From: AAAI-83 Proceedings. Copyright ©1983, AAAI (www.aaai.org). All rights reserved.
Episodic Learning
Dennis Kibler
Bruce Porter
Informationand ComputerScienceDepartment
Universityof CaliforniaaltIrvine
Irvine,California
Abstract
- Stage 1 learninginvolvesunderstanding
&.en each availableoperatorshouldbe
applied. The concernhere is with
learningthe enablingconditionsfor
individualoperators,withoutknowledge
of the other operatorsin the solution
path to providecontext.
A system is describedwhich learns to compose
sequencesof operatorsinto episodesfor problem
solving. The system incrementally
learnswhen and
why operatorsare applied. Episodesare segmented
so that they are generalizableand reusable. The
idea of *augmenting
the instancelanguagewith
higher level conceptsis introduced. The
techniqueof perturbationis describedfor
discoveringthe essentialfeaturesfor a rule with
minimal teacherguidance. The approachis applied
to the domain of solvingsimultaneouslinear
equations.
- Stage 2 learninginvolvesunderstanding
J&Y each operatoris appliedwith
emphasison the sequencingof operators.
We refer to this as episodiclearning.
Episodicsegmentationis the groupingof
operatorsto form an episode. Episodes
are discrete,reusablecompnents for
plan generationand each simplifiesthe
problem state.
1. Introduction
With the aid of a teacher,juniorhigh school
studentscan learn to solve simultaneouslinear
equations. Operatorsthat are appliedin solving
these problemsincludemultiplyingan equationby
a constantand combininglike terms. The students
are alreadyfamiliarwith &
these operatorsare
applied. Moreover,the teacherassumesthat the
studentsunderstandbasic conceptsabout numbers,
such as a numberbeing positive,negative,or
non-zero.
The main featuresof our approachto episodic
learningare:
- segmentationof operatorsequencesinto
meaningful,reusable episodes.
- augmentationof the instancelanguageto
includehigher-orderconceptsnot
present in the traininginstanceitself.
Our system,nicknamedPETr2 incrementally
(definedin [7]) inducescorrectrules from the
traininginstancespresented. The rules are
correctin the sense that at any point in the
learningprocess:
- perturbationof a traininginstanceto
create new instances.
2. Related Work
- the knowledgeis consistentwith all
past traininginstances.
Relatedwork in stage 1 learningincludes
Neves's [lo]systemwhich learnedto solve one
equationin one unknownfrom textbooktraces. The
system learnedboth the context (preconditions)
of
an operatoras well as which operatorwas applied,
althoughthe operatorhad to be known to the
system.His generalization
languagewas simpler
than ours in that a constantcould only be
generalizedto a variable. The programLEX [8]
uses version spacesto describethe current
hypothesisspace as well as concepttrees to
direct or bias the generalizations.
As it is not
the main point of our work, we keep only the
minimal (maximallyspecific)generalization[ll]
of the examples.
- sequencesof rules (episodes)are
guaranteedto simplifythe problemstate
if they apply.
Learningrules for applyingoperatorsinvolves
two stagesof learning:
1Thi.sresearchwas supportedby the Naval Ocean
System Center under contractNO6123-81-C-1165.
2Please refer to [41 for a completedescription
of our approachincludinga PDL d;scriptionof the
learningalgorithmand multipleexamplesof its
use. PET is implementedin Prolog on Dec2020.
Availableupon request.
MACROPS [2] is an exampleof stage 2, or
episodiclearning. This system remembersrobot
plans that have been generatedso that the plan
can be reusedwithout re-generation.The plans
are stored in triangletableswhich recordthe
191
variablesin the instancelanguage.
order of applicationof operatorsin the plan and
how their pre-conditions
are satisfied. The plans
are generalizedto be applicableto other
instances(as are episodes).
4.2. Generalization
Language
FollowingMitchell [81 and Michalski[61 we
have concepttrees for integers,equationlabels,
and variables(figure4-l). Basicallywe are
using the typed variablesof Michalski[61.
While effectivein learningplans, MACROPShas
difficultyapplyingits acquiredknowledge [l].
The centralproblem is that the operatorsin a
MACROPSplan are not segmentedinto meaningful
sequences. Any sequenceof operatorscan be
extractedfrom the triangletable and reused as a
macro operator. A sequenceof lengthN defines
N(N-1)/2macros. However,few of these sequences
are useful. MACROPSoffers no assistancein
selectingthe useful sequencesfrom a plan. If
sequencesare not extractedfrom the triangle
table then the entire plan must be consideredan
episode. This resultsin a large collectionof
opaque,single-purpose,
macro operators.
