From: AAAI-84 Proceedings. Copyright ©1984, AAAI (www.aaai.org). All rights reserved.
The Use of Continuity in a
Qualitative Physics
Brian C. Williams
hrtificinl
Inlclli~encc
1.aboratc,ry
Massachusetts
Instltutc
of ‘I’cchnology
‘I’hc ttbility
is cstcntial
to reason about
in analyzing
role of continuity
for analyzing
behavior.
ill qualitati\c
the bch;lvior
steps ncccssary
physics
MOS
21 qualit:itivc
‘l’hc discussion
and the rcl,jric)nship
then
how
the
in a system
analog
of the rcnsoning
using
‘I’cmpornl
b-
focuses in on the ilsc
quantities
physical
over time
discusses
that exhibit
ovcrvicw
simulation
bctwccn
in describing
ckcnts
paper
circuits
begins with a hricf
A~wlysis.
dcrivativcs
This
and its application
to pcrfi)rni
(‘I‘Q)
of confilllli[y
order
systems.
of IIigital
‘I’hc discusGon
Qu;llitati~c
a scrics of complex
physical
;tnd their
quantities
Gnd
higher
ch,mgc
Figure
OVCI
1 : I<C Circuit
time.
Assume
that
IN’I’IZOI)lK’I’ION
bc positi\,c.
‘I’hc
to
;Ibility
csscntial
to
fCiISOI1
and troiiblc-shooting
physics
is to
the
f2w ycnr’s
last
cool\ ing
:~nd
Klccr
interactions.
3lXNlt
this
21 fri~mcwork
for
in
mcch;u~isms
1’84)
atId
to n qtlalitiltiVC
of the KXOliillg
dlYlWillg
type
hoth
algcbr;l
begins
large sign:11 behavior
of MOS
on the USC of continuity
their dcriviltivcs
initially
il
into
physical
One
rclcvnnt
type
overview
for cxamplc,
cirrrcnt
through
discussion
then
‘I‘hc second
the behavior
quantities
and
of physical
quantities
OVCI
time alld
behavior
vicwcd
of a circuit
;IS
in rcsponsc
a set of intcrvi\ls
operating
regions.
Analysis.
is best illustriitcd
pnrallcl
RC circuit
Annlysis
to nn input
in which
‘I’hc qualitarivc
which
deccribcs
reasoning
causnl
over time,
dcviccs
by :I simple
exhibits
the
move
whcrc
process, mod&d
cxamplc.
the following
Figure
‘1’0 provide
qualitative
through
is Cnusill
for the circuit
time is
circuit
diffcrcnt
by ‘I’Q
I shows
350
produces
n positive
corresponding
to
Prop,\gation.
term cffccts
of
“V, R’dccl-mxs for an interval
of
‘I’his type of reasoning
iI mcchnnisln
and
is rcprcscntcd
its resulting
for ani~lyzing
behavior
as a network
is dcscribcd
circuits,
is nccdcd.
of dcviccs.
by a dcvicc model
lSmce VJ,~ is a decaying cuponcn~ial. it is positive
02ro at 00.
behavior:
instantnncous
mark the cvcnt;
is modclcd
Analysis.
each type of &vicc
~--
a
0:ro.“
hcing
arc required
the
the !ong
dctcrmincs
for cx,rmplc,
rc;lchcs
the circuit
the description
C(ILL~~~S
which
across the resistor,
in ‘I’Q An;\tysis
inputs.
CVClltUillly
by ‘I’ri>nsitioll
Qll;~litiltivc
dctcrmining
. . .*’ ‘I’hc mcchnnism
type of rcnsoning
thcsc qualitative
break
types of reasoning
in\,olvcs
volt:tgc
which
focuses
bctwccrl
I-CilCllCS
interval.
to ;I set of Ilrijllnr)j
the resistor
awl
stops flowing
to mu,
‘I‘wo
rcnsoning
“,A positive
(V,N).
dCCrCilSC
CVClltiially
by iI scrics of cvcnts such as V,,
CilCll
during
the
time.
‘l’cmporal
of
illld
to
the resistor,
state at xcro volts.
intcrvAs.
circuit
response of lhc circuit
simulation
for ;iii;ily7iiig
tilC
the CilpacitOr
Lhrough
Of lime illId
Lhc currciit
is nlarkcd
scrics of time
current
LilC CilpacitOr
intcrv;ll
or V1~’ moving
positive
to illlaly/C!
iii rcasoiiing
:I brief
steady
this type of rcnsorling
‘I’hc
circuits.
with
;I
‘I’his description
(dc
(I:orbus.
