From: AAAI-84 Proceedings. Copyright ©1984, AAAI (www.aaai.org). All rights reserved.
QUALITATIVE REASONING
WITH HIGHER-ORDER
DERIVATIYES
Johan dc Klccr and IIanicl
G. IIobrow
lntclligcnt
Systems I.aboratory
XJ’ROX
Palo Alto Rcscarch Center
3333 Coycjtc I Iill I<oild
Pal0 Al to, Cillif0l~llia 94304
Figure
Considcriiblc
;ibout
phycic;rl
systems
I~II)\+ II. 1953,) (J’~rbu~.
1984iI) (Williams,
cvcnts
which
I lOWC\Cr-,
of the
1984b).
CnOLl@
that
impossible
wc h,ivc
may
time
built
Ill-own,
intcrv,lls
PilSSCS
thC
IlilS
iimbiguity
pnwcrful
nAic+
illusive.
‘I’his
1984b)).
of
lllOdC
Of bCgovern
first
analysis
sight
0, and [z] =
it
addition
and
of
qLl~~lllitiltiVC
law
the prcssurc
alotte.
[F] =
[PI.
on many
all our
cxnmplcs
illustrntcd
FOr
cxaniplcs.
from
in Figure
pLlrJ>OSCS
a simple
Of CXJ?~illldtiO~l.
fluid-mechanical
WC
regions
prcssurc-regulator
aJ>plicablc
cxamplc.
change
for
small
time
Using that theory
for iIll dcvicc
ilnd idcntificiltion
1982) prcscnts
Scales
the
using
it is possible
causal
quantities.
of the ncgativc
a theory
to dctcrminc
cxplnnations
fccdbnck.
of causal analysis
prcssurc
rcguJ:Hor
input
close?
regulator
prcssurc
Dots
cvcnts
flow
occurring
rises indcfinitcly,
will
the valve oscillate
when
over
a longer
the direction
WC
address
the valve cvcntually
a sudden
input
scale.
the
through
F =
of the prcssurc
each charactcri;rcd
SJXlCC
iff z > 0. [z] =
to 1liIvC
0 iff
;I lilndmiIrk
is dcscribcd
cxccpt
result
Of k
by qualitative
for
the
cast
of
‘I’hc
is ambiguous.
3 constriction
kP)
to
is proportional
is rcprcscntcd
regulator
by diffcrcnt
equation
for flow ([A] =
cotnplctcly
of
change,
time
when
of intcrcst
quantity
k].)
cotnponcnts
signs
hilt
(Figure
qualiLltivcJy
as
1) has three opcmting
open
but the prcssurc
Ot’EN:[A
thcrc
[F] -= [P] no longer
([PI).
is no longer
= A,,,,],
complctcly
is applied?
88
=
01, [Q] =
0
[F] -= [I=]
=
When
0).
lhc
valve
holds as the nrca
0) no matter
when
any restriction
[P] = 0
< A < A,,,],
=
,Innlogously,
across the valve is zero ([PI
If the
CLOSED:[A
equations.
0) and flow is zero ([F]
across the vaivc
WORKING:[o
prcssurc
thilt
OllC
k drops out as it is alwavs positive.
the prcssurc
as an
for their
In this paper
+
is stl’;lightf~)rw~rrd.
closes, the qu;llitativc
1.
and Brown,
intcrcst
space of [z -
across it (i.c.,
tmnsitions
[z].
is [z] =
1934a) (Williams,
drilW
a\ilil;\blc
(dc Klccr
vnlucs whcrc
of 5 is dCllOtCd
of dcvicc
v<~lL~csconsisting
(Williams,
iff z < 0. (Notice
opposite
‘I‘hc val\c
it out
-
Arithmetic
C~UilliOllS.
bilscd 011 thcsc laws and tcstcd
progrilm
1982) of particulnr
quillitativc
(flWl1
arc landmark
valLlC
~J‘hc bchilvior
tIlc gross-time
irtforttlnlion
‘I’hc points
‘l’llc qlMlitiltitC
Regulator
;lltcrn;\ting
by points
WC use the qualitntivc
it
makes
the cast,
govern
z =
uscs
cillctlltls
SCJTKltYltCd
(t:orbus,
which
At
is not
which
rely otl qualiltrtive
iI computer
lilWS
i1ItCWillS
occur.
well understood.
of qualitative
IilWS
and
ilnd prediction
lillldillllC~ltill
Jaws.