Branchingwithin an episode is made impossible.
In either case, combinatorialexplosignmakes
planningwith the macros impractical.
integer
/
\
non-zero
zero
/
\
\
positive negative
0
/I\
/I
1 2 3 ..e -1 -2 ...
Figure 4-l:
b
variable
/\
x Y
a
Concepttrees
We permit generalizations
by 1) deleting
conditions,2) replacingconstantsby variables
(typed),and 3) climbingtree generalization.
Disjunctivegeneralizationis allowedby adding
additionalrules.This coversall the
generalizationrules discussedby Michalski 161
except for closed intervalgeneralization.
3. Operators for Solving Linear Fquations
The operatorsapplicableto solving
simultaneouslinear equationsare describedin the
followingtable:
4.3. Rule Language
Knowledgeis encodedin ruleswhich suggest
operatorsto apply. Rules are of the form:
rator
Semantics
ccPnbinex(Eq)Combinex-termsin Eq.
combiney(Fq) Combiney-terms in Eq.
combinec(Eq) Combineconstantterms
in Eq.
deletezero
Delete term with 0 coeff.
or 0 constantfrom Eq.
ReplaceE42 by the result
=-wEql,Eq2)
of subtractingEql from IQ2
ReplaceEQ2 by the result
addU%&W)
of adding Eql and Eq2
ReplaceIQ by the result
mult@WJ)
of multiplyingQ by N
4. Description
3
<score>-- <bag of terms expressedin
generalization
language> => <operator>
(The score of a rule is describedin section6.2.)
PET does not learn "negative"rules to prune the
search tree, as in [3].
5. Perturbation
Perturbationis a techniquefor stage 1
learningwhich enablesa learningsystem to
discoverthe essentialfeaturesof a rule with
minimal teacherinvolvement. A perturbationof a
traininginstanceis createdby:
Languages
- deletinga featureof the instanceto
determinewhether its presenceis
essential.
4.1. InstanceLanguage
The instancelanguageservesas "internalform"
for traininginstances. We adopt a relational
descriptionof each equation,so the training
instance:
- if a featureis essential,modifyingit
slightlyto determineif it can be
generalized. Perturbationoperators,
which are added to the concepttree used
for generalization,
make theseminor
modifications.
a: 2x-5y=-1
b: 3x+4y=lO
is storedas:
{term(a,2*x),term(a,-53),term(a,l),
term(b,3*x),term(b,4*y),term(b,-10))
For example,given the problem (a) 2x+3y=7 (b)
2x+3~-5y=5,the advice to cambinex(b),and an
empty rule base, PET first describesthe rule as:
where a,b are equationlabelsand x,y are
(term(a,2*x),term(a,3*y),term(a,-7),
term(b,2*x),term(b,3*x),term(b,-5*y),
term(b,-5))=> combinex(b).
31t shouldbe recognizedthat MACROPSwas
designedto controla physicalrobot,not a
simulation. For this reason,the designers
thought it importantto permit the planner to skip
ahead in a plan if situationpermitsor to repeat
a step in a plan if the operationfailed due to
physicaldifficulties.
Now PET perturbsthe instanceby modifyingeach
of the coefficientsindividually.This is done by
zeroing,incrementingand decrementingeach
coefficient. Same of the instancescreatedby
perturbationare:
192
(i)
3y=7
2x+3x-5y=5
(ii)
2x+3y=7
3x-5y=5
4 PET understandstW0
applied.
selectingan operator:
(iii)
2x+3y=7
2x+4x-5y=5
reasonsfor
1. By applyingthe operator,the problem
Since ccmCnex(b) is effectivein example i, PET
generalizes(minimally)its currentrule
conditionswith this exampleyieldingthe new
rule:
state is simplified. In the domain of
algebraproblems,a state is simplified
if the number of terms in the equations
is reduced.
{tem(aJjcy),temb,-7) I
term(b,2*x),term(b,3*x),term(b,-5*y),
term(b,-5))=> ccmbinex(b).
2. By applyingthe operator,the
preconditionsof an existingrule are
satisfied. The rule being formed for
the operatoris then looselylinked
with the rule that the operator
enables. If more than one rule is
enabled,then multiplebranchesthrough
the episodeare allowed.