Of CillCtll~lS
a quAitativc
and the rclntionship
in describing
ontologics
;I system
(‘1-Q) An:llysis,
has ~CLW
ccntcrcd
f0r iIll
’ 121 this poinl
has rc;lchctl
iICfOSS
voltage
iI positive
to CiiSCllill’gC
hegins
the
‘I-his C~NISCSthe vottagc i\crOSs the resistor
producing
tlCCKXSCS
zero.
Over
for cxprcssing
steps ncccssclry in performing
(Ising ‘I’cmpor;11 Qu;llitativc
physics
thcorcms
which
V/h
of it qualitative
the role of continuity
‘I’hc discussion
lCVC1 is
analyzing
rcnsoning.
dcvicc
~IWCSS ccntcrcd
from il fCW simple
physics.
of
qu;\lit:ltivc
for
the uCC of ;I qLl,llitiltiVC
‘I‘his pnpcr cxnmincs
Cll2llgC.
qiinlitativc
modcling.
One objcclivc
for
Brown,
1983). thro~lgh
systems.
a theory
includes
nt tllC
behavior
tasks as designing.
physical
provide
uhich
nbout
such
pcrfrmn
tl
at instant
(Vl,v) is positive.
n rcprcscrltntion
Quantitatively,
‘I’hc
and
for t <
function;llity
the interactions
CO and rcnchcs
a
of
between
model
devices
con$ts
associated
are described
with
the dcvicc’s
and their dcrivativcs).
behavior
by a set of network
of a set of algebraic
relations
terminals
‘I’hc relevant
laws.
bctwccn
(c.g.,
current,
cquntions
C2USAL
A device
YIiOP,~CA'I'ION
state variables
voltage,
constraining
Causal
charge
Propagation
set of qualitative
the circuit’s
forward,
in the above cx~~mplc are:
Ic,
=
I,<R
Resistor
=
Cw
Capacitor
Model
--I<”
KirchotYs
Current
I12 =
Model
(rcfcrrcd
to as prir~ory
cffcct on other circuit
a qu,\litativc
stn,lll
that (V,,v)
I .aw
inputs
signal analysis.
is positive
t’ropilg;\tion
C,~usal
at instant
products
laws, to dctcrminc
quantities.
cxplari;~tion.
result
as
it is given
VI,\ i\s the primary
Using
the following
a
their
‘l‘his may be vicwcd
7 In the IiC
tl.
when
interval
couscs) arc propagated
using the dcvicc models and network
instnntancous
V[,
occurs at the start of a time
(whcrc
cil1IsC,
“A
--) B”
l*CildS“A Cll1ISCS
ICI”):
‘I‘hc behavior
and
of the overall
dcvicc
models
and
circuit
is infcrrcd
is cxprcsscd
of V,N in the l<C circuit
behavior
into
intervals.
separated
rcprcscntcd
Stiltc
tion
signs.
ncgativc
tivc
((-)
consisting
=
-),
is unknown
‘I’IIc qualitiltivc
laws
[VIN] =
‘I’he
(X)
state
cquiviilcnt
and
subtraction
=
two
?)(dc
‘I‘l~.\NSI’I’ION
Klccr,
using
Resistor
circuit,
numbers
order
Iiighcr
order
Model
[I,,]
=
--[I,,]
KirchoFs
Current
detail
Llsk.
of behavior
which
For the analysis
adcqLIi\tc
LO exinninc
to rccogni/c
saturation
1979),(Forbus,
and I:IWS;IW:
dcriviltivcs,
ovcrnll
the circuit’s
state variables
quantity
interest
“the
of
voltage
interval”).
marked
WC only
number
on the lcvcl
in the particular
first and second
maximums
‘I’hc
dcpcnds
MOS
by a SC~LICIICC
behavior,
of
and inllcction
keep track
within
or “the
qunntitics
not
predicts
the
dcscribc
how
dctcrmincs
Current
I .ilw
WhctllCr
Analysis
making
points
(c.g.,
the
to the
region
charge
is moving
ON).
quantities
/rmsi~iotr into
will
interval.
