M.
Comm.
pp. 113-126.
(6)
NEWELL
19,
3,
No.
March
1976,
M.,
and Infinite Machines".
"Computation: Finite
N.J., 1967.
Prentice-Hall, Ergleicod Cliffs,
(7)
MINSKY
(8)
PIAGET J.,
"S truc tura i sm",
Harper Torchbooks, N.Y.,
(9)
Inculryl
Empirical
1971.
HARARY,
NILSSON N.,
Methods
"Problem-Solving
Intelligence".
McGraw-Hill, N.Y. 1971.
(10)
in
Artificial
Committee on Systems Concepts for Fire Protection
FIRE
NATIO NAL
1974,
Nov.
Structures,
in
PROTECTION
ASSOCIATION, Boston Mass..
(11)
(12)
SEARCH
(13)
GOLOMB S. & BAUMERT L.
"Backtrack Programming",
J.A.C.M. Vol.12 No. 4, Oct.65,
----
pp.516-524.
BITNER J.R., & REINGOLD E.M.,
"Backtrack Programming Tecniques",
C.A.C.M. Vol.18 No. 11, Nov.75, po.651-656.
----
KNUTH 0.,
of
Efficiency
the
"Estimating
Programs".
Vol.
Computation,
Mathematics of
Backtrack
29,
No.129,
Jan*75,
pp.121-136
----
SWIFT
J.D.,
"Isomorph
Rejection
in
exhaustive
search
77
technIques".
Amer. Math.
(1960),
Op.195-200.
Soc.
Proc.
Symp.
Math.
Appl.
----
WALKER R.L.,
"sAn enumerative
combinatorial
problems".
Math. Soc.
Amer.
techniaue
Proc.
for
Symp.
a
class
ADOI.
Math.
10
of
10
(1960),
pp.91-94
(14)
RITTEL, H.,
in
corfigurations"
"Theories of cell
and Planning,
Oesign
Environmental
in
Methods
Moore, M.I.T. Press, Cambri dge, 1970.
Emergirg
ed. G.T.
-9,
79
3.-
SPACE AND FUNCTION
ANALYSIS.
This chaoter exoands
the
S.A.R.
methodology to
formulation of
their
general
principles
of
the description of spaces, the
standards
functional
and
analysis
the
of
relations.
in
It describes
about
of
the
can be
functions
elements, their
detail
defined in
relatlons
proceeds then to present
a process
along
alternatives,
possible
"state-space*
and
generating
and
how
a site, a set
terms of
their
standards
positions;
and
that enumerates all
the
lines
the
the
*problem-reduction*
of
for
methods
alternatives.
3.1.-
Formulation
3.1.1.-
of
Standards:
Site:
The site constitutes
place elements according
the
environment
analysis, the
or a set of
spaces that define a
where a
environment
to certain rules. At
functional
room
first,
is
where
the scale
formed by
room or an area
we
of
a space
within
a
function can be performed.
As container
for this function,
standards
aboLt
80
the
of
can be defined at two main
site
is called the
what
terms
and another in
ACTUAL SITE,
of
SITE.
the FORMAL
3.1.1.1.-
a
of
proposed as part
defined
be
represents
site
of
terms
in
or a space being
of
Characteristics
design.
under
area
the
space
existing
given
a
consideration,
can
ACTUAL SITEt
actual
The
site
In terms
one
levels:
this
or MATERIAL
its SPATIAL
ELEMENTS.
SHAPE,
define their
or
figure
(3.1a,
of
height
these parts
through
3.1b).
all
of
spatial
descriptior
the adjacencles,
Shaoes
RELATIONS.
include the
components.
length,
the total
overlappings and
done
is
site
between
relations
width
of how
The representation
parts.
in
to rectangular
rectangular
Dimensions
of
each of them
for
and
restricted
together to form
fit
the
and
been
have
combinatior
any
figure
space,
total
DIMENSIONS
application,
this
and
the
form
that
areas
set
the
SPATIAL ELEMENTS we can describe
As
them,
such as
shown
containments
In
figure 3.1.c.
As
physical
like
MATERIAL
contrapart
blank walls,
ELEMENTS
that
we
can
describe
delimits the spatial
access walls,
windows,
etc.,
thte
elements,
and
define
81
OI~(A4~
AP.7445I
A1'A//e
;/,
again
for
F/A2t'
each
of
them,
besides
its
their SHAPE,
TYPE,
5-.Z
DIMENSIONS
and RELATIONS.
The actual
site
figure 3.2.
might
be
thought
as
the
bul I t
82
total
space
that
represented,
correspond
in
can
the
dimensions,
shapes
stands
for the spatial
spaces
the two
material
and
their
between
function.
kinds of
elements.
FORMAL
The formal
site,
or areas that
It
as a set of
can be
nodes that
elements together
and
or material
3.1.1.2.-
the sense of
a
graph notation,
to spatial
and
contain
a
links that
set of
relations
with
that
exist
figure 3.3.
SITEt
on the other
we might say,
hand,
do not exist
represents
at alI
built space, but are conventions used
in
for the
formulation of standards.
As
situations
in
the
described
case
of
rooms
In
the
housing
with the S.A.R. Methodology, we can
83
several
functional
layouts
variety,
furniture
pieces
look at
their
the
of
tend
they are centered
or
MARGINS.
elements
possible arrangements.
furnitu
lets say a kitchen where
the
along
group
represented
as the site
As an actual
dimensions
formal
site
the wall
No
and
particular
made a
general
'site*
has
statements
general
for example, where a
of
Its
walls,
to
would
be
connections,
we can describe
elements
(figure
been
statement
its
spatial
3.4a).
As a
a MARGIN ad]acent
where kitchen furniture may
layout
where
site,
ire and appliances tend
we can specify a ZONE and
"W1'
MARGINS
figur -e 3.4.
site,
physical
and
only on one
necessary
in
and
positioned.
A room,
accomplished
be
can
function
ZONES
make
can
we
Through them
about
or can be,
are,
of
actual
the
in
areas
can
we
ZONES
scale,
(1)
"functional
those
functional
housing
the
at
done
A system of
represents
in
spaces,
describe such schemes of agrupation,
Is
as
in
layouts.
most equipment
talk,
along
aligned
space, surrounded by circulation or operation
To
among
grouped in
to be
time,
the
spaces,
living
most
In
wall
most
They are,
ways.
certain
that
and notice
be
defined yet,
about
to
nositioned.
but we have
possible
layouts
84
along the wall
"W1'.
A zone can help
POSITIONED,
define what
have.
margin
are the
we
If
agree,
statement
a
represent
lengths
be
zones
in
possible
fit
such
Margins
can
be
a
must
would
layouts,
on the
can
system of
in a
in figure 3.4.,
the site
might
pieces that
elements
that
positioned
about
can
the zone can help us
time a restriction
represent at the same
of kitchen
then
to
instance, that
for
elements
in margins,
end
adjacent
different
margins,
zones and
always
a
us define where elements
and
DIMENSIONS
site.
(figure
3.5).
Zones
conventions
and
about
used
to
the POSITION and DIMENSIONS of
represent
elements
85
In one direction.
A system of zones and margins can have different
widths,
the
it
in
across its whole length or width,
or
run along one boundary of the space, cross
extend
middle,
(figure 3.6).
simply cover one part of our actual site.
define
To
dimensions
In
S.A.R. concept.
which
length
conventions
on
and
positions
the opposite direction, we can again use a
A SECTOR Is a part of a
can
be
zone
and
margin
specified. If for instance, in our
kitchen example, the wall *W1'
has a window and we want to
say that some of the furniture pieces must
always
be
In
front of the window, then we can break a zone into several
parts,
one
of which has a length that corresponds to the
86
10=76202~~
length of
.--.Z
:,
the window,
and
stands
Ir-;;E
for
the
sector
where
-.J1
OP
V.
I
-
those ecs 7,e
those pieces can be positioned. (figure 3.7).
A zone,
as a
formal
construct,
has
only
one
87
dimension:
The
width.
length and width. A
sectors can have both dimensionsi
zone in an actual
length
from a combination of
actual
dimensions of the space
such case, we can say that a
site
take
can
sectors, or take it
where it
zone
from the
is positioned.
with
its
its
length
In
and
=V'k1
744'1 f
74C.3
width defined,
has only one sector equal
A system of
a ZONE DISTRIBUTION.
to itse lf.
zones, sectors and margins
It stands
for
a set
of well
Is called
defined
areas within a room, which can be used to describe general
statements
element
on
how
elements
layouts are acceptable.
3.1.2.-
Elements:
are
positioned,
(figure 3.8).
and
what
88
The elements
that
equipment
said in
the
defined
in
assumptions
or
3.1.2.1.-
material
we
can
units
the
of
of
elements used
of
terms
their
for each of
and
these
or equipment are
constitute
RELATIONS
any combination of
can
their
in
furniture
them.
have
3.1.2.2.-
USE
The use-space
or
around a physical
can
be the space
is
For
each
As
or
several
one
its
among them.
the
shape
of
limited to rectangular
them.
ways, but
length,
that exist
pieces,
site
element
an
assembled In different
to define
be
UNITS:
of
the restriction
elements
can
describe the SHAPE and DIMENSIONS of
to
3.9
SPACE,
pieces that
Similar
figures or
function
or
As was
function.
layouts of
defined
PHYSICAL
together with
units
a
1,
furniture
might be described.
pieces,
physical
be
their USE
The physical
element
perform a
of chapter
can
certain RESTRICTIONS
actual
of
it.