Klccr
of
lS)SZa) (Williams.
is filirly
proven
Quiilit;ltivc
rcnsoning
1984) (dc
IIiscovcring
six fundanlcntal
of a dcvicc
in qualit‘ltivc
cxJ>l;lnation
change.
bchuvior
to formulntc
WC ~;IVC
and
IIcscription.
the inhcrcnt
idcnlificd
behavior
Klccr
time
dckicc
this gross scale time
ilppcars
(dc
made
1982) (I IilycC, 1979) (Kuipcrs.
occur over short
WhCll
h;lvi()r
has been
progress
1 : t’rcssurc
what
the valve
to fluid
flow
is
(F]
‘l‘hc other
sensor which
opening.
with
important
measures
diaphragm
causes
position
such
on
The
on
spring
(5)
(A)
[A,,,,,., -
A] =
the
(PA,I,,,I,,,,,,II,,,).
is A,,,,
[A,,,I,J] -
moves
force
Thus.
the
is proportional
Al,b~,rrr,.,r,~r thus
-
[A] =
contexts
by
(1) Start
(2) Solve
to Z.
‘I’hc
(3) Identify
of
the
intcrcstcd
prcssurcs
in developing
of thcsc quantities
dcfinc
and flows
a model
dcriv;\livc
diff‘crcntial
for ~hc V:IIVC
=-T -aPot‘,.
quantitatiic
‘I’hcsc
equations
quantitative
and
equations
rcl,lting
in any one
associated
[:c]
KIILIC
of
dF
Step 4 is Cxpandcd
Construct
StiltCS
dcrivcd
from
form
of
from
dcrivativcs
if
to
dcrivc
from the previously
and prcssurc
which
WC tried
arc ~CIlCKltCd
(4b) Gcncratc
the
pAal
([F’] :-
[PI)
transi-
g0 to (2).
all of the states
transitions
a scqucncc
bctwccn
of time.
So
of intervals,
CilCh
(Rule
to the simpler
qualitative
gcncratcd
(0)
illtC_SlYltioll
[z,,,,~]
=
the
from
rules
fOl*
[z~,,, rr.r,l] + az,,,,,,,!.
m;ltCh the
the
for
current
gcncration
VillUCS
L’millUilJ:
stdtC to
potential
6.
and
each more cxtcnsively
V~l/UC
next
cquntion
in step 4a.
311 transitions
ilnd cxcmplify
Rule
(continuous)
the plausible
~~o~l~o~~tri~di~to~~ysliltcs (I<u~c I) which
WC summari/.c
explain
0):
tllC qLLllitiltiVC
device quantity
descriptions
Cheek
information
using
and test scqucncc.
of succeeding
testing
hcrc
and
aftcrwnrds.
IlllISt
Ch:lllgC
continuously
over a transition.
qualitative
given qualitative
equation
WC
i\nd difTcrcntiatcd,
relation
WOUICI
of
rclnting
get
the
to a state which
flow
l
aF = aP
I<ulc
is inconsistent
(2)
Itlstnttt
system CilllllOt
with rcspcct to the qUillit;ltiVC
change
and IIO other
inst;rntancously
‘I’hc
uvuidntiw.
ruke.
Changes
~a11
chilngcs
from
hilppcn
transition
equations.
zero
happen
at an instnnl.
is incorrect.
By definition
aP =
of prcssurc
LIP,,, -
l
ap,,,,.
Presumably
Rule (3) Dcrivafive
prcssurc-regulator
dclivcrs
as prcssurc
to ;I load
an output
incrcascs,
i3PoTL1=
since
which
aF.
demands
to dctcrminc
rise aP,,
=
-.
sol;ltion
This
+:
the quahtativc
apt,, =
+.
indicates
response
Using
thcsc
although
the drop
across the valve
incrcascs
dccrcascs
thcrcby
the amount
the qualitative
achicvcs
completely
reducing
solution
this behavior
ncithcr
for reasoning
it is
of the output
prcssurc
how
for
which
flow
and Brown,
WC provide
(5) Hi$ercor&r
(6) Chatlge
is non-zero
WC list out
I-Iowcvcr,
such as whcthcr
See (de Klccr
Rule 0 also applies
W/P.
ChCltlgP
RlllC
to dcrivativcs.
2 ills0
ilppliCS
t0
Rules
deriva/ives.
0 and 2 apply
to all
of dcrivativcs.
a liulc
rises,
the prcssurc-rcgula?tor
behavior
hcrc
rise.
Rule
prcssurc
orders
the arca available
addrcsscs
of causal explanation;
about
and
nor its gross time
closes, opens or oscillates.
for a discussion
the output
itlSlOtl1
dcrivntivcs.
+,aP=+,aF=+,aA=
apout =
that
to an input
(4) Dcrivnrive
more
l
possible
cotr/ittui/y.
the
0 RUlC
flow
thcsc
rcciirsi\cly
idcntifics
possibic
description
successors using rules 2 through
aP.
the
illlilly7Cd,
;llgorithm
into the followings gcncratc
0 Rule (1) C’orrlrtrtficliort
that
YCt
is a simply
threshold
the
0
Notice
rhrcsholds.
next states from
state ;Ind iIll
iI pilrti;ll
each significiint’
aP -+ aA
equation
aF = aA{-
to lhcir
SOITICstate.
(4il)
> 0.
this rcduccs
in each quantity.
arc moving
Ciln be in Cach state for an interval
of thu dcvicc
with
(4~)
$$,P
A and P arc always positive.