Ordering
;I
transitions
tions bctwccn
dcvicc
on
from
qualitative
cnp32itor
to /cro
or
towards
which
regions)
ON
often
dctcr-
ano~hcr
region
only
first, marking
discusses
transi-
hcrc is easily cxtcndcd
other
aud
or
subset of these
or boundary
this article
dcscribcd
bctwccn
Recognition
is mobing
whcthcr
region
is dccrc;Gng
the boundary
dctcrmincs
across boundaries
operating
tlic
new region
Although
tions across zero. the mcchilnism
and their
tl;lWitiOllS
positive
dctcrmincs
ilnother
‘l’ransition
that more than OIIC quantity
rccogni/.c
the
and may bc vicwcd
Recognition
towards
positive
zero, or a mosfct
the end of that
in the circuit
of quimtilics
is moving
‘I’rimsition
it possible
;I qllillltity
from
of
OVCI’time,
into two steps: 7‘rmrsi~iotr I<ccwgtti/iotr
is broken
mints
it
not
or
rcsponsc
chngc
quiintities
large sign;11 analysis.4
or boundilry.
fhnd
instantilnco~ls
regions (c.g., moving
of
in rcsponsc
to a set of inputs
and the qualitntivc
than
zero (c.g.,
is dcscribcd
to
transi-
in (Williams,
a single
mosfct
During
an interval
qualitative
region
is in saturation
and the beginning
transitioning
is
values of
during
‘I’IIANSI’I‘ION
each
(c.g.,
‘I’hc
the
Transition
I~I’C’OC;NIl’ION
basic
assumption
Ordering
underlying
Transition
Ifccognition
qualitative
3C’ausal l’ropagatiorl
IS Guitar to dc Klccr’s Incrcmcntal
Qunlitativc
1979) c\ccpL that the cpantltics
bclng ptq~gnted
arc not rcstrictcd
but ma) lncludc C~:IIIII~ICS
and higher order dcn\ativcs.
reasoning
systems
ilnd
is:
of the next is
bctwccn
regions.
%his differs from cxlicr
qualitative
dcrivnti\es
(de Klccr. 1979).
Model
1984).
intervals
‘I’hc end of the interval
by one or more
-+ Kirchofl”s
of
analysis
WC llilvc
circuits
for each interval.
remains
is positive”
-l<csistor
=
a qu‘mtity
OFF
and
in the RC cxnmplc.
‘I’hc circuit’s
=
[q]
I.aw
to cutofT’) at the end of a time interval,
boundary
for the lirst and
volt:tgc.
the ;in:ilysis
must bc obscrvcd
For simplicity,
first dcrivativcs
dcscribcd
and
current
in
[w]
dots
as a qualit,ttivc
I.aw
may also bc crcatcd
of pcrformitncc
minimums.
bchnvior.2
of
Model
two qu;rlitaGlc
or not
Capncitor
uscci
-- CiIl)ilcitor
and Trutlsiliotl Orderirrg. ‘I’mnsition
[%;-I
derivatives
=
Analysis
bctwccn
Model
=
dCriVit~iVCs
Current
[ ;&“I
Propagation
but
‘l‘rnnsition
is
and a ncga-
models
[IO]
Model
/\NAI,YSIS
Causill
a
towards
higher
-KirchoIf’s
arc
positive
multiplica-
ncgntivc
of the i\bovc
[h]
set of cquntions
~tnalo~ous
=
of open
‘I’ransition
An
-1 Resistor
[Icy,]
[Xl.)
the sum of a positive
((-+) + (-)
[VW] =
=
can
variables
;I set of relations
of
4 Given
[II<,]
scparatcd
bctwccn
is denoted
into
the sum
while
of intcrcst
as ;I scqucncc
the circuit’s
of addition,
cxan~plc,
+ (-)
number
1983).
and
sign, using zero as a boundilry
For
time.
or regions
is rcprcscntcd
instnnts,
arc then combined
&Am
on
n quantity
intervals
(‘I’hc sign of a quantity
vnriablcs
qualitiltivc
‘I’imc
by
by their
and ncgntivc.
which
a set of open
by ;I set of bound;lrics.
of
for t > 0
the space of villucs
t,tkc on is broken
the network
is:
Vtp~ = Vtllrlrale-+
Qualitatively,
from
as a function
Anal~ais (dc Klccr.