Elements
the
pieces
of the oossible
terms
UNITS
the
needed to
are
while performing
PHYSICAL
are
shown
In
figure
physical
units,
for each of them we
have
width and height.
SPACE:
is the space that
Is needed
unit to be able to use
we need around
a table
In
it.
order
along
Use-space
to
sit
89
down,
the free soace
needed
free
open
to
arawers,
for equipment
parts
Use-space
several
The
lines
for
rectangles
themselves
to
in
several
form
physical
or the space that should be
left
to move around.
be
described
also
Physical
certain
3.9b
pieces.
complex
as
one
or
delimit
use-soace
They can be related
use-spaces,
or
among
related
to
or adjacencies.
RESTRICTIONS:
units and
ways. For a
of its physical
figure
units through position,
3.1.2.3.-
in
can
the space
rectangular shapes with DIMENSIONS and RELATIONS.
dotted
specific
needed to swing doors open,
use-spaces can be
given piece of
restricted
furniture, certain
parts or use-soaces could be overlaped
by
90
parts
different
of
furniture
how
Independent on
Is
elements
through
stated
(i.e.
definition of standards
elements),
we
pieces.
can specify at
relation
the
the
next
3.1.3.the
level
between
two
In
the
issue
the piece can or cannot be overlapped
physical
units or use-spaces of other elements.
with this
restriction
minimal
access side
is
furniture
of each
piece, if
we can define
between
Relation
for each
either
by
Together
piece
If
a
required and by how much.
3.1.3.- Relations between elementst
Elements can be
in several
located related to
one
another
ways.
Relations between elements, called
here 'element
91
constraints*,
relative
positions
in
theory
that
three variations
of
one-
-
ADJACENCY.
-
OVERLAPPING.
-
CONTAINMENT.
express
and
satisfied In a
example,
specification
can be described
certain
functional
whether
*element-2*,
*element-2*
*element-3*,
or
and that can
only three -In
relate two
elements
conditions
that
layout. They can
"element-1*
whether
about
be
fact,
are supported:
relations
Element constraints
time
any
this application, however,
In
tested.
are
should
should
be
*element-1*
at
should
regulate,
be
adjacent
contained
should
the
be
for
to
by
overlap
92
"element-4*.
be expressed
Relations can
we can
of
the
RelatIonst
Simple and Compound Element
3.1.3.1.-
statements
constraints in
'simple'
one
in
in two ways,
list
a series of
form
'ELEMENT-RELATION-ELEMENT*, as was done in the
last paragraph.
This
clear
is quit e straightforwards and
and simple.
formulation
of
all
important,
But,
relat ions
to
or
"Compound'
it restricts
the
equally
of
lists
long
the
we
way
we
t tis option,
can be
relations
logical
connectives
denied
by
exoressedt
are
AND'
simple
'OR*,
With them
'NOT'.
there
relations
about
think
is
a
*compound'
option.
second way in
constraints.
linked by
constraints
and
one
select
might
anothe r if we could have the
To include
which relations
to
the way
elements,
constraint over
hand,
the other
havin g to be satisfied, constraints which
rarely corresponds
between
on
in this sense
of
capable
we can formulate
being
relatlors
such as:
SI.. "*element-1*
shou Id
never
be adjacent
be
adjacent
to
cr *element-2*
*element-4*,
to
but
"element-1',
or
'element-3*
*element-2*
should
neither
should
93
element-3* be contained
by "element-4'".
Compound relations have
of
Instead
(1),
notation
to obscure their advantage.
bisarre
sufficiently
perhaps
a special
saying:
element-1 relation element-2
as
where,
inside
the relation
and
by it.
element-2)
element-1
(relation
related
are
statements
now as:
expressed
first
simple
before,
said
we
So
that
a pair of
then the
parenthesis
list of
we
elements
define
that are
the statement:
***element-1* should be adjacent to *element-2*"
is turned
into the statement
(should be-adjacentto *element-i* 'element-2*)
or for simplicity
e2)
The reasons
for
although
clearer,
user, if
(adjacent el
perhaps
we think that
themselves
other
this
not
as
statements
connected by
*and".
elements:
statement-i
and
say:
"'statement-1'
and
justifiable
might
can
the case would be
The relation "and'
statement-2. So if
'statement-2'",
become
in terms of
in a relation
elements
relations,
inversion
we say
be
a
in
for two
has
we want
two
to
94
(AND statement-I statement-2)
format
with the same
relations
we
if be each statement
and,
simple
substitute
as
(adjacent el e2)
(adjacent e2
e3)
then we have
of
"AND',
a
them
relation
binary
nested
binar
the
two elements,
with
is a relation
that
by
formed
relatLor
or a compound
relation
e2 e3))
ei e2)(adjacent
(adjacent
(AND
'ADJACENT',
each
with tte
format:
ei e2))
(relationi (relation2 et e2)(relation3
(relationi elementi element2)
element
Compound
blocks,
the
relatlons,
hierarchy
of
can
complx relations
nested
binary
other
together with the
be
unary
like:
at
building
"CONTAINS' binary
the
relations
relation
as
use
"OVERLAP* and
'ADJACENT",
which
relations
as
bottom
to
a
and 'OR',
"AND*
"NOT*,
of
form
more
(52)
Which corresponds to our previous statement 51.
A
can
be
relation
graphical
visualyzed
representation
in
on each node,
terms
and
for
of
of
a
each
these
relatiors
tree, which shows a
node
two
or
one
95
(C4U*x-WNA='
[R
52:(AN
E54-)
ev e ))
(40- 14sWr A3 J4))
(coornavrp
as
elements
successors
respectively.
,
would
The statements
look
help explain the
Or
show them as
of
like the
or
binary
in
(5-1)
unary
,
and
following figure 3.12,
the system of
nature of
if we don't break our
a
Jj)
E0 j64
line statement,
in
04Z)
which can
oarenthesis.
blocks,
building
it would
relations
look
but
like the tree
In figure 3.13.
Simple
relations
several
satisfy,
to
that
alternatives
combination of
compound
Both
list
one
constraints
of
represent
can be accepted depending on tthe
connectives that
Statements
satisfled.
represent
constraints
iInked
by
we use.
*AND'
have
to
be
both
relation-cne and relation-two, represent
96
F/#A(Awf,/z 11
,-6(=44:23
-"r
constraints
that
we want
5W4)
(A#sii
satisfied in a
54}
functional
layout.
97
Our
list of
a
to
then
equivalent
simple
of
formulation
first
statements
statements
be
would
linked
only by
AND's.
can
them
have
In
present
the same
at
time.
by
preceded
Statements
their
then t he other cannot be
the two satisfied,
of
one
we
if
imply that
Exclusive OR'S
layout.
the final
two
for
that
either or t he two, or both of
we can satisfy
statementts,
impl y
OR's
Inclusive
*exclusive-or".
or
*Inclusive-or*
as
ways:
two
in
considered
be
connectives,
0R'
linker by
Statements that are
"NOT *
simply
reverse
relations.
final
For our example
as
considered
alternatives
accepted
as
if
then,
there
(figure
3.14)
which
combinations
valid
3. 11,
OR's
inclusive
possible
figure
in
If
appear
the OROs
w ould
are
be 7
be
would
in the final
arrangement.
the OR's
If
three
3.1.4.The
between
then only
the
first
would be considered acceptable.
lists
through
were exclusive,
location of
*POSITION
an
Position of
RULES'.
elements in
elements
in the
the si te.
site
Is
defined
These rules specify a relation
element and a site where it can be
positioned.
98
I--
Z -
(Amiqret'r
(r"as0/r
4:-
(r
(*4M
£1
El4) 54
Z 54)
r
(gm~mW
6,-(9 A2~egur
46*)
F3 51
5i ff4)
£14)
(4404rrrr-5
Position rules can relate elements
at
different
levels.
Elements can be
the space that forms the actual site,
to
located
*room-x',
the
site
simply In
they
can
be located In zones within the room, *zone-1*, o r they can
be positioned in sectors within the zones,
Similar
pos ition
rules
to
can
rul es, or 'compound*
the
be
relations
expressed
pcsition
*sect o
between
rules.
elements,
*simple r
as
r-a".
The simple
position
re lation
bet ween an element and its site would be:
elementi is
positioned in sitel, or
(PUTON elementi sitel)
In the
same notation that we
used
fo r
element
99
constraints.
The compound position rules wii11 use
connectives
or
element,
several
locations
alternative
rules that define several
the
to form nested position
and *NOTO
'OR*
'AND*
again
for
an
alternative elements that a space in
zoredw
MAPerr.,0a
;E4 4zZxwV))
7
~
(F//~4r('{/7
2/6/4
3/F
the site
can contaln,
like the rule
in
ano its tree representation
3.1.4.1.-
figure 3. 15
(figure 3 .16).
Definition and
Levels of
Exp ansion
of
Position Rules:
rules
Even though
both
different
from
are
express
relations,
element constraints
position
In several
ways:
-
relating
First,
elements
they
relate
to elements.
elements
to
site,
vs.
10/
4W ---
Sw
4/%(p/;'s
za#a463M )
, 4/S
(PrQ/
o
-
Second,
zones
and
def ine
relations
three I evels.
these
blocks*,
(4 4Z
element constraints,
To
correspond
position
rule
this
6o'
can use any
using
the
dlginr
with
can
(P.R.)
these
of
combination
same
terms
as
in
z1=17)
to describe a position standard as:
generate
to
as a space
between elements and site at any of
A compound P.R.