=
‘I’hc
varinblcs.
IlOt
L~C initial
2:
the qualitative
arc aF
both
StiltC
simulation
Ibr
thlls
-dt = p+
AS
arc
thcsc
is P’ = AD,
equation
rcguIi)tor,
sensor
ChilllgCS
which
set of the possible
IlCXt
device
sIi)tcs
equations
loop
this WC need to
for the qunlitativc
mode of the pressure
a
~~-OITI
‘Ik
the time-lint
az.
In the WORKING
aA
t:or
equations
basic simulation
StiltC.
those quantities
nondctcl-ministic
them.
WC arc
Just ;IS WC write
of ;I quc\ntify.
[$]
regulator.
‘I’hc
may
over time as follo:vs:
some initial
(5) I:or C,lCll
qualitative
how a change
in the others.
vnluc of z we write
or ilbbKViillCd.
and
to predict
cilllscs changes
a qu,rlit;htivc
the qualitative
of the ~I‘CWIIT
valve
0) or OPEN
tions.
This
bctwccn
if the prcssurc
passes the
A ,,,,,,), the qualitative
change.
for the qualitative
(4) Construct
-+-
rc,lchi~blc
\ ~IUCS
with
‘the
qualitatively
the behavior
the behavior
time
causes A to reach threshold
-
and
hcncc
Conversely,
enough
+ causes A to reach threshold
such as this equation
the relationships
(aA =
CLOSE
to drop.
If
(aA =
distance
USC
incrcascs.
change
analyxcs
“[ j-1” instead.)
‘l‘hc abobc models dctcrminc
for flow continues
arca
(P).
[-+I -- [A]. (WC usually
Filluc, but in ambiguous
to a
cxcrtcd
the
complctcly
to a force
to the prcssurc
A irI,s,rrrl Ir,l
drops,
So incrcascd
diaphragm
bahnccs
(ks)
prcssurc
the spring.
is proportional
valve
for flow
as a qualitative
force
the diaphragm
by the
arcn available
USC
down
arca.
comprcsscs
arca obstructed
the output
F,, and pushing
valve
arca available
is the
to set the size of the valve
decreased
that
the prcssurc
the spring
WC
of the prcssurc-regulator
pressure
The sensor acts by converting
pressure
[P] =
component
the output
lo all zero &rivn/ives
at some instant
rules
cxamplcs
of diffcrcnt
RUI,P:(O)
: \‘AI,UE:
cannot
3 through
i&yitttpossiblc.
cvcr bccomc
5 scparatcly
from
A quantity
identically
zero.
0 and 2 to give
lcvcls of analysis.
it
1984)
a framework
WC define
its gross time behavior.
continuous
CONIINUI’I’Y
continuity
for
qualitative
if the value goes from
or from a point
an interval
to one of its two neighboring
variables.
A
to its bounding
intervals.
change
is
point(s),
In the quantity
SI M ULA’I’ION
As the input
pressure
rises, the output
continues
to rise and the
IA significant
variable.
87
quantity
dciincs
component
operating
or is an indcpcndcnt
state
space used hcrc
(in cithcr
the continuous
direction),
The
but
continuity
rule
in any transition
changes
not bctwcen
description
equations
in step 4n. ‘l’his often avoids having
transition
occurs first ((Williams,
of
the values
and the partial
poLcntial
x=0
transitions
to dccidc which
possible
1984a) LISCStransition-ordering
in his
Figure
WC Ciltl
though
KilCll
/cro.
cquntion
LISC
=
‘HIC
0.
into
-[O]
$- [+]
valve
tllc
flow
can’t
is dccrcnsing
tllc11 [A] =
is passive
equation
0;
then
[F]
== [P,,,,t]
[A] =
-[PC,“,]
CIOSC,
i.e.,
‘I’hc behavior
ncvcr
it will
by the VLIIVC
SO
[PC,,,/] =
$- [-I-]
regulator
by [F] =
WC get
way.
and s1nallcr
arca npproaches
Every
dccrcments
zero asynlptotici1lly
if one sees that
the closed
substitutes
An altcrnativc
to argue
incrcmcnt
(i.c., bccomcs
arbitrarily
nrgumcnt
state is inconsistent.
algorithm.
provides
transitions
a computation~llly
to inconsistent
tnorc
states.
envisioning
algorithm
idcntifics
step 4 only
considers
transitions
close to
to gcncrating
idcntificd,
orphaned
all states is that when
one can easily
notice
12
[F,,,,,,,]-
the spring
on
force
iFI rtcfton].
algo-
to the
loop,
states.
states.