to first dcrivativcs,
4Allcrnnti\e
appronchcs IO dcscribc the beh,lvior of quantities mom
qualitA\c
region
boundarxs
hake bcm proposed by (dc Klccr and Brown,
19X4). (I:olbus,
1983) and
(Kulpcrs,
lYX2a).
focused only on first
351
The behnvior of real physical syslenrs is corltitzuous.5
MOW
system
prcciscly,
which
intervals.
thcorcms
hithin
two
is the
dcscribc
In this
provide
section
how
qualilntivc
thcorcms
the interval
is both
of intcrcst,
continuous
while
which
thcsc
move
arc then
bctwccn
then
tl f
t2)
it will
used to dcrivc
that a quantity
it will
will
to move
Inordcr
is continuous
to 0 will
the second ;IssIIIIIcS thnt a quantity
or instant
illll~rvill
during
some
lo lhc next.
of
of sonic quantity
how
a quantity
If, li)r cx,rmplc.
time,
will
over time, a set
changes
from
a quantity
it bc positive,
the next time interval. 7 ‘l‘hC Illlcrttlrtiinlc
during
tllilt:
the behavior
interval
from
finite
arc scparatcd
by a continuous
interval
c to 0, traversing
interval
of time
of
by an
the interval
to move
from
of an open
(c,O).
moving
Similarly,
from
of time,
a quantity
of an open
of
where
0 to some positive
interval
Convcrscly,
function
{(tl,t2)
time
wc can Sly that a quantity
lcavc L at the beginning
open
0 to e
arriving
moving
interval
from
and arrive
E
at 0
1. If some
Thcoro,l
VrllUC
for
This
interval
some
closed
notion
will move through
of time,
and a quantity
interval
of continuity
of
time
will
(possibly
is captured
with
the
rule:
Continuily
or ncgativc
this is th;rt a qllilntity
an open
nn instant).
following
is positive
zero
during
a boundary
OII
for only
one
way of viewing
region
Rule
quilntity
Q is positive
(ncgati\c)
during
an instant,
it
StiitCS
will
positive
Klllilill
(ncgntivc)
for
sonic
open
interval
of
time
I
immcdiatcly
lf f is continuous
on the closed
number
bctwccn
f(a)
point
in [u, 01 for which
X
Inluitivcly,
this
cross a boundary
when
the positive
the posilicc
interval,
St;itc
means
and
J(X)
that
moving
lllily
it cannot
[a, b] and if 1 is any
then
=
thcrc
from
one
must
qunlitativc
will
region
instant
moving
intcrvai
cxnmplc,
transition
dlawn
the
that
arc continuous
Value
‘I‘hcorcm,
1x1~ ccii Lllc rcprcscnlations
O~WII ir1tcrwl.r.
region
Also,
recall
regions
qimitity
An
that
(Q)
for sl,rlc
and
by Causill
wcrc positive
positive
following
the
can also bc dcrivcd
be
illld their dcrivativcs.
v,lri,iblcs
can
;lnd tinic.
tl
(interval
ltulc,
12).
following
of time,
Propagation
during
WC predict
during
They
it
interval.
or ncgativc
or ncgativc
that
the open
may.
howcvcr,
12.
instant
marks
or from
;1 quantity
;I boundary
moving
from
to an open
by the
wc dcnotc
is positive
-I3 --
at son-it
tiinc
exists some finiic
and 0. rcspcctivcly.
instant
tl
open interval
(Q@tl
=
hnctmn
f
IS continuous
iT a smnll
and 11.UC can Lccp lhc chnngc
\1i1;111ch:lllgc 111f(x).
holdmg the change 111z sullic~cu~lv smsll ” (I.oomis,
chngc
tn f(z)
in 5
of quantities.
at the relationship
‘l’hc following
(f)
througliouI
(a, b), then
c whcrc
dccrcasing
the
products only a
(instant)
1977)
[a, b], and
if f’(z)
(a, O), then
f
is ncgativc
throughout
thcsc two corollaries
in terms
is an
with
of n state variable
of its vnluc
and its dcrivativc.