'bulIoing
structured
simple
a
sectors,
urtv
a site
for
r440a)
all
rule,
the
possible
however,
layouts
that
we have to know the
101
*ell
positions that
detailed
the site.
of
way
In a sImilar
constraints,
element
and
enumeration,
for
to the case made
compound
there is a conflict between how much
should
we
information
For
*zonel*.
wittIn
take
*spacel*, and
positions have to be defined at the most
all
level
can
*e2
the positions that
enumeration,
can actually have within
the
give
way
a
in
think
we
permit
to
standard
positional
about
constraints.
rule,
For a position
in the
conflict
this
Each
defined
P.R.
the
possible
at
any
p.r.'s
level
of
the
that can be
Implies
all
relating
its element to each of the parts
level
If,
Z2,
be solved
following way:
-
has one
can
formed
the
that
site
by
site
down.
for example, a site has two
zones:
Zi
and
then the p.r.:
(PUTON el sitel)
Implies both:
(PUTON el Zi)
(PUTON ei Z2)
-
vs.
specifically,
As the rule
I.e.
is defined generally,
Z1 or Z2,
i.e.
site,
then we can assume that
102
either of
to
link them
to
with an *exclusive-or*,
proceed
can
the two positions is valid, and we
form the new rule:
~V
which simply says,
if
we want
positioned
in
any
*site1* then "e1"
'site1'
zone,
two
in turn, has
sector s,
for example: Z1
s-/
//rdA
(lg (r,--04/
S2, and Z2 has S3 and S4,
in
valid
*S2*,
or
In
"S3*,
parts
of
each
has Si and
))
/ $$9)
5f 54 )))
that
or
"ei'
*e1*
In
turn
or
In *S1',
*S4*,
into:
would all
'et*
be
positions.
By
general
we
*e1*
X
then the rule would
which again assumes
in
S/)
C/
6C
two
of the
as a valid solution. If
accepted
be
would
positioned
*e1*
can
possible
level
avoid
automatically
expanding
a
rule
from
to its constituents in the following
having
to define
one
levels,
each and everyone of the
positions that an element can have.
Our original
103
rule at
the section, can be expanded then
beginning of
the
(Q0tc( PCX
(0
5/
53)f
E/
%3)
(P-Q
{o
A position standard can be expressed also as:
where
if e*
and
*e2.
e1*
say:
any place of
positioned
the
It
the site, as
our
reans: "put
all
as
long
in sector "S2"",
we did before,
all
for all
stands
the expression
element
el site)
(PUTON e2 site)
lets
the elements
*e2*
Is
in
not
and by a simitar procedure as
e+
Is
first
elements in the problem definition:
(PUTON
elements,
expanded
into
104
linked
then
we
because
*AND*,
by
want
all
sat isf led
(
and we
which
sectors
Si
and
have
f or
S2
#(/7:w Az 6/!)
the resulting rules
a site
with two zones,
ZI
and Z2,
in ZI, and only one sector S3
t'Pri~u f/
(w
(Plr#
S
))
6/5))
5Z ,i)=)
two
In Z2,
105
intot
would be converteo
3.1.4.2.-
Position
A third
difference
from
results
constraints
Rules
between
the
p.r.'s
element
and
two or more
having
site
both
by
shared
Is
sector
one
overlapping zones where
Over lapping Zonest
in
areas.
position rules were
If
defined
at
problem,
as
sectors, this would represent no
of
to
we would have
in each sector.
levels,
level
the
If we
however,
which element
then
always
always simple and
at
rules
compound
allow
elements that
the
list all
we have to define
go
several
a way to find out
goes where.
For
a
example,
problem
following
the
with
def inition:
Site:
21 with Si
Elements:
el
and
and Si, Z2 with
S2 and S3,
e2,
(Qa (Pt!EQW
(pc
cm~ss
Position Rule:
can
-be
positioned
does not
in sector S2.
EZ
zz)1
'3 Zi/)) )
3 ZZ )))
which
elements
If we think of
P.R.'s as
define
MITLbraries
Document Services
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Cambridge, MA 02139
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107
subsets of
describing
elements
that can be positioned
in a
site then,
Zi can have
(el or e2 or e3)
Z2
(e2 or e3)
can have
and if
(AV
we
expand
the
(er
X/ .5/1)
(/f
4/'4
rule
41P
(Ol
its
sector
SO
(Par
S/)
(ar £Z 52,)
(m
(pgtog(CX
(P17 5 )/
(P4r [
4Z
(e
into
s6))
(4r5 Sz
z
(pr1 55 )))
defini tion:
then the subsets
Si
different
can have
for
each sector would
(e1,e2,e3)
S21 can have
(e1,e2,e3) from ZI
S2 can
have
(e2,e3) from Z2
S3 can
have
(e2,e3)
where
S2
subsets
bet
is
of
undefined
because
elements that can
it
has
be positioned
two
in
108
it.
these
Having
zones. If
both zones Zi and Z2,
defines the
only be those that
In S2 can
elements
can
then the
decide
a
elements in overlapping
position
to
how
on
convention
we
however,
subsets,
Intersection of
appear
In
S21 and S22
position rules for S2.
S21 =
(eie2,e3)
=
S22
(e2,e3)
S2 =
(e2,e3)
(4'
(#/T (4ire' ,E/ S/)
(
(e~
(0/
52 ))
50
5Z
5
(Pgen4W
/
5 )) 3 ))))
,(parNOw(fss - S-4
2
3
(#2(PCgvx
and the position rul e would bet
To expand
can
proceed
overlapping
Intersection
as
then
sector,
of
the rules of
we
we
rules,
di d
have
and
Intersecting
we
3.1.4.1. but for each
in
to
zones,
check
select
first
for
the
those positions that
109
satisfy this test.
system
3.2.-
Standards:
A set
of
of elements
to the spatial
The
functional
parts of both site
relations
that
by
defined
a
and relations. The elements correspond
correspond
that regulate relative
rules
standards is
regulate
and
first,
locations,
furniture
pieces.
to element constraints
and
position
second,
absolute locations of an
element
in
the site.
In a more formal
consists of
manner,
a
set
of
standards
a 4-tuplet
S, E,
R, P
where:
parts that
'S*
form
-
*E'
Is the set of
-
"R"
Is the set of
is the
form
*P*
the
set of
compound, between elements
-
actual
and
formal,
the environment.
parts that
use-space,
spatial,
spatial,
physical-units
and
furniture pieces.
desired relations,
simple
or
in 'E*.
Is the set of desired positions, simple or
compound, that relate elements
In
"E*
~ with
elements
in
110
"s*.
A set of stanoards
L
graphs
,
which are
elements In *S*
that
to relations
correspond
the
of
in
*R*,
possible
to one of
correspond
together with
positions
in
*P*,
in
graphs
which
one of
enumerated.
a
In
or
"P*,
L,
the
by
to
edges
or both.
varlant,
furniture
the desired
for a given
contains all
'R'
the
where
the desired combinations of
To evaluate a set of
L,
linked
and are
or
layout,
A functional
one
formed by nodes that correspond
or both;
or "E",
possible
of
set
a
implies
is
links
relations
combinations
of
S,E,R,P.
standards,
functional
the
layouts,
subset L*
has
to
of
be
111
Enumeration:
3.3.-
a
do
a
have
not
layouts
functional
Is a valid or an
layout
a
when
membership. We
we
but
have,
members,
of
list
have a
are the poss ible
what
priori
that a standard can
Judging
for
instead a criteria
but we have
know
we do not
the set L*
For
for
criteria
invalid
furniture
arrangement.
To enumerate L*
then,
we have
which
qualify
layouts according to t his criteria.
From
the
possible
graphs,
possible
all
graphs
our "solution space*,
the variations that
To do
solution-space
might
reject,
we need
into
different
equal ly
Not
analized
that
rules
( figure 3.17)
th e points
Importance,
to the same
comb inations
conf iguration.
and
rules
these
having
situation where all
same
where solutlors
need
we
the
partition
as soon as po ssible, "chunks" that do not contain
any solution at all.
the
to extract
we have
"chunks"
important,
of
L
set
the
that
rules
functional
as
L*.
of
are members
this,
and
exist,
L
in
all
construct
to
of
variations
mean
in the design
therefore,
and
Ievel
would
detall,
of
this
all
having
have
criteria
have
checking
criteria
a
to
be
all
the
on
each
112
express
Wh en these rules can be defined, we can
We can state,
necessary.
a
are
systematical ly
worth
are
layouts
developed,
we can
construct
needed
more
next,
at.
looking
of L+
members
look for
the
if
know
to
order
in
satisfi ed
of
time, and check that some conditions
the
at
one
layouts
of detail,
levels
At different
becomes
this
terms of some conditions.
its validity In
layout, or
be,
the possibility
them,
through
We can
might
case
the
whenever
space
this
of
port ions
solution-space.
whatever
reconstruct
construct or
entire
our
the structure of
the m
through
We
can
to evaluate a
standard.
figure
3.3.1.-
3.171
Overview:
The S.A.R.
relations,
and
parts
formulation
of
standards,
with
Its
provides a way for expressing these
rules.
sequentially
by
variants
furniture
of
The generation
can
a solution or
constructing
carried
on
* graph",
where we add one element to the site according
the
position
satisfaction
rules,
of
the
positioned elements.
and
we
element
check
at
be
each
constraints
step
between
to
tte
the
113
Pr/60/W 3,/7
At
as a
carried on
of
the most simple
layouts
'depth-first*
this generation can
the
search, and
represented
be
can
level,
be
enumeration
as a 'State-Space*
model
where:
-
the
graph
being
constructed
represent
our
*state-descrlptor*,
-
the
*state-operators*,
position
and
rules
constitute
the
114
-
the
relations
against
criteria
which
between
states
elements
are
the
tested,
as shown
in
are
.44
9,
'
figure 3.18:
figure
3.18:
At a higher
can
be
used
to
level,
decompose
subproblems
which
alternative
combination
position rules,
with
can
make
of
the compound
the
problem
the
search
Position
can be considered as
several
subproblems
descriptions, as shown
figure
3.3.2.-
in
po
i nto
expressed
sepa
as
figure 3.19:
3.19t
Description
of
rules
different
s impler.
rules
a
sition
the process:
Each
in compound
rate
problem
state-space
115
P01=4N 41 -C
(AkP (ac (02 (Wf
'-5-
v
A-1!