-
o++
[F.,,,,,‘,]
0
-
-
0
+
o--
=
I‘,lblc
the
12
have
aF,,,r,nq
=
-
-
1
+
+
1
+
+
++o
0
-
o+++
0
-
--0
I‘,lblc
AMI~IGUI’I’Y
power
of the remaining
the diaphragm-spring-stc1n
prcssurc
The
balances
the
a momentum
reducing
its equilibrium
Figure
causing
the prcssurc
the valve
This
2 illustrates
and shock absorber
of
it to move
below
again,
2 : State
is correct,
the
modes
operation
of
spring,
the
it should
pushes it back:
thcrcby
a
Splitting
6
1
8
+
+
+
+
-
0
-
-
of
four
qualitative
(XC ‘I’ablc
9
10
11
12
+
++o
o--
-
+
+++
+
-
o++
+
F.qwtions
6
7
8
9
10
11
12
13
123
123
123
1
1
1
123
123
+o-
+o-
+o-
-
-
-
-
-
-
+o-
+o-
--0
+
+
I)--
+
-
+
+
+
++o
-
By
Dcrivativcs
has built
position,
up
thus
a mass situated
[F,,,,,,]
rule.
prctations
which
state in which
or oscillation.
ing its lcro
on a spring
(i.e., friction).
possible
88
WC notate
aF,,,,,,
about
combinations
--
-
WC get from
.
6-1,
6-2,
in aF,,,,,,
-ax
can happen
of transitions
=
t3Fsprr,zy =
aFfrlctton
+
ambiguous.
the contradiction
equations
and 6-3.
have three
Dcrivativc
so state 6-3
In state 6-3, cvcry
Movcmcnt
of the quantities
of the derivatives.
arc somctimcs
work
since [ZC]=
which
aF,,,,,,
state 6’s derivative
alnbiguous
threshold,
no information
themsclvcs
how much
unintcndcd
mass-spring
[F,,.,],
dv =
2 gives the values
[v]. ‘I’ablc
but
the
thcsc in tcrprctations.
by the dcrivativcs
-[F7,L,155], d
For cxamplc,
arc only
Howcvcr,
by the equations:
-
one of the interprctntions
possible
1984).
bctwccn
---au =
that the dcrivntivcs
intcrprctntions
As it has overshot
ringing
aFfrlCtlorL =
avoidance
Klccr,
is govcrncd
‘t’ablc 2 illustrates
by the prcssurc
valve
arc dctcrlnincd
-[VI,
Note
by the
(dc
by tnoving
which
ve reasoning,
arc thcorctically
remaining
intcrprctations
aFsprt,rg =
force and
but by the same reasoning
producing
details:
bc.
bctwccn
If the
producing
as it closes; howcvcr,
past its equilibrium
what
the csscntial
regulator.
the spring
the force cxcrtcd
the
by cxnmining
incrcasc,
force acts against
whcrc
force
the spring
overshoots
prcssurc
gains velocity
the position
restoring
of the pressure
the output
valve slowly
it rcachcs
rules arc illustrated
fragment
incrcascs.
force on the diaphragm.
friction.
system
or qualitatively
(intcrprctations)
+
to hIas-Spnng
1
+o-
=o
au =
sets of states and
0
345
123
0
aFjr,,,,,,
-
1 : Sdut~orls
1
ad-
all legal transitions
-
[F,r,<.hL>,,] ==o
‘I’hcn
Another
-
system oscillates
time
‘I’his
solutions
0
by the spring
J’,pr,7,y -t- F,r,l.l,,,7,
possible
z =
and z > 0 to bc
singlets.
QUAI,l’I’A’I‘I\‘P:
input
at equilibrium,
ma
-kz
is mod&d
sake. dcfinc
the mass i? provided
F,,,,lhS =
In many casts of qualitati
‘l’hc
F =
for the spring
For simplicity
-8~.
345
0
of climinnting
legal dcvicc
unrcachablc
net
[Fsprrng] +
aF ,,,R,, =
vantage
I,aw F =
by Newton’s
I.aw
1).
contradiction
mctlwd
to legal dcvicc
with
absorber:
=
Hooke’s
‘I’hc resistance of the shock absorber
cqL1;1tions hiIs thirteen
reasoning.
As a precursor
all nossiblc
8~.
and i3F =
‘I’hc
shock
[Km,]
the
the envisioning
clcgant
--[VI.
--[VI
[“I-
been
System
is unncccssary
Thus,
for. iIll sorts of sophisticated
to the simulation
and
prcssurc
thcrcforc
=
as the m;~ss position
the valve
in input
WX;
in valve
‘l’his asymptotic
that
[F]
3F =
bccvmcs
0.
to the right.
it is possible
of the 1nass is dcscribcd
or qunlitativcly
a contradiction.
XTO but ncvcr rcachcs ;Icro).
avoidance
for
is closed
load
close in the following
causes sm,1llcr
tllilt
to prove
the sensor
the prcssurc
cannot
rule
If the valve
[r]
In
this
the arca availi1blc
Substituting
rithm
2 : Mass-Spring-Friction
instead).
cvcn
[0] =
equations
by stclj 4. ‘I’his climinatcs
gcncratcd
analysis
description
dctcrminc
In many cnscs lhcre arc no possible values
with both the qL1alitativc
gc1lcri1tcd
discontinuously
gives il partial
algorithm
‘l’hc qualitative
arc consistent
may change
AVOII)ANCE
of the rcmC1inilig quantities.
which
or -
states.