Qualitative
for all
is constant
on (a, b). If j’(z)
increasing
function
(a, b), then
f
is
on [a, b].
the bch,tvior
and is captured
function
on [a, t’] and diffcrcntiablc
throughout
Ry combining
ns sn~nll as WC wish by
to zero
Lhc interval.
is positive
on
to 'I'Q Annlysis:
is equal
the
quantities
of the Aleall Vdue
intcrcst
which
information
bctwccn
two corollaries
has a dcrivativc
2. I.ct f bc continuous
If some
(t, 0) separating
at the continuity
by looking
Vi~IlICS of z in an interval
region.
by the
to looking
1. If a function
an
is rcprcscntcd
ndclition
T/~cor~~~t/
(‘l‘hon~~s, 1908) ;lrc of particular
I<CCi~ll
Thcorcm,
“‘l‘hc
remain
to m-0 at the insl;uit
by using
;I relationship
the range of ‘I state varinblc
c > 0), then thcrc
SConlinult>:
imnlcdiatcly
time
posi/i\~c (0, 03) ;md /lcgcr/iw (- 00, 0) scpar,ltcd
zeru. which
boundary
WC dcduccd
state variables
must
the next
for time consists of a scrics of i~s/r~rr/.s scpnratcd
to :I bound,lry
open
the open
bc ncgativc.
quantities
Intcrmcdiatc
that the rcprcscntntion
open
interval
following
the first pilrt of the Continuity
each siatc varinblc
In
ISy assuming
of
Using
some open
the instant
to the RC cxamplc,
nljt~0~5
open
In the above
zero during
that all of the circuit’s
tl.
that instant.
Q is zero during
;uid ‘I’iiiic
\‘:iriahlcs
results
remain
Returning
cross zero when
or zero during
one
will
following
2. If sonic quilntity
1977)
quantity
regions.
bc posilivc
is at least
1. (I .ooniis,
a continuous
and ncgilti\rc
quantity
howcvcr,
interval
f(b),
‘1’11~s cnch state variilblc
to another.
bctwccn
by
points
at the crrd of the open interval.
nn
for dctcrmining
some
at E at ~hc end of the interval.
‘I’hcorcm
to dcscribc
& is described
lcavc Lcro at the brgirr,riug
remain
of rules is nccdcd
distinct
that
take a finite
Another
\‘;~luc
two
take
value c. F~urlhcrmorc,
the
and diffcrentiablc.6
‘I‘hc Intcrmcdiatc
zero (any
If WC assume
time.
and
the Cotl/irruit)’ Rule and
quantities:
It~lrgn~liot~ Kulc. ‘I’hc first rule rcqiiircs
over
over
the intuition
quantities
thcorcms
functions
of & from
open interval).
a physical
of sirnplc
of continuous
WC discuss
value
dcscribc
arc a number
‘I‘hcsc
regions.
about
which
the behavior
in dctcrmining
qunlitativc
rules
functions
Thcrc
that arc continuous.
of calculus
time
it
during
At the qualitative
by the following
Integration
the previous
Icvcl,
Value
the Intcrmcdiatc
is described
over an interval
instant
this is similar
(interval)
to intcgmtion
rule:
Rule
Trnnsikions to Zero
1. If a quantity
?lhc
noki!ion
(u, b) dcnolc~~ Ihc open inbmal
!hc c~oxcd mlcr\a!
bc~wccn u
hclwccn
a and 6,
wh~lc [a, b] denoks
md b inclusive.
352
is positive
and dccrcasing
(ncgntivc
and increasing)
over an open time interval,
then it will move towards
that
transition
interval
and possibly
xro
to zero at the end
during
of
the
in tcrval.
2. If a quantity
increasing)
is positive
but
over an open
to Lcro and will
not
time
remain
dccrcasing
interval,
positive
(ncgativc
then
(ncgativc)
and
it cannot
during
not
transition
Next
the
is increasing
(dccrcasing)
interval
and was ycro during
positive
(ncgativc)
4. If a quantity
during
is constant
7cro during
during
the previous
some open
instant,
then it will bc
during
+
some open time
instant,
then
interval
It is intcrcsting
to note
the dcrivativc
instant,
that a quantity
(Q)
[$~]@tl
interval
(IX),
during
bctwccn
two
(c.g., in a resistor
the
at soinc
cause
during
Q
dcrivativc
a change
during
In
and
its dcrivativc
in current
the
next
that
tlic
above
cast,
by a qualitative
instantaneously
number
intercstcd
of higher
also bc applied
dcrivativc.