()0/r C Z/,9
zz)V
(Q4'
(c
(A (41r 4 z)
A/11
At
4/ 0ftz
/l cZ P/f P
0,477M!W3
I
(A!T~
%3
A
(foPr z/0
(Rir C/)
A76245
511V
4tC4r6 $~9
The process
task of
with
description of
different
for
levels
this hierarchy will
outline of
in a search
of
be
down
of
the
smaller
complexity.
done,
first
The
in
a
the problem reduction steos, and second,
functional
detailed example.
will
breaks
exhaustive enumeration into a hierarchy
problems
quick
for generating L*
layouts
presented
through
Generalizations and- definition of
be made along the way as it
a
terms
becomes necessary.
3.3.2.1. Problem reductiont
The solution
space of
be constructed through the
rulest
-
GENERATION
RULES
a
functional
application
of
standard, can
two
kind
of
116
- TEST RULES
partition
the solution space into subsets that may contain
TEST rules
solutions.
through
check the existence of solutions
*operators'
*constraint
and
systematically expand and preclude regions of
In
and
Generate
GENERATION.
by
produced
those partitions
Test,
or
alternatives
produce
rules
GENERATION
criteria',
the solution
space.
Generate
corresponding to
are
rules
of
kinds,
different
two
levels, simple and compound,
the two
that
oosition rules can have:
-
TRUTH TABLE, and
-
PERMUTATION OF ELEMENTS
For compound
position
rules,
we can explore
alternative position that are acceptable for
different
element
(figure 3.19) through the construction of
TABLE,
as
explained
in
3.3.2.2..
rules, we can explore the different
can
the
take within
OF ELEMENTS,
For
an
TRUTH
a
simple position
locations
an
element
its zone or sector through the PERMUTATICN
or the variations in absolute positions.
Test rules are also of
-
POSITI8NAL,
-
DIMENSIONAL
and
two different kinds:
117
These are operations that check
DIMENSION of
elements
in
site,
a
Positional
Tests are
-
EVALUATION
OF POSITION
-
EVALUATION
OF ELEMENT
EVALUATION
tested by
OF
RULES,
Tests
-
ZONE
-
SECTOR DIMENSION
-
MARGIN DIMENSION
as
elements
include
the size of
a margir,
tests fort
an
element
to be
against
against
the
both dimensional
CIRCULATION
between
by
In a certain zone,
and
or remaining
length
both
the
oositional,
elements
absolute
margins
in the
position
if
or it can be defined by
relative position if assigned to be through the
use-spaces
the
the
as in MARGIN DIMENSION.
site. It can be defined either
assigned
by
CONSTRAINTS.
in SECTOR DIMENSION, and against
Important,
is
can be tested
DIMENSIONS
length and width of
An
the
CONSTRAINTS
of a zone, as in ZONE DIMENSION,
of a sector,
and
the EVALUATION OF ELEMENT
and check
constraint
a
POSITION RULES, and relative oositiors
Dimensional
width
by
regulated
thet
Absolute positions of
can be
as
standard.
functional
the
the POSITION and
in a given
different
layout.
118
a preference order between
These operations have
can
attempt
not
a zone until
elements in
of
permutation
we
Instance,
For
themselves.
we know what are
the position rules that assign such elements to
try,
to
dimensions of
find out
them
or
if
the elements against zones
that
location can be,
Only then
not.
constructing
hold
the
all
that the overall
in fact,
we can permute
will
these different
is broken down,
check
to
and sectors,
occupied
by
we kncw
elements
check
and
valid dimensions,
arrangements,
that the
the
margins
elements in the adjointing
zones, and
pattern Is respected.
circulation
The different
task
decide
elements we
relative positions are being satisfied, that
can
to
locations are known, we have
can have valid positions and
while
zone.
before doing any permutations or changes.
Once such
the
location of
valid alternative
first what
the
rules are compound, we have
position
these
If
a
levels
into which
are then in order of
the
Importance:
TABLE.
1.-
TRUTH
2.-
EVALUATION OF
3.-
PRECLUSION.
4.-
ZONE DIMENSION.
5.-
SECTOR DIMENSION.
enumeration
POSITION RULES.
119
6.-
PERMUTATION OF ELEMENTS.
7.-
MARGIN DIMENSION.
8.-
EVALUATION
9.-
CIRCULATION.
corresponding
"'1.v
CONSTRAINTS.
OF ELEMENT
to
expansion
the
pruning
or
P###w"1
4 Ive
PWINAr
M941/Nt
AT#17N
operations as shown in figure 3.20.
fig ure
3.20:
EXA MPLE:
The
how
these
possible
following example
operations
layouts
will
interact
for the standard:
to
be used
to
enumerate
describe
all
the
120
SITE:
ELEMENTS:
R EL ATI ONS: (
Oea'$/4'
AfC
pWeM/
Z
(4a4sNe CZ5< eM/r
Z2)
(0e
zz)
f I gure 3. 21:
{4VO (4/vr'
4
The first operation
to be
POSITIONS:
enumeration
process
functional
applied to
solution
layouts
that
first partition that
the
all
the
a
given standard can
we can make
compound
rule.
we
can
have, then
to
the
Implicit
in a
corresponds
alternative position rules that
this,
for
stands
space
possible
To do
start
would be then:
the
If
z
Truth Table:
3.3.2.1.
the
6#1r V /3P
4(44D (sm
(g
are
consider
each
*building
block" In a compound rule, as a simple relation that is or
is
not
satisfied
in different alternative position rules.
To each simple relation,
TRUE or FALSE, according to whether or not
have
these
positions
lets say
we can assign a *value*,
satisfied
in
solution space that we want to explore.
the
we
decide
region
to
of the
121
All the
these
relations
different
combinations
can have,
represent
position rules.
level
of
actual
layout,
we
the
pursued among
al I the subdivisions
a solution space
that can be made out of
Without
the
at
general
having explored yet any
what alternatives should be
first
decide
that
values
of
different
permited
position
the
by
rules.
Compound
depe nding
simp le position rule,
be
TRUE
represents a
valid
or
FALSE
or
the combinat ion of val ues
whether
on
can
statements
an
for
each
InvaiId
posi tion rule.
The
alte rnative
assigns TRUE
that
the simple rules
of
ch,
elf<
F/667
the
solution
space
into
of values can be expressed then
combinations
by a TRUTH TABLE,
each
of
subdivision
0
V
FALSE
values
to
the possible c.ombinations.
all
in
or
//
c c
01z
122
figure 3.22:
In
our examole
these possibilities.
the different
Truth
or
enumerate
all
0
to
in
31
there
binary.
be
can
simple rule
a
with True o r
As such,
there
are
values,
simple relations there
relation
with
So
a number
that
be
As can
from
rules,
large compound
probl em*,
*counting
they
1 values,
0 or
statement.
and for
added to the compound,
two
False,
r epresents
each column
alternatives from 2 to 2.
insofar
"oinary counters"
Tables are
figure 3.22,
in
seen
cross
that
as columns
alternatives for a compound
the
or 1, and
for each r ow.
values
"count*
is
single rule appear as a
possibilities appear
along alternative
of
by
and the Truth Table
side,
take the values Tru e or False, 0
can
that
left
matrix where each
represented by a
there can be 32
3. 22),
The position rul e is represented
tree on the
a horizontal
row
(figure
each new
Is,
increases the number
for
one
simple
of
relatIon
for a compound relation with two
are four
values,
three simple relations
for
there are
and so on; running into the enumeration or
compound
a
8
values,
*counting*
of
large numbers very easy.
For the time being,
in
mind
but
no
this problem has
solution has been
Implemented
been
kept
to reduce
123
to
the
direct
combinations
starting
such as
likely
*
in
POSITION
figure
the
of
some
problem
in the
definite
relation
for
oartitions
of simple
OF
that
compound
in terest us.
tha t
relations
have
a
out these
find
the
TRUTH
connect ives
that
in
sense.
In
expressed
together
-
the
EVALUATE each of the columns
logical
As
have a
out these
find
Position Rulest
for the compourd statement. To
value
alternatives, we
tie
rules,
EVALU ATION
in
valid combinations
are
rules
These are combinations
TABLE
those
towa rds
RULES.
Only
TRUE
How to
3.22.
3.3.2.2. Evaluation of
position
be
first valid co mb natior,
from the
is
combinations
valid
values
to produce valid Positi on
our 'counting"
column
of
assignment
must
could
alternatives. One possibility
this generation of
simole
3.1.3.2,
the
into compound
relations
statements
meaning:
for each,
have
to
be
AND*,
*true'
,1a
the
two
to have
4 /
element s
in
the
the whol e relation
124
evaluated to "true*,
for
-
elements have
each exclusive *0R*,
to be
either one
the two
of
"true' to make the compound
statement
*true".
both
for each Inclusive
elements
being
'OR',
one
produces
*true'
of
two
the
or
a *true"
compound
element
makes
relation.