Step 4 of the siiniilatio1~
next stdtcs.
0 and +
and -.
is: no quantity
bctwccn
RULE:(l): C:ON’I’RhIXTlON
potential
arc bctwcen
+
quantity
to the
is approach-
for all quantities.
first, or happen
intcr-
rcfcrs
As we have
together,
need to be considered.
all
As there
arc 4 possible
those
transitions,
15 possibilities
thcrc
are 24 -
arc rcaliyablc
1 possible
bccausc
choices.
Only
12 of the resulting
3 of
states
This simple
often
rule eliminates
used for transition
f(y)
co-occur.
others
holds
All
ordering.
Figure
gcncratcd
sonic
of
I~;I\,c any information
about
bctwccn
rule
avoidance
the
possible
rules
(Williams,
1984a)
for a threshold,
transitions
of this spccialiLcd
by the algorithm
ass:nnc all transitions
cxamplc,
as well.
by the contradiction
3 illustrates
transitions
For
if z and y arc heading
at the threshold
applications
are covcrcd
don’t
the need for more sophisticntcd
1984b) uses the rule:
and z =
k > 0, thus if $
zero.
arc contradictory.
(Williams,
x =
in z and y
as well
states
solutions
so this happens
and [La3,]
=
first.
--,W,,ass
=
and
all
dcrivativcs.
state
WC first
arc possible.
All
with
arc
unknown,
not
bc
$- 2nd
bccausc
-.
More
that
All
intcrcstingly,
bctwccn
transitions
by rule? 2 through
the number
indicates
which
shown
6.
Each
in
this graph
numbcrcd
rule climinatcs
that
Thus
bc
0 to +.
two st;ltcs
and bc-
stntcs labclcd
rule
imposes
and
bc-
a scnsc of
to state 6-1,
For [F,,,,,..]
must
n-l
change
to state 6. aF,,,,,5,
must
obey
transitions
but
must bc +
to change
back
bc in state 6-3.
For,
from
which
is
of this rule it is possible
to
-
to +
is not possible
-/-, dy =
2, cvcn
rule
between
change
0 arc impossible.
law of qualitative
1-4
a diffcrcnt
their
whcrc
unless
both
dcrivntivcs
arc
&c =
to prove
0, ay =
As a conscqucncc
transitions
This contradicts
cast (b) of
process theory
analysis
it is possible
if
a situation
states 5 and 6-2 arc impossible.
JIy rules
minimum
if the
may
intcrprctntions
continuously
this
(i.e., system
bctwccn
and thus WC product
can be
arc is impossible,
dcrivativcs
arc ambiguous.
quantities
the equality
rule, thcrc arc still a large
from
system
even
: l)l~~l~I\‘/\‘I’I\‘C:
INS’I’AN’J CJIANGl’ RUM:
-/- and 32 =
of impossible
1982).
a continuous
State 5 can transition
As a conscquencc
oscillation
unknown.
System
within
to 5, LiF,,,,,.5,5
must change
dcrivativcs
RUIX(4)
number
(Forbus,
the
cannot
aF,,,,,
bc -
in
bctwccn
f-or state 5 to transition
out by the rule.
prove
both
on the state diagram.
to transition
of their
climinntcd
state
vary continuously.
transitions
so that [F,,,(L,5b] can change
avoidance
first.
to cast (a) of the
has conscqucnccs
Although
must still
All
arc impossible
ruled
rule
rule has consequcnccs
to zero, aF,,,, ,, must
6-3
continuous
computed.
the quantities
not vice versa.
contradiction
occurs
process theory
This
inputs.
intcrprctations.
twccn
applying
-
is cquivalcnt
-
CONTINUITY
must
dcrivativcsf
well-behaved
direction
After
Thus,
so [TJ] =
not 5. ‘l’his
D~RIVA’I’IVE
derivatives
n-3
of the Spring-Mass
time
+
rule of qualitative
This
3 : State Transitions
short
0, &J =
equality
change
for z to drop for
than 0 in an arbitrarily
In thd case of state 3, [v] =
to 4 but
twccn
Figure
it will take some time
grcatcr
3 can transition
RULE(3):
using just rules 0 and 1. As we
first order
period,
> --00,
y bccomcs
as many
rule.
the 2nd order
J-lowcvcr.
than
(I’orbus,
1982),
hc dots.
that
oscillation
rcquircs
a
of 8 states.
it.
I NS’L’ANTS
RUIX(2):
INS’J’AN’I’ CJIANGK RULE:
In state 3, the mass is not moving,
pulling
to the left.
right.