order
(z),
analy7ing
dcrivativcs,
bctwccn
during
cxprcssion
then
If a bccomcs
positive
Rule
next
for the next open interval
at
instant
SOIIIC
for 12. Similitrly,
positive
incrcasc
in 5, making
it positive
for 12. ‘I’hus llic
Integration
propagntc
the
dcrivntivcs
As
bctwccn
cffccts
down
direction
of changes
towards
WC hnvc
seen
a quantity
or away from
zero).
each quantity
the lower
above,
order
is moving
with
If a quantity
a chain
from
Rule
higher
and
in the
still
the next
If, howcvcr,
interval.
a quantity
then
respect
Rule
to zero
is zero and increasing
the quilntity
(A) is moving
cannot
it may
or may
not
St~ppos~’ some other q\lantity
bccomc
for
/cro,
describes
(e.g.,
which
quantity
an open interval
PXO.
As a result
of all qriantitics
into
from
zero.
of
the
each quantity
rclntions
to dctcrminc
the criterion
cvcry
which
is
bctwccn
qunntitics
of consistency
quirlitativc
relation
and
is used
grows lincnrly
relation
quantity
with
(Q) is unknown,
by ‘l’ransition
associated
the
involved
with Q, along with
in that
then its
liccognition.
rclntion
In this
the directions
can
somctimcs
be
Q’s direction.
qualitntivc
relations
negation.
used
addition
and
operations
Transition
place
constraints
on the direction
A complctc
in (Williams,
1984).
in modeling
devices
multiplication.
Ordering
‘I’hus
contains
(c.g..
a few cxamplcs
zero)
involved
consists
for
a set of
toward
of each quitntity
provides
of operation.
HI,
by the end
of the interval.
Ortlcring
rules
and
of
each
of
which
transition
in the operation.
of these rules for cnch type
list of Transition
Ordering
rules is prcscntcd
Rules
where
Ic is a positive
‘I‘hus
quantities
conslnnt)
then
WC need a mechanism
will
reach
instant,
the same direction,
zero
of time.
the other
(i.c.. A =
arc cquivnlcnt
o\ cr the open
interval
simply
also holds
for
dots
WC have
m,\y transition
divided
(they
the set
(i.c.,
its direction
quniititics
a quantity
A alld C arc moving
+ k?B.
to zero before
that
transition
of intcrcst
1) they arc moving
transitions
at the same time.
check on equality.
A =
with
-I&),
since
in
to 7cro
‘I’his
may
‘I’hc above
rule
negating
a quantity
respect to lcro.
is the sum or diffcrcncc
is mot-c intcrcsting.
C =
arc moving
353
must
quantities
k,A
WC know
as a consistency
negation
not challgc
then
and 2) if one of the quantities
quantity
bc viewed
continuous
1) those which
use a
transitions
zero for some inlcrval
or set of quantities
Recognition
the
the
bccomcs
1954)
uses 1) the direction
of a non-zero
thcsc
Transition
ORI)12HING
of Transition
IMrow,
then this solution
bc dctcrmincd
‘I’hc cast whcrc
‘I‘RANSI’I’ION
ilnd
the
with
or dccrcasing
must transition
towards
ycro
Howcvcr,
cxponcntinlly
the
towards
(B) rcachcs ycro first and H causes I,$$ to
then A will not reach
determining
;irht during
reach
from
in Ihc system.
quantities
and the following
of time,
Value
all sets of possible
criteria.
to consider
cast,
If the signs of two continuous
during
0 when
to jump
thus this solution
Klccr
satisfy
worst
Ordering,
of relations
‘I’hc next section
ordcl
[I?] =
lntcrmcdii~tc
to zero, and 2) the qualitative
status (c.g.. can’t transition)
to locally
and
derivatives.
Integration
the
IJ causes
and its dcrivativc
along
‘I‘hc
(12). then it will cause
it positihc
If
of the other
(tl).
two
grows
as a set of constraints.
first
‘I’ransition
equality,
(dc
Ordering
rcspcct
used to dctcrminc
involves
in 21, making
uses the relation
may
the above
need
only
Transition
cast a qualitative
(a) of a mass (where
arc constant
which
If the derivative
a
+
the
to this is to CnumCr;ltC
for
systems.
but
can
with
t state variables.
with
number
order
high
being analyzed
(v) and accclcration
a) and that iIll three quantities
includes
Integration
the
the system
suppose
v&city
the
and
which
an incrcnsc
an
approach,
continuity.
to that
solution
large
thcsc quantities
causes a change
a system
each dcrivntivc
For cxamplc,
the position
& df -
in
direction
arc inconsistent
(e.g., Q is caused
Can transition,
direction
WC arc
(their
moving
0).
test each
for
which
violate
of quantities
can transition
in vol tagc).