*true*
relation
these simple tables.
are nested
'false*
a
and viceversa.
*ANO,*0R*,'NOT*,
to
a
for each 'NOT*,
-
in compound
are evaluated
When
several
statements,
then
according
*AND*,*OR' or
we first
find
'NOT's
out
the
125
values
"lower"
relations
the hierarchy,
in
the
the resulting values as values of
then
the
the
for
next
the
elements
relation up, and continue doing so until
and
relation,
final
have
the
whole
pass
for
we reach
statement
4
JoeC
<'v4
as shown
evaluated,
in
figure
3.23.
figure 3.23:
In
of
example
to
in
figure
figure
3.21,
rule
standard
the position rules
that
evaluate
26,27,28,30,31
and
columns:
3.24. Only these
relations represent valid
position
our
truth-table generated
are only
*true"
shown
the
at
the
side of
these combinations make any sense
32,
combinations of
alternatives
left
for
to
of
the
the table,
continue
as
simple
compound
and only
exploring
126
for
possible
in the
columns
root,
If
we
branches going
out
layouts.
different
matrix as
we can preclude
/0
then from further
3 4
5,
7
O/
the regions
that
/ /
extend down
the
tree
our
consideration all
r303/32
524 25627
/
0
/
'
00
011
_VW74
of
89 /'///2/3/4/5//8/.92ZI222
--A/
of
think
those paths.
figure 3.24:
Through compcund relations we
into
problem
elements.
the
different
possible
Through the assignment of
for each
problem. Through the evaluation of
considered valid
can
that can be made
all
decide which of
we
truth values,
the possible decompositions
a
locations for the
explore
we can
decompose
can
these
the alternative positions
values
should be
and continued being explored.
3.3.2.3. Preclusion of repeating elements:
127
across
manner
look
different position alternatives, we would
the
compound
as
26,
branch
now Into
a left-to-right
example in
In our
Moving
statement
true.
evaluated to be
This statement and
the rest that have passed
now
checked
are
our previous test,
all
of
PRECLUSION
for
reoeating elements.
As can be better seen In
brach
30,
little ahead, there are s ome cases when compound
element,
true,
to
evaluated
be
to
here *bed*,
In
positions for one element
can
element
An
Even
be
not
though evaluated to
makes no sense when
The
test
different
and
Z
zone
environment:
assign
but
zone Z2.
in
positions
valid
of
accepted.
in two places at the same
this
the
This repetition
compound
Inter preted as a real
for
rules can
times the same
two
cannot be, obviously,
TRUE,
a
lumping
branches
time.
statement
position rule.
with
repeating
4 S, /4, /00, 7Z4
elements, would preclude then columns, or branchesj30 and
32,
from further considerations and reduce the search for
functional
In
layouts to branches,1 26,27,28 and
figure 3.25.
figure
3.25:
3.3.2.4. Zone Dimensions:
31,
as
shown
129
/
(PW
"--
'6
34
'
After
!V
the site
position
The
possible spaces
now
of
in
first
elements
is assigned to only one of tte
the environment,
this
that
at
width-
the
least one of the
Is
equal
or
check.
assignment Is correct.
zones
or
If,
of
sectors
element.
dimensions
larger to the width of
is going to be positioned, and equal
width of both zone
or
and adjoining margin.
element and
elements
for Instance,
allowed to end in margins, then we have
only
to
and we have
a rule that defines so, and second,
is
should be considered.
site
position
can be positioned.
is an agreement on how the dimensions
there
it
there
of
repeating elements, there is only
Elements can be positioned in
if,
k
II
combination
where each element
dimensionwise
if
one
selecting
l/
1Z
1
/;' +fl
rules, valid and without
in
3/.YZ
//
:44
(4tgi-)
one part
17
.--
ru )
are
to check now
-length
or
the zone
where
smaller
than
129
figure
For
suficclently
3.26:
our
small
example,
{j
MM-449AM
F
test
as shown In figure
the
Zi
and
four
is passed by al I the
4/M4W#N4'6
branches
zone
to contain any of
any position. So this
-olow4w7#
both
3.27.
Z2
elements
are
in
remaining
13o
figure
3.27:
Sector DimensIonst
3.3.2.5.
Valid positions
the other
along
have to be ch ecked
in zones
dimension:
Elemen ts,
length.
be lo cated in zones Zi and Z2, for branche s
31, w Ithout repetition
now
we know, can
26, 27,28
and
and fiting within t he wid th of
both
zones and margins.
test
The
the
for
to
assigned
elements
check s
Sector Dimensions,
al so
a zone can
fit
I engt h, or along the length of the sector where
if
all
along its
have
they
been assigned if this would have been the case.
Sector Dimensions checks
depending on how they are
or widths,
the
of
such part of
figure
the
our
zone or sector,
sum
of
lengths
between
It
is
widths
or
all
length
of
all
than the corresponding dimension of
the difference is occupied by an empty
length or width.
As a convention, this space is
entity.
the
of
3.28:
smaller
space with that
exceed
lengths
site.
the
When
is
positioned,
In a zone/sector does not
elements
elements
that the sum of
treated
not broken down into several
elements but appears
as
one
unit
as
one
empty spaces
that
can
be
131
In
changed
in position but
this
Under
further
expansion,
7
RKECM.0N
~~a#
,V'rMS
P/w-
I loi
keeps always its dimension.
test,
but
branch,31
all
is
excluded
from
rest continue as valid
the
23
T,6M
I'A'-
132
options where
furniture
layouts might
figure 3.29:
.1 *
exist.
133
3.3.2.7. Permutation
of Elements:
we
have SORTED each element
before,
said
these options, as was
For each of
in the rule to only one valid
the site.
and possible space in
of
assignment
this
For branch 26,
to
elements
site would bet
Z2 with closet and desk.
Z4 with bed.
interesting
two
present
sort
This
characteristics:
into
produces a CLASS of furniture
1.-
It
2.-
It permits sorting the
constraints
that can be used for
constraints
LOCAL
or
GLOBAL
element
layouts.
pruning criteria.
We know that as
I.-
already
a valid
be represented
This
as:
layout satisfies one
fit
in
they have been assigned, and
the only sector
If
that
the width of
they also
fit
the zones
where
length
the
of
that each zone has.
we forget
for a
furniture variant.
each
position
alternative
moment
that
elements
the zones can switch positions, we can say we
found a
have
we
which schematically can
furniture layout,
elements
Its
rule,
and sectors are
margins,
is without considering
that
concerned,
far as zones
element
If
can vary
on
its
have already
the other hand,
location in
within
we accept
the zone, we
134
can say then
We
layouts.
space where several
rule
of
and
indeed,
found,
have
found
have
that we
a
a region in
arrangements share the
furniture
of
CLASS
the solution
position
same
assigred to the same sector
are validly
or zones
a site.
The different arrangements
formed
the
by
permutations
Zone Z2 can be,
each zone.
for
of
the element
are
CLASS
this
In
locations
in
Instance, either:
and Zone Z4 can be either:
r---
1~10
these
The combination of
equivalent
several
generates
same elements in
EQUIVALENCE
different
arrangements that have the
constitute
the same zones, and that
of
CLASS
locations,
furniture
layouts
and
in terms of the
relation *position*.
Exploring
*top-to-bottom*,
we
our
solution
have
partitioned
from
space
the
set
of
all
135
are
yet
positioning
any
exponential
explotion
putting the
bed in the
put
we reduce
at all,
corner of
lower
tr led
then
Z2
to
so on, for
and
Z4,
the
we star ted
could have had, had
we
the upper corner of
in
desk
the
piece
furniture
our arrangements
*merging"
that can be tested without
2.2.4.2)
*re.
branches
into
By
by similarities.
grouped
layouts
where
CLASSES
into EQUIVALENCE
layouts
possible
each possible combination.
have
we
class,
terms of our
class
by
generate all
class would be generated
1.-
The
elements
In
branch
by the
in
our
that can
layouts.
the
26,
equivalence
foil owing representation:
*state-descriptor'
plus the positioned
model
state and a series of rule s
example
or
t ree,
we
representation,
the equivalent furniture
For our
e quivalence
the permu tatlons of
We build a combinatorial
State-Space
an Initial
each
in
to construct now all
the site.
elements in
layouts
enumerate the
To
Is
site
formal
the
elements.
2.-
The
Initial
3.-
The
goal
is t he empty site.
state
state
is
the
site
all
with
the
positioned.
4.-
The state-operators are
the
list
of
simple
136
prosition
column
with
rules
that
true*
values
£3- 4
)
Where
elements
llq
c
the operators simply state that
a
can
be
-jY
F-
formed
by
the
In zones Z2 and Z4.
should be
4
A11, S
ZE2
be
the Truth Table
we are exoloring:
V4
should
in
two
possible arrangements of
That
formed by two elements El
a desk or a closet,
and
layout
an
arrangement
in
Z+
and E2 either of which
that an arrangement
for Z2
137
kqA
A9
Ai6O~
3~33:
SA
138
should be
either
formed by two
elements E3 and E4
El
following 4
figure
can
be
a Ded or an empty soace.