Envisioning
transition
stnrtcd
‘I’hu~, the mass has moved
predicts
the wall.
to the left in rcsponsc
on the mass.
l’hus
to its equilibrium
position
balances
To transition
have
friction.
moved
close
whcrc
the weakened
from
to the
successors.
pulling
to the Icft,
the mass must
to its equilibrium
is
‘J’hc
In stntc 4, the mass has
to the spring
In state 5 the mass has a velocity
force
as fihr as possible
states 4 and 5 as possible
from state 3 to state 5 is impossible.
moving
no net
Any
but the force of the spring
have
spring
it towards
but
position
before
formally,
any change
some time)
[z] =
changes.
states) and intervals.
dcrivativc,
11~ modeling
satisfying
single instant
have to
it
from
instant
of a quantity
+,ax
=
-
in any quantity
to zero.
and
[y] =
Consider
O,r3y =
from
back
+).
and its dcrivativc
Marc
think
what
non-
gcncrally,
if arty
As a conscquencc
the
to instants (corresponding
to momentary
than one zero quantity
has a non-zero
of them changing
happens
one at a time or all
as a scrics of instants
scnsc of causality:
to following
ay can change
any change
If more
WC get consistent
As a conscqucncc
is constant
0, 3.~ =
the state is momentary.
WC can cithcr
intuitively
at the same instant
a quantity
(e.g.. [x] =
for time is expanded
at once.
began to move.
More
ontology
is
force pcrfcctly
3 to 5 the mass would
zero quantity
close
there
moved
state in which
zero is momcntnry
by grouping
transitions
with
we get an
thcsc instants
simultaneous
in a
changes
time interval.
of rules 2 and 4, if [z] changes
to 0, so any tcndcncics
from 0, no other
to change
will
persist,
zero happens
two quantities
(at
+.
+,
As [x] =
*Some transitions corresponding to operating region shifts (not nmbiguitics) need to
bc handled with some cart. 1:or cxamplc. a piccc-wise linear model has undcfincd
dcrlvativcs at the joints.
and the ultimate
effect
remains
note that if some non-zero
the quantity
transition
nor
its dcrivativc
is considered
Unfortunately,
intcrvnls
using
ball.
the moment
At
a%ment
moved,
the
has zero
0, a~
L3z =
0. ‘I’llC
velocity.
result can bc proven
may
not
with
order n is ncccssary.
11o
from
about
instants
solving
is a second
to
[%urrentl
+
‘I’hcorcm).
solution
ax,,ezt
(which
first,
it rcquircs
next;
second.
state
The
is any
dificult
or not.
Hc
after.
intervals
and
cast is handled
as possible
difficult
by considering
Pz
Mean
the second
the current
case occurs
all states that
0
3
a2Ffr*ctron
=a'LFsprrny
= -
0+++
- -
-
-
a"F,,*,,
=
-
0
-
-
qualitative
order qualitative
makes
higher-order
aox =
This
qualitative
dcrivativc.
arc not dcfincd
change
dcrivativc
Ps
=
for
ax =
iff a%
must
[$$I.
bc defined
For brevity
linear
a state
DifTcrcntiating
a linear
of the equations
computing
equation
dots
higher-order
products
0. This
This
altnczt
Recall
v(t)
not change.
As thcrc
to
the rules can bc summnri/.cd
Tcro happens
if they dill’cr
and
is
rule
and
all
orders.
first.
in any
0.
known
whcrc n+l
Pz.
A state is
is the qualitative
‘I’hcrcforc
a(&))
The
in terms
of the
derivatives
in operating
QUA LI’l’h’l’IVl+:
is easy.
so the
arc finitely
many
climinntcs
tlic
all Pz
expansion.
3) cannot
whcrc
transitions
‘I’akc
over
is no possible
is in state
1.
1.
[v].
all the states (thcrc
expansion
way
around
‘l’hus,
for a
arc
zero.
some
time
As all the derivatives
bc zero cvcrywhcrc.
remain
from
for cxamplc
arc continuous
v as a ‘l’aylor
all
arc zero in state
region
so there
go
it and
In state 1, v and all its dcrivativcs
WC can write
the transition
as rule
to one
of v arc
if the dcvicc
is in
in state 1 and has al~trz),s beers irl sfaie
from
6-3
to 1 is impossible
/.
as v is non-zero
in state 6 and zero in state 1.
use
equations
rule
1. because
by the Taylor
its dcrivativcs
state 1 it will always
I)I’RIVA’fIVES
is non-zero
‘I’his
to state
the dcvicc
%I’RO
to the same caveats
arc zero.
12-1
justified
ALI,
a quantity
to occur).
when
‘I‘0
(subject
zero, v must ncccssnrily
that
equation.
wc somctimcs
a linear
CiII1 transition
back.
at all dcrivativc
= 0 and a”+’ z #
whcrc
of its dcrivativcs
states 6-3
‘I’hc
systems,
state 6-1-l
transition
dcrivativcs
from
NO CHANGK
A transition
in the
=
calculus.
the valve
tililt
SINJW
of the system.
from
in terms of lower
as is done in conventional
was illustrated
of State 6
a7h,,,,,,,,t
-1 a~+~~~~,.vrrrr,l,
772first non-zero
states arc difl’crcnt
RULK(6):
by an
Ix].