If
arc not
is unknown
[A] =
(c.g.,
which
number
Instead,
causal
relations
transitions
moving
interval
is similar
(they
subsets of thcsc quantities
of sets of possible
=
open
the
and
indcpcndcn
suppose
which
number
similar
off zero ins~lntnncr)usly.
moves
rclatcd
of a quantity
for
whose status
to 1) quantities
crossing
simplest
intractable
the
leading
without
transitions
of the
(11) (i.c., [&]@tl
inStitllt
positive
transition
and thus arc discontinuous
‘I’hc
that
during
cxamplc,
P’OI
12.
quantities
two parts
it bchavcs
Q to incrcasc
;I quantity
dilfcrcnt
how
bccomcs
I-urthcrmorc,
Q is also positive
bctwccn
parts
%clu
$2
it will
positive.
rcl;\tivnship
at
If
0).
in the first
affects
the same intcrvnl.
is resting
=
then
and bccomc
while
in the last two
that quantity
0 and
that,
of 11’1~quantity
can’t
[B]) and 2) q uantitics
to -
and was
it will bc zero during
which
WC want to dctcrminc
without
‘I‘hcorcm
the interval.
the previous
2) those
set of qualitative
[A] =
time
interval.
thus
zero) and 3) those
the following
Off Zero
3. If a quantity
affects
towards
transition
Transitions
following
zero)
is unknown).
instant.
rule
towards
towards
For example,
of two other
assume
Lcro and B is constant,
If A, B and C arc positive,
C and C can bc eliminated
then A will
from
that
whcrc
transition
the list of potential
transitions.*
On the other
A, and finally,
bcforc
hand,
if B is negative,
if B is zero,
kIA).
the same time (since C =
Also, consider
C arc positive
and B is negative
If B is known
to bc constant
must also be moving
Finally,
A and/or
for multiplication
same time;
but the direction
zero and will
othcrwisc,
C will
of C is unknown.
towards
zero, then
kc)
dcrivativcs
C
Both
Causal
that,
if
and mosfct
and C won’t
transition.
cxtcndcd
make
Thus, Transition
Ordering
1) factors
the qunntitics
at the same time and 2) crcatcs
sets according
to which
transitions
other
Transition
tion
Ordering
mechanism
values.
Or&ring
bctwccn
to the one
similar
If as the result
mined
that
1) all
the same time,
bc toward
interval.
the
remaining
zero,
then
Othcrwisc,
or the
sets of possible
help
being
rcsolvc
system
transitions.
the remaining
at
how
to
capture
Marc
provided
can
try
of
state
the
techniques
that
transitions
WC have dcduccd
resistor.
Next
it must
bc dctcrmincd
to zero
at the end
transition
lntcgrntion
Rule
is moving
towards
[I,?,]
and [I(*,]
‘I’hc
using
lmknown.
pro.
is moving
WC know
towards
In addition,
that [w]
=
WC dctcrminc
[?&I.
howcvcr,
since
[gyp]
KCI,
that
their
can
In
into a system for designing
intuitive
Qualitative
or its higher
the long
bctwecn
notion
arc
to dcvclop
while
of thcsc changes.
modeled
qualitative
been
response
order dcrivativc,
a few
of continuity
have
incrcmcntal
term cffcct
quantities
able
Analysis
the
by
rules
continuous
to dctcrminc
regions.
‘I’hcsc
rules
and integration.
Howic
Ramcsh
Shrobc,
I’atil
Rich
Zippcl.
dc Klecr,
Johnn
for many
and Dan Weld
insightful
Dannicl
comments.
dc Klccr,
1
J. and
Order
Arf$cial
[3]
dc
Ijrown,
to appear
Quali tativc
I’rocecdiqs
in
Itr~clligctm, Austin,
Klccr.
J.,
“Causal
‘1X-529,
Milss‘lcllusctts,
and
Na(ioml
August,
Ilascd
on
with
Higher
C’or@rertce
011
1984.
‘I’clcological
MI’l’ Artificial
Scptcmbcr
Rcnsoning
r~f hz
‘I’cxas,
Physics
Irilclligetrce.