Applying these
first
which
then
E2
layouts
(3.34).
then
operators to the
E3
as shown
and then
in
E4,
initial
we
the bottom of
layout,
generate the
our
tree
In
139
way,
elements
however,
zone.
same
within the
F/Mc
assigned
to
can
The
be
bed
piece
which
as:
smaller than
corresponds
09
II
th at
and ther is a
decide on
any
of
layout, and
f
$3
change the operators E3 and E4
The elements
fit exactly
know
we
OR
[JemI~
F76OME
(Z4),
to its four 90
In the furniture
appear
to
positions
its site
Therefore we can
remaining empty space.
these
be
33
Z4
is
could
example
degrees rotations. From SECTOR DIMENSIONS,
this
same
differently
positioned
for
the
in
positIoned here always
are
Elements
in zone Z2,
Into
desk and closet,
on the other hand,
therefore they can only be rotated
140
180
degrees, which keeps
the
zone
tree
like
their dimensions along
flIB0J
0
ir/Ajj
t he same
9//,El
O
H
-~
~E
liDo
~
changing the operators El and E2 Into
produce a
Which could
the
one
Qqeierating
partially
represented
64 possible
combinatorial
in the
layouts with sirrilar
following
oicture,
positions.
141
2.- Classes of
sort
the
constraints.
arrangement
element
If
furniture layouts
constraints
into
help
Global
we think of a furnitdre layout as
formed
of
different
then we can
break
arrangements
down
our
us
or
also
Local
a
at
room
the
previous
zone-sector
level,
State-Space
representation into the following model:
142
room levelt
-
state descriptor = same,
-
Initial
-
goal
-
state = same,
state = same,
=
state-operators
assignment
of
zone
arrangements generated by:
zone level:
-
state descrlptor = Z2,
-
Initial state = empty Z2,
-
goal state = complete Z2,
-
state-operators
=
assignment
Z4Z
A764
4Ai;a
elements to zones as?
zone level?
-
state descriptor = Z4,
-
Initial
-
goal state = complete Z4,
state
=
empty Z4,
of
143
-
state-operators
Zr
-
=
assgnm1e"t I
of
-
Iw~
I74J34*u
elements to zone as:
woul d
which
,OAC
produce
the
following
room
404O~
le
state-operators:
In
apply one of
the generation
we
would
the operators at the room
proceed
level,
first
i.e.:
to
144
ROOM
-
but in
out
first
Z2,
order
Z
to do
this we would
an arrangement at
this operator,
therefore
we
zone Z4 which can be used as
to
have
construct
o
at
which produce:
the
zone
by
we can apply to
3-40f)
to get: {90,d4)
level: (344
(1oe)
form:
and continue with ZZ In
with:
It
a
applying the operators
that
find
to
have
a
(5-4of
similar
way,
first
145
9
.1e
165A4'b 4
and
then:(3,4'I)
E0
to form:
This
descriptions,
operators
same
for
of
representation
where the result
of
equivalence
class,
and
one search oroduces the
level
up,
permits
us
the next search one
State-Space
NESTED
produces
the
to
the
sort
146
in the following way:
constraints
A
CONSTRAINT
GLOBAL
in
elements
relates
zones or sectors,
different
same
the
In
A LOCAL CONSTRAINT relates elements
zone or sector.
As wiil
first
the
at room
State-Space
relations that apply between
those relations
third
help
include those
while
would include
for
and
constraint
th-e
element
Z4.
These sorted
DIMENSIONS
Z2
in
that apply to element
OF
criteria for
elements In Z2 and Z4,
State-Space those relations that
positions in
can
would
level
the second State-Space
in
constraints
the
EVALUATION
section
the constraint
(3 .3.2.9),
CONSTRAINTS
ELEMENT
the
be seen in
and
CIRCULATION
us prune
constraints,
Element
branches
are
In
with
the remaining
tests ttat
exploration
our
MMARGIN
of
the
Solution Space.
3.3.2.8.
Margin Dimensions:
When two or more zones share
them,
might
are
the
elements
larger
than the
margin
between
can be positione in each zone,
that
overlap portions of
a
the margin
width of
As the functior of
if
their
dimenslors
the zone.
a
margin
is
precisely
to
allow the position of elements with different widths, when
a
furniture piece extends beyond
the width of
its
zone It
147
the margin.
occupies a portion of
common
conflicts
margin,
is
the elements
fit,
elements
have to
we
check
overlapping
this
If
For overlapping
overlap.
elements
the
then
both
margin
the
than
larger
is
the sum of
if
a
than the margin then
or equal
the sum
if
occurr:
end within
zones
opposite
might
smaller
overlappings
dimensions
in
two elements
When
permited
Is
or not.
the
generation of
the
As
room, zone
levels,
against
previous
conflict
that stops
branch
In
would
the search
possible would pass it
continue
tree,
only
our search all
passlog the tests
the
of
arrangements
two
way down
the 64
we
case
in
conflict
any
a
further.
in branches 26,27,28 all
withou
Is
there
from going any
for example,
the test while
pass
if
see
to
dimensions
margin
the
test
arrangements
7
assign an
or sector, everytime we
its position, we
element to
of
layouts proceeds at any
to the bottom of our
EVALUATION OF
and
CONSTRAINTS
CIRCULATION.
3.3.2.9.
Evaluation of
Element Constraints:
Element constraints,
expressed
by
compound
like
statements
which
can be TRUE
FALSE,
depending on particular combinations of
values
in
rules,
however,
their
simple components.
the assignment of
Different
are
roles,
position
or
TRUE-FALSE
from
position
these values Is not
done
MITLibraries
Document Services
Room 14-0551
77 Massachusetts Avenue
Cambridge, MA 02139
Ph: 617.253.2800
Email: docs@mit.edu
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DISCLAIMER
Pagelas been ommitted due to a pagination error
by the author.
149.
through
a
*blnary
counter',
alternatives to explore, buth
actual,
present, re Iat Ions.
,Wq-
or
a
through
decision
the
on
which
evaluation
of
150
defined elerent
constraint
The evaluation
like
done
bottom-up,
that
statements
as in
If
the
when all
to wait
for
from
Is
compound
simpler
so
to
relations
from the
compound
the relations, however, can be evaluated only
values are set
a
complete
a
that
but
series
of
each
for the results
branch 26,
(OR
is
State-Space
as was
said
representation
In any
other
that
the relation
zone
to
element constraintst
this
would mean
(adjacent desk bed)
(adjkanti
have
nested
the constraints
evaluate
layout
a
if
.
or sector arrang
( AND
then we have
or FALSE,
to TRUE
layout,
can
we
without having to wait
In
constraints
element
(3.25).
figure
representations, we can break
before,
the
is set to FALSE.
the evaluation of position rules,
through
generated
of
among
satisfied
not
is
the relation
present elements then
is
it
TRUE, if a
the relations have a value
it does,
If
not.
then
the site
some relations to other existing elements
either satisfies
or
is positioned in
an element
If
be
&ak ch'set ))
broken
down
into
the
following
151
room:
(441
(4,(t'g*;#n
d't/k d"S9)
z one Z 2
117011V
zone Z4
We
'gg4 A4')
can do
f/ekwe
transformations
the
this by applying
7
~-e/,#77ffi
shown
in the next tables:
tree
which reduce our
into several
that have
each
trees,
at
to be satisfied
do not
possible
satisfy
layouts
to
of
the
the
constraints
element
to the relation
one corresponding
pruning away all
that
zerieZ
level
of
the site.
in
those combinations
constraints
12,
from the original 64 that we
if
reducing
and
there were no
could
have
26
branch
the
further tests
had
in
figure
152
153
0
0U
3.39.
In
produces
this
valid
case
the State-Space
of
zone
Z2
always
arrangements because its two elements are
154
and
always adjacent,
When
adjacency bed-desk.
the case of
not
and
that
Z4
in
constraint,
over
it
search, when we don't
have
still
a
relation of
like
positioned,
branc 26,
then
we
do
assumed that the
is
the element
constraints.
values
long as we can have 'true'
As
at
comes
the
a piece is not
have priority
position rules
however,
preclusion
the chair, in our example
evaluate
with our
X2
between
level,
global
therefore
we stop. True
we proceed
constraints,
problems because the desk
*adjacent'
EUl
In:
to the closet as
after
Therefore,
check
blocks
the access to
its use
space.
we
have to
checking element constraints
for CIRCULATION.
3.3.2.10 Circulation:
we positioned has to have
Each element
its
use
element
space.
that
is
IF
a spatial
in a zone, then we treat
it as any
access is
other element, and specify the relations
have with the
If
to
defined as
this
located
acces
furniture Pieces it will
the circulation
that
it
should
as
access
serve.
Is defined simply
155
use space withour any particular specification,
every
to
then we have to check that there Is a chain of use-spaces,
or
be
can
access
this
which
through
spaces
leftover
solved.
last
This
we position a
Everytime
path. If we think of the
a
finding
graph,
furniture piece we check for this
a path
precisely
be
a
that
spaces
use-spaces, margins or
leftover
are
linked
bh
a
certain
that
minimum.
us to reach
around, and that should allow
we can walk
each piece
of
of
adjacencles
furniture
the room.
in
Finding a spanning tree for a graph
problem
solved
to
has
path
of
series
spaces
links
the
of
that goues through some
of
tree
A spanning
that graph.
its nodes. Our circulation
of
but visits all
for
a problem
then
path is
circulation
a
as
constructed
being
layout
finding a "spanning-tree'
is
case.
second
test corresponds to the
with several
Is
well
a
alforithms that can be used.
(2).