For
form
dcrivativcs
no scnsc.
quantitative
derivatives
=
(2,4) Any
point
higher-order
IIcrivativcs
cannot
contradictions
Howcvcr,
orders.
dcrivativcs
ang,,,,t
discontinuity
to all dcrivativc
3 : Higher-Order
but that 6-2-l
is no change
4 apply
+
(1) Avoid
order
next states.
Rules 0 through
a"+'F tnass
=
and
PV
+o---
momentary
l’hcreforc
[z,,,,,]
-
happens
it is non-zero
satisfy
-
3
‘I’wo
are:
problem
if [z] =
-
1
=
state is momentary.
at an instant,
-PF,,rass,
~~~~~~~~~~~~
derivative.
the current
must altcrnatc.
3
In terms of higher-order
interval
what
whether
the models
p~~,,~~~,
succinctly:
at the instant,
to know
to tell
3
+
Value
formulation
=
=
for state 6.
1
stale 6-2-1,
if the
as [z,,,~]
the
diffcrcntiating
LM-*v
PF,,j,
0
‘I’hcsc second-order
and
that
in the following
next
illStilIltS
to tell whcthcr
so the only
+
Table
1984a) (Williams,
from
-
aFnLas.9=
(0,3)
Williams’
avoids
13~ rules 2 and 4, if &E is non-zero
interval
that as the spring-
instants
structure
thus
Notice
non-%cro
with
happens
possible
and vclocitics
of dcrivativcs
will be non-zero
two problems
that
can
from SCCOII~ dcrivativcs.
proven
what
requiring
it is always
bc
if thcrc
knowing
what
=
3
-
+++++
in the equations.
by (Williams,
-
1
dcriva-
orders
for
=
This
easy to tell
2
a&mng =
a2v =
WC arc guaranteed
it is conscqucntly
is momentary
axiom
cm
*a*,an-1x7Lczt
integration).
=
in a finite
+++++
aF/rrcttun
though
is [z,,,,r(]
is linear,
form:
the solutions
1
of z(t).
forces
rule
3 summarizes
au =
move out by itself.
integration
If and only
cvcn
easy to represent
system
+ a7'+1J?~,,,,,
but
dcrivativc.
mcntioncd
three
system
suggested
the
xnezt, %d,
(by
order
‘l’ablc
a
have
order of the system which
is to bc gained
mass-spring
anflFj,tctm
Qualitatively,
-
it is often
for second dcrivativcs.
rcwritc
drops
it cannot
fill’ illStililtS
rcfcrcnccd
for
be moving,
of rclcasc
expansion
mentions
is in state 1, it cannot
An alternate
it can’t
one
this rule is that higher-order
only
ball cxamplc,
thus rcquircs
1984b)is
Suppose
is the first non-zero
the variables
the
it is
derivatives.
-Pv, a"+'Ffrrchon
= -ant-lv
[z,,~~~) =
can be proved
accclcration.
n is the qunli&ltivc
mass system
system
(it
integration
PZ
h
order
does not change their essential
So [z] becomes
Fortunately.
equations
information
‘I’hc dropping
-.
applying
directly
‘I’hc spring-mass
ncgativc
=
using the Taylor
bc known.
be dctcrmincd
instants
to t.hCSC equations,
all higher
and the
equation
‘I’hcorcm).
qlLllitiitiVC
correct
‘1%~ difliculty
in the instant(s),
is rclcascd,
and
a2z
sohtions
to
neither
interval.
it is. At the moment
0. and
=
for
the ball
dcrivativc,
integration
Value
whcrc
anhL,rent
lx current] +
tives
qualitntivc
Mean
thcrcaftcr
[z] =
can change
is invalid
+
In fact, it is intcrcsting
has a non-zero
in the following
the
[hLrmLt]
immcdintcly
the same.
quantity
form
quantitative
evkt&n(wt),
illustrates
applied.
90
vs. QUAN’l’I’I‘A’I’IVE
solution
the qualitative
Qualitative
to
i.c.. a damped
shtc
reasoning
the
spring-mass
system
is of
sine
wave
(Figure
diagram
after
all the rules have
obtains
a qualitative
43).
Figure
description
the
4b
been
of the
=
behavior
of the mass-spring
system
without
recourse
Figure
to quantitative
This
methods.