J. and 13obrow, I).. “Qualitative
1)crivativcs.”
Recognition,”
J.S., “A
in nrf$cral
Reasoning
Intclligcncc
in
Circuit
I .aboratory,
Camb.,
AIM-664A,
Ml’1
1979.
above.
the capacitor
model.
model,
arc also moving
[4] Forbus,
Artificial
K.13..
“Quitlitativc
Intclligcncc
Process
I .&oratory,
l’hcory,”
Cambridge,
Massachusetts.,
May
1983.
zero, since [Tel]
the resistor
and [$$L]
be
arc
bc dctcrmincd
dcscribcd
towards
from
and
cannot
derivatives
[6] Kuipcrs,
Ihc Ntrrioml
August,
II., “Getting
~‘otlJiwrrce
the Envisionmcnt
OII Arlifcid
in l’rucecditrgs
ltight,”
Ittlrlligrrm,
Pittsburgh,
of
Penn.,
1952, pp. 209-212.
which
towards
zero.
[7] I .oomis.
I,.,
(“rtic~rrlus, Addison-Wcslcy,
I<crtding.
Massachusetts,
1977.
Finally,
they
it to
systems.
zero.
is moving
from
the
that VI~J
WC know
rule for cquivalcnccs
zero and [Ic,]
the
By applying
-,
argument,
Rule,
that [9]
it is dcduccd
equivalences,
=
of each of thcsc quantities
Ordering
through
or not any quantities
and
[%l]
the
12.
and [&SF]
Integration
far that
interval
towards
allowing
physical
arc cur-
whcthcr
Using a similar
‘I’hc direction
For cxamplc,
+
of [!$‘f],
the
the ‘l’ransition
arc both
=
arc also moving
direction
dctcrmincd
using
to [V,N]
thus
across it and is discharging
of
physical
move
[2] dc Klccr.
will
in an input
variables
one’s
Conllucnccs,”
voltage
and
being
circuits.
dctcrmincs
dctcrmincs
been
HP:l’I~:l<~NCFS
has a positive
of Temporal
WC have
I thank
Bobrow,
[l]
capacitor
information,
complex
MOS
is currently
for the
each
qunntitativc
sets of possible
to the RC circuit,
about
Propagation
Analysis
assuming
occur
cxplorcd.
Returning
Analysis
at the end of the current
transitions,
remaining
Causal
functions,
is known
will
quantities
may bc cxtcrnally
potential
which
occur
oscillators
pcrformancc
components
‘I’rirnsition
By
qualitative
rules it is dctcr-
transitions
of thcsc
an ordering
remaining
rcntly
potcntii~l
the transitions
propaga-
in propagating
thcsc infcrcncc
and 2) the direction
and low pass filters,
more quantitiltivc
of a system to a change
using a constraint
used
of applying
been
RLC
thcsc
Huh
rules arc applied
have
of many
is being incorporated
high
Analysis
transitions.
Two
the Transition
and their
the behavior
Qualitative
predictions
‘I’Q Analysis
Transition
predict
such as high
prccisc
discussed:
Applying
currents
into sets which
an ordering
prcccdc
the voltage,
and
‘I‘cmporal
to incorporate
and dchugging
transition
Propagation
circuits
more
addition
Thus
and used to correctly
hOOtStl-ilpcircuits.
to Lero at the
12.
are zero at the next instant.
implemented
A.
WC know
transition
A nor B is transitioning
ncithcr
zero at the end of interval
at
A and
reach zero before
(c.g., A x B =
to zero, Ihcn
transition
the case whcrc
and A is moving
towards
B transitions
then C will transition
A and C will
then
will
potential
since
all
all of the
transition
transitions
quantities
to zero
exist,
at the
each of thcsc
arc qualitatively
same
time.
quantities
Since
will
81r inslead WChad snid that C transilloncd to /cro firxt ~hcn A would
liom
~IUT to minus
--)
This
violales
wlhouI
crossing
!hc lntcrmcdiatc
/cro (i c . [A]
Vnluc
‘lhcorcm
=
[C]
-
[II]
and. thcrcforc,
equivalent,
=
no
other
transition
to
have IO jump
(0) -
cannot
(-)
occur.
=
Iv1Williams,
Ij.C.,
“Qunlitativc
in Jrl~ficiirl /tll~Yligcvi~y?.
Analysis
of MOS
CircLlits,”
to i\ppcar