We apply this as
a
prunning
criteria
In
the
following way:
-everytime an element
spanning
path
or
valid
circulation.
tree
for
is added,
it,
if
we
construct
we succeed we
a
have a
156
our
brancn
26,
this
come
furniture variants
which are
elements
In
layouts
for
tree.
combinatorial
With
we stop any position of
fall
we
-If
which
4
to
down
for
possible
our
test
of
one case
the end of our
long
When we apply this
12
represent
position
our basic
rules,
and
search.
complete procedure
to all
the
other branches
7,
27
layouts
and 28,
that
we end
up
with
the
19
our standard permits.
27.-
e
*
eW
~e %S
ol/m
Ine
m
/Afendr
26 -
basic
fdrniture
57
158
4.
COMPUTER SYSTEM:
A computer system was
the process
defined in
operation
computer
of
SPACE
and
or be
FUNCTION
programs being implemented
generation
of
Basic
to
carry
last chapter. It can
"furniture sl-u fler",
an independent
the
the
implemented
Variants at
be used as
incorporated as
ANALYSIS
at
on
M.I.T.
in
for
the Housing Level
the
the
by M.
Gerzso.
From the user point of view, the system
as
having
two
definition of
of
existing
relatlors
one
corresponding
querying this Irformations,
in the site,
such as sizes of
et
.,
and
to the
second
for
cuerying first
the
for
furnitdre,
adjacencies
Implicit
configurations
furniture variants.
Internally
-
the
partst
standards, and another corresponding to
process
or
main
appears
basic
it
is organized in
front end "SARCASM",
a
variants
program,
or
four modules:
user
written by M.
Interface
of
Gerzso and
M.
Gross.
-
a Relational
-
a
-
and
RDB
set
Data Base,
"RDB",
manipulation
LANGUAGE",
the SEARCH programs,
routines,
OQUERY
159
4,1:
O6C43A.
4.1. Relational
standards
The
during
needed
the
process.
the
Information
the
1r
for
a
RDB(1)
constitutes
which
for the system. There
Is only
generation
rules used in the
(in the Permutztion of Elements).
A small,
for
formulation,
Information
representation
another
Base:
enumeration process and the resulting
configurations are stored
the maln bank of
Data
in core, ROB was
furniture shuffler
written
specifically
and the S.A.R. Basic Variants
program.
In
this
as entities and
after
the
kind
of data base,
their relations. As
description
of
S.A.R.
Information
Is
Is aulte obvious
and
the
stored
now,
enumeration
160
process, both our standards
are
basically
,%/*7iE4a
the simple
/
that:
entitles
52A,(c(~
i*
function
Where we have
have a
entIties,
list
"state-descriptors'
and relations. For example,
would be represented
site
our
and
lists of
In our ROB as:5/)
entities,
one
for
tt~e
another for our elements entities, and we
of relatiors,
in
this case
binary
relations,
one for the position rules, the other for the adjacencies.
161
AO-744V7
E
56
r
5 2
A RDB consists
Information,
entities and
is
lists
before,
of a general
a
provides
it
lists of relations.
to
up
us.
search process,
as:
What
can Input
We
or we can input
of
way
//E4-)
in
the
logical
representatlon
structure of
In
those
a standard as
we did
we
put
during
our
that
have
concerned
with
elements
as El and S1.
the site
This general
of
-
Where we can keep track of
been positioned
lists
cefining
state-descriptor,
a
for
reoresentation
is
the data, rather
contents, we can change our
descriptions
than
as
its actual
we
please,
162
include
partial
complex
as
formulaticns
relations or construct new
as
retrieve
desired,
lists out
existirg
of
ones.
made
relational
through
this
description of
Precise
together with a set of
structure
algebra or relational
can
be
calcu Ius,
operations which can be applied
to
retrieve or
In our case the RDB
was
implemented
with
the
following data structure:
where we keep the entities and
as
two
entries
for
entity
contain
which
lists
external
lists,
and
40
their
relations
respectively
entries
for
20
relaticn
lists
In
entity
site,
such entr les we keep
chains
el ements)
basic
data
or the relation chains, such as
(i.e.
position
rules,
t-e
about
name
(i.e.
adjacencies ),
a
ena
maan
(re~w ~~474v)
fe/*eader~
I
/A4 MM#{
by
(kf,4wr
____I
r
(/f0Ar
I
doeA4/?#
__
I
4
.9
Yt/,,r~,,
hill"2~~ad'
Aea,"
~~ ~
po
,~r
I
)
rhcA
d-dA
~~&~U
L~ /I~W'%
or
4
p
7774
-61-.
1
I
-~
r-<-~
-~
163
Rewwt;?..
Rea4/Oay as
r
17ZIIZ
A 4r P )Wa.. A4-rW
ansa
164
pointer
end
of
to the beginning of
of
format
the
were not
and
For each
Infcrmation
each
for
entry has a name,
the previous or
for
of this
to the next entry in
each element
list has
that we want to store.
entl y
to
the same
additio al
list,
link it
flexible
Information
as
to
list.
The
besides its name and type.
been to be as
Each
list
property
a
in
entry
have an
down, and two pointers that
In a property
kept
list, we
a type, a pointer
further
,explained
and
type
as
such
thought
were
relations.
entities
chain
slots that
all,
at
used
to - he
and a pointer
list
plus additional
each list,
necessary but
each
possible
Is
idea
In
terms of the elements we use for our reDresentations.
From the description
Information
can
be divided
can see that the relational
in 3.1 we can see
in the
part is
that
follcwing way:
taken care of
our
and we
by
the
165
and
entity
is
information row,
here
we
can
have, spatial
*property'
link several
which
the
structures),
this
For each atribute
keeps
actual
track
we have
a
of
the name of the
dimensions,
restrictions,
data
(in
and th-e needed pointers to
In the property
In
list.
property
'atributes' that an entry might
examplet
for
etc.;
graphics,
in
or nonspatial.
entry
Information,
incluced
particular
the
while
lists,
relation
the
different
data
elements
further
list.
By subdividing information
store different kinds of elements
keep their different data
in
In this
way
in the entity
can
we
lists,
different entries of
ard
property
lists.
In the other part of our
relation
list,
being related by
RDB,
we have
for
each
an
entry representing a pair of elements
it.
In this
entry we donot need
to
store
166
the
elements again, but we store
elements, a reference
information
lists
where
of
location
they
are.
two
the data entry
Several
in
in
delete
In the
routines
*id"
for such
them In
tte
doing so we avoid
*1d'
is
the
core, and we have
location In
other Items are poIrters that
previous and next pairs
retrieve,
By
to
our case this
therefore a pointer for each element
The
an
that can help us get
redundancy of Informatlor. In
address
Insteaa
link
the pair.
our pair to the
chair.
were
written
or query elements
to
and relations,
Insert,
as shown
figure 4.6
and in more detail,
With them
we can
relations(PUTRDB);
delete
(DELENT)
pairs
relation
or relation
pairs
in
retrieve
the values of
combined
queries as will
all
they
In sert
are;
elements
entries
in
relation
th e
Information
lists
relation
some r elations
be h own
in
(PUTSPC),
(VALNAM);
In QUERY
lists
(DELREL)
lists
or
or
(DFLRDB);
or
LANGUAGE.
make
-
-
-
&-
-
-
-
KA 11reel77 W 5
-
U
U
r4~
P
1
I
dl.
W.
AO
AWWW
1'
AMP
I
I
III
U
I
1
0~
A
1~JL
I
U
I
-I
AO
AOAW
do
LiEJ
Wat
--- -- -
Gdbkv/
168
programmers manual
not
these
of
detalIs
The
included
in
as
easier
understand,
to
data
Implementation,
languages
manipulation
4.2.
Is
are
Spatial
flexibility,
the
ease
irdependence,
of
4.2 are
of
precision,
ease
clarity and
the data
(mieically present
in 4.3).
Representation?
representation of
aroung
L.Teague's Ph.D.
representation of
thent
the ones in figure
The spatial
organized
(2) Thesis on "The
Relationships
Spatial
layouts
furniture
In
a
Comouter
building design".
System for
Teague
network
like
tables
simple
in
these Thesis.
The advantages of an RDB(i)
use,
are
routines
of
rectangles
as
a
larger rectangle. Based
in
spaces
describe
within a
Tutte's network representation of
it
extends this description
the
following representatlor.
It
organization
express
and
network theory for
in
to
in
a
squared
building
rectangles(3),
three dimensions.
network
terms
the
By using
spatial
makes therefore available the results of
the analysis and synthesis
of
spatial
169
A
f3
A
relationships.
In this network, spaces are described by
or
*directed
links"
wtich
'fIow*
correspond
'arcs'
to
thIe
I
7170
which receive
describeo as 'nodes'
left
the
of
*flow*
the
origin for
between
adjacercles
The
space.
nori'ontal(xy)
the
(z) or
vertical
'arc
space,
flow"
for
tre
sides of two
soaces are
on ore side
the
lets say, ano
the
of
dimsnslans
which are the
'n
soaces
arc'
right
the
I
side,
like
mode
This
to
Instance,
for
of
Eastman's
opoosed
repres erta t Ion
(4) or
Yess io's
(3),
was selected
two reasonst
1.-
Its
limitatiors to rectangle
SECTORS
In
rot
f3ct,
nave
complex
sfaPes,
they
are always
rectanIles
because
in
the
end,
ZONES
we
though
composed of
d3
shaoes
Interfere with the orincicles in the methodoloqy.
ever
for
restrict
the analysis to
ortogonal
soaces,
ard
ard two
directions.