The major effect is to delete the conditionon the
x-term of equation(a).
The operatoris not effectivefor exampleii,
and the negativeinformationis (in the current
system)unused. Generalizingwith example iii,
the rule becomes:
term(b,2*x),term(b,pos(N)*x),term(b,-5"y),
term(b,-5))=> combinex(b).
And after all positiveinstancesof the operator
c&inex(b) have been generatedby the
perturbationtechniqueand generalized,the rule
is formed:
PET adds a rule to the rulebasewhen the
purposeof the rule'saction (the "why" component
of the operator)is understood. The first
operatorsfor which rules can be learnedare those
which simplifythe problemstate, such as
combininglike terms. Any operatorsapplied
before combinecannotbe understoodand PET must
"bearwith" the teacher. After rulesare formed
for the combineoperators,subtractcan be
learned. For instance,sub(a,b)appliedto:
a:
b:
yields:
a:
b:
term(b,pos(M)*x))
=> cambinex(b).
Essentially,perturbationis a techniquefor
creatingnear-examplesand near-misses [13]with
minimal teacherinvolvementupon which standard
generalization
techniquescan be applied.
Mitchell'sLEX system [91 uses a similarmethod
for creatingtraininginstances. Note that in our
techniquewe perturban instanceand try the same
operatorthat worked before. The perturbationis
used to guide the generalization
process. In LEX,
perturbationis used to generatenew, possibly
solvable,problems. Heuristicsare needed to
select appropriateproblemsas the cost of
applyingthe problemsolver is high. See [5] for
more detail on our perturbationtechnique.
2x+3y=5
2x+=1
2x+3y=5
2x-2x -ly-3y=l-5
Now PET can learn sub(a,b)for reason (2) above: a
rule which is alreadyunderstood(fora combine
operator)is enabledby the subtraction.An
episodecan be formedconnectingthe rules for
sub(a,b)and combine(b).
6. Episodic Learning
6.1. Importancefor Learning
As we noted in the MACROPSuse of operator
sequences,unless the system can select
meaningfullyuseful sequencesfrom the set of
candidatesequences,combinatorialexplosionmakes
reuse of generalizedplans infeasible. We define
an episodeto be a sequenceof ruleswhich, when
applied,simplifiesa problemstate. Our episodes
are "looselypackaged"to allow branching. Rather
than storingan entireplan for reachinga goal
state from the start state,we segmentthe
solutionpath into small, re-usable,generalizable
episodes,each accomplishing
a simplification
of
the problem.
6.2. ScoringOperators
A simple scoringscheme connectsrules into
episodesand resolvesconflictswhen more than one
rule is enabled. A naturalscheme is to score
each rule by its positionin an episode. The
rules for the ccmbineoperatorsare given a score
of 0. The rule for subtract,which enablesa
ccnnbine
operator,is given a score of 1 (O+l).
Intuitivelyrthe score is the length of the
episodebefore somethinggood happens (i.e.the
equationsget simplified).When selectinga rule,
PET selectsthe one with the lawest score among
those enabled. Ties are resolvedarbitrarily.
4"Understand"is used here to mean "know-how"
encodedin productionrules. We do not mean to
suggesta deep model of understanding
which might
includecausalityand analogy.
Episodic,or stage 2, learningis concerned
with these sequencesof rules and their
connections. These sequencesare learned
incrementally.Learninga rule for an operator
dependson an understanding
of J&Y the operatoris
193
7. Augmentation
(analogousto our test for predictiveness),
then
more backgroundinformationmust be "pulledin"
until it is associated.
A descriptionin the instancelanguageis
basicallya translationof a traininginstance.
This descriptionis in an instancelanguagemore
appropriateto computationthan the surface
languageused to input the instance. In complex
domains,more knowledgeneeds to be represented
than is capturedin a literaltranslationof a
traininginstanceinto the instancelanguage.
In the domainof algebraproblems,augmentation
serves to relateterms in the instancelanguage.
For example,a relevantrelationbetween
coefficientsis productof(N,M,P)
(theproductof N
and M is P). Augmentationof the instance
languageis necessarywhen the terms or values
necessaryfor an operation(theRHS of a rule) are
not present in the pre-conditions
for the
operation(theLHS of the rule). For example,
considerthe