4b dots
is technically
Friction)
-
However,
halt.
perhaps
1984a)
dcfinc
the
Although
of
force
cxamplc,
rapidly
a spring
ncvcr
stop
moving
Behavior
that
13clow we give an English
4.
system
starts in any other
( “*”
If the system
indicate
description
state it cycles
through
thcrc,
oscillation
states 2 through
the system cannot
(2) ‘I’hc mass is to the right
whole
system
of equilibrium
is stationary
(5)
Near
spring
right
(7) Mass rcachcs
spring
equilibrium
has become
(8) Spring
begins
(9) Spring
to compress,
is completely
(10) System
begins
weak,
equaling
but momentum
equilibrium
to the left,
movcmcnt
spring
force
carries
it
mass stationary.*
towards
dominates
spring
(13) Mass reaches equilibrium,
has bccomc
weak,
cqualling
the fundamental
but momentum
stants and moving
laws capture
what
to zeros.
would
laws of time-like
contradiction
‘I’hcsc
othcrwisc
carries
behavior:
avoidance,
simple,
rcquirc
with
Mittal
but gcncral
moving
Such
approach
(true
0
the
for non-
to dctcrmine
a more
John
Sccly
commcntcd
sophisticated
Brown,
early
J.S.
Scvettth
Itr/entniiorml
Age. cditcd
“Getting
from Structure,
qualitain-
Procccditlgs
Brian
drafts.
1984b.
infcrcncc
tcchniqucs.
91
” Tufts
Proccsscs,”
ott Arltjicial
in Experf
S’yslnt~s in
Edinburgh
University
Proccedittgs
Itttclligettcc,
pp. 209-212,
Reasoning About Causality:
University
Analysis
Cortfcrettce
Working
of MOS
of /he
1982a.
Deriving
Papers in Cognitive
Circuits,”
in a Qunlitativc
on
pp.
Intclligcncc
Physical
Right,”
“A
Envisioning,”
Intelligence,
Joirrt Cottfcrcttce
Manifesto,”
1984a.
USC of Continuity
Narionai
Brown,
Artificial
1982.
about
by 11. Michic,
the Envisionmcnt
18, May 1982b.
IIC., “Qualitative
of the
on Arl$cial
Process ‘I’hcory,”
of I/W
in Ar~ifcinl
Itttclligcttcc,
[9] Williams,
B.C., “‘l’hc
and powerful
sophisticated
“Qualitative
Noriotral Cot!fcrctrcr
on Ar!$cinl
[7] Kuipcrs, I)., “Commonscnsc
it past to
off
Cotlference
I’rocmiings
Scicncc No.
[8] Williams,
integration/continuity,
discussion
AIM-664.
Cnmbridgc:
M.I.T.,
K.I>., “Qunlitativc
Reasoning
Press, 1979.
[6] Kuipcrs,
IL
force.
PRORI,l’MS
WC prcscntcd
rcquircs
I,aboratory,
[4] Forbus,
Behavior
tivc
landmark
information
2
might
if WC knew
some fixed
Sanjay
Nnrional
/he Alicroelccrrotzic:
the right.*
OWN
state
time.
would
Of course,
bc enough
Irtrclligrt~ce. pp. 326-330.
1981.
[5] fklaycs, P.J., “‘l’hc Naive Physics
equilibrium.
friction.*
(12) 1;riction
fruitful
Forbus.
of de
434437. 1982.
[3] Forbus, K.I).,
system dccclcrates.
comprcsscd
rightward
the velocity
is
restoring
but the system
Qualitative
Physics Based on Contlucnccs.”
to appear.
“Foundations
of
[2] dc Klccr.
J. and J.S. Brown.
force.
position,
that
with
I,aw,
there
ambiguity
whose
asymptotically
this
end,
‘I‘hc
deduce
[I] dc Klccr. J.. “How Circuits Work,” to appear.
1984) dc Klccr,
J. and
(dc Klccr
and Brown,
equilibrium.
past.*
(11) Near
and Ken
Proceedings
dominntcs
end.
end of
friction.*
(6) l:riction
-
4b must
and dccclcrating.
at the cxtrcmc
force
is mandatory
an axiom
Icave.
pulls back the the mass towards
equilibrium
than
would
of
physics.
WC had many
motion.*
(4) The spring
was greater
thcrc
by assuming
a spring
no oscillation).
sort
(Williams,
rcachcd.
us to
Hookc’s
(i.c..
and
ACI(NOWLEI)GME:N’I’S
13
Williams
state which
to zero
or some
1982)
will
design
correctly
quiescence
in
and if the
instants).
(1) A quicsccnt
(3) The
of the states indicated
starts in State 1 it remains
allow
not
to the right
springs)
qualitative
Figure
dots
producing
constant
pathological
and Quantitative
interval
out
friction,
states of Figure
particular
WC could
damped
equations
towards
(Forbus,
of cxistcncc
still obeys the qualitative
asymptotically
spring
any
and
approaches
tells us that the oscilla-
arc cvcntually
momentary
analysis
For
out
thresholds
thil t
qualitative
of Coulomb
Law,
amplitude
that a transition
mechanics?
this problem
no guarantee
ends.
physics
to quiescence.
Hooke’s
kind of qualitative
a model
that all approached
Law,
common-sense
What
the common-sense
“quantum”
transitions
Newton’s
decay in oscillation
tion must cvcntually
is possible:
possible
(using
the exponential
qualitative
4 : Qualitative
include
zero asymptotically.
model
Figure
not
correct
Arlifcial
to appear
Physics,”
In~clligettce,