2.-
Its network description blends
Itself
ilte
171
well
our
with
and
RDB
the
network
becomes
one more
relation in our data'fo r config urations.
One
represent ing
was
change
as
spaces
switched soaces as node s
mad e,
arcs
sides
and
and si des as
graph wit h geometric ch aracterI stics
dimersions
amount
for
each
a nd
node,
representation,
is some
of Elements),
1iritat ions
to keep
spaces are rectangles always.
a
convention
representation.
add
on
In
corresponds to the generation
get
an
L
shaped
room.
* free'
corner,
space,
and subdivide the
S 0,
room
Teague*s
that
as
all
(th
we establish
this
*squarec"
we can see how when we
is
layout
o f I ayout
we do
o f c hair,
I nto
*room2*
ele gart
Permuta ticn
1
by
the
In branch 28)
then
to the end of
*room' and
way
we
is extend the
our
*room1*.
The same happens with chair-use-space
roomi"
ard
the resulting
he does,
0 btain
Wha t
i.e. SouttEast
shape
layouts (at
examp le
our
chair-physical-unit
our
to
t rac k
to
how
as
this graph seauenti ally
when we corstr uct
we have
Having th en a
links.
such
we
having at each link the
is done during generation of
as it
nodes,
as
of
spaces.
of adjacency between two
There
Instead
however,
and
we get
and *room*
When we put
the bed-p.u.
then
we
just
reduce
172
O4AIR.0
Om~IR.LS
1!41 -,
173
'roomi'
and
"room2*,
CHAIRJ
"IR.UsKO
which d sappear with the position of
PL
Uq!PK
PU
KD.PUi
c"IAR.F
OHAJR.L6
DESK.LU4 D~s
'ED."U
the bed-use-space.
The network In
the right
side
should serve to Illustrate that there
tree at each
to another.
level
of
our
figures
is always a spanning
which allow us to move
from one element
174
until
we get the
COAIR.
final
CMAIRiG
layout which
CM . A
a
is
basic
furniture
DEgK.
PU
PU
CMAiR.
J
SK AO DESI
CMAI.A
V
lu
ED.Pu
CLOSU
EDAIS
CM
CLOS
var Iant.
As a layer in our ROB there
is a set or
that keep track of this NETWORK control.
corners
of
overlapping
adlustments
spaces
of
fall,
other
what
spaces,
in our representation
is
and
outines
They chec
the
ffahe
where
containment
the
or
necessary
as shown before
4.3. Query Language:
Through this module, I shoLld say, pretentiously
called, QUERY LANGUAGE, simple aueries can be
constructed
47f A0,9
7%Ar
4 Q/Af
-7-- ~eAf~e-o1/6~e4s~
175
60w~evgy
out
of
combinations
of
MEMBERSHIP, INTERSECTION
new
information
in
basic
set
and UNTON,
to
the R01. Together
operations such ast
retrieve
or
with VALNAM in
form
our
176
previous routines,
relational
they constitute a reduced version of
we
which
calculus,
similarly to our previous rules,
the SARCASM
I
(Is
and
we
exDress,
following queries
tbe
in
enti
relationname entityl
of
2
y2)
the
for
look
under
Pair
relation
the
relation
the
logical
same
members
connective
This query would be answered
before.
used
have
)
45=,Q47
AA'AA'C7-46M~C
('/
"relationname", and OR is
that
to
use
syntax:
('CII
where
can
a
fourth element
TRUE after we positioned the
in our
layout
1 or branch 28.
(W4P4DAA
(6/uAe A4,op|awr.-7o,W
example,
A different
e lements
elements by
of
the
the
to
the watl2
two,
adjacent to the
Will
to
left
of
left
and
get
find the
list of
the value
the value of
the bed,
will
produce a
of
will
awO
set
of
al
all
tte
Inter sec tion
elements each of them
the bed and by
the wall2.
return the elements that satisfy any of
the
177
-ae'
~bp' i 1 ~ z))
two relations,
or
*adjacency'
'by",
performing
a
set
uni on.
o~e4ne~'M
The routines ttat
do this
work are:
4.4. Search:
Search
is a
recursive,
backtracking
proceduce
178
which
carries
enumeration process defined before.
on the
each
to
Its parts correspond
of
explained in there, coordinated
This
possibilities
we
procedure.
the
solution space
our
operations
by a general
exolores
procedure
in
9
the
of
tree
by
applying
the
following principles:
valid
starting at the root,
alternative
among
the binary-counter or
-
If
one
level
or
have
of
furniture
after advancing one
level,
and
doesn*t
find
a solutior
successors at the next
-when there
and
solution,
starts to
-
demanded,
If
of
it
look for
we do,
piece.
it
checks
backtracks
It
it starts again,
level
looking
if
we
to
the
If
It
for the
down.
are no valid successors to extend
backtracks
other nodes In
when we have
it succeeds
It
TRUE-FALSE
tries to find a next successor.
level
possible
alternative,
assignment
a
previous
In
of elements.
acceptable
an
a solution or not,
one
down in the tree, and applies the
the positioning
-
for
the successor nodes, whether
respective operator,
values,
looks first
the permutatior
It finds
then one
advances
it
In
a
solution,
to
a
a previous level
different branches.
and
its search and ends
only
one
is
the process;
179
when
are
all
successors,
asked
it
continues
lookin
for
valid
advancing, backtracking and recording all
other solutions until
there are no more branches
left
explore.
Its general
parts are
and the operations of
thent
enumeration correspond
tte
to
18o
Chapter
(1)
Four,
Martin
Notes:
J.,
"Computer
Data-ease
Prentice-Hall,
(2)
Organization",
Englewood-Cliffs N.J.
1975.
Teague L.,
"The
a Computer
Representation
System
of Spatial
Relationships
for Building Design".
Ph.D. Thesis,
Civil
Engineering, M.I.T. 1968.
in
181
(3)
Tutte W.T.,
Rectangles",
"Squared
Proceedings IBM Scientific
on
Problems,
Combinatorial
Computing
White Plains,
N.Y.,
SymposIum
IBM Data
Processing Division, pp.3-9
Eastman C.,
(4)
"Representations
Communications
April
(5)
for Space Planning",
of the A.C.M.,
Vol.
13,
No.
4,
1970.
Yessios C.,
and
Structures
"Syntactic
Proced ures
f or
Computable Site Planning",
Ph.D.
Thesis,
of
School
Urban
an d
PubIic
Affairs, Carnegie-Mellon University, Pittsburgh.
5.-
CONCLUSIONS:
With this
there
is
possibe
one
arrangement,
generative method we can
or
we
can
find
configuration or
fin d
out
all
the
out
if
furniture
oossible
configurations for a given spa ce.
We can show then wha t
layout
consequences
are
182
functional
a
in
implicit
in
ELEMENTS
or
the
configurations
If
that
can
a
it
with
the
S.A.R.
OF
CHART
for a rectangular site
where
dimensiors
func tion, ther %e
shou I d have to contain a
construct
new
disapoear.
the minima I
In
the
which
observe
which configurations
or
we are In teres
space
LAYOUTS,
and
CON STRAINTS,
appear
change
the POSITION RULES, or
or
SITE,
the
and
definition
our
of
then with the four parts
parameters
standard. We can start playing
we sho w
CRITICAL
in a
matrix
form, a set of rooms w ith a certain increment in the x
y
and we disp lay
dimensions,
can ho
dimensions that
(sequentially)
the first com b ination of
standard,
d ou r
layouts that
in
or
f1r s t
the
or
all
are possible.
I
The C HART no w is used
represent
to spaces, and as such it I s the first
of
to
principles
in
our
produce not
one,
precise
exist
With
supports.
the
functional
but all
the
of
mathematical
.
norms
t he
design
t he
S.A.R.
can
and
be
now
be able
to
that
arrangements
CHART.
sense in this
"f .
we
norms,
the furniture
in
ot
definitions,
This exhaustive caoability Is
the
steo
apolicatior
formultlon
in each room of
the
in mind when he assigns dlirenslors
an archi tect ha
that
to
not
Thesis, but
*oroved"
merely
'felt'
in
183
that
It might
be true.
explorations.
arrangements. If
arrangements,
we
have
f or
tool
a
for the
unelegant way, but
to
experience
exciting
v ery much
been made
from
mixing
and
needed,
were
they
analyzing spatial
grabing concepts
manner,
oersonal
for
norms, or
standards.
The generation thas
hoc*
a
design standarcs
dialogue
meanlngf ul
a
new spatial
formulating
differert
we can talk a bout economic perfor mance of
and
preferences
the
to
factors
we show the consea uences of our
If
can
c ost
we assigr
more
desirable*
less
or
of
terms
In
layouts
obtain
configuration s
"desirable*
space.
we assign preference values to position
If
we can
or relations,
Inters
for
basis
a
however
have
We
time
being
to
'ad
olaces
as
in a very
perhao
them
able
be
different
in an
been
h as
it
enume ra te
an
design
alternatives.
If
fields,
left
it
it
started
poisI-1tIes
barely
that
Interests
did not end with answers
in all.
many questions
terms
from
its
permit
S
touched.*
How
to
In
difterert
for each, b ut
design
In
enu meration
of
a
define
exhaustive
Instead
Thesis,
but
How to advance In the direction of
th Is
a nrnhlem
If I
.
not solved
in
this
184
Theory of Spatial
manipulating
Configurations
themmight
not
worth to continue explor1rg.
If .
.
be
and
our
reasoning
clear now but
In
certainly
185
APPENDIX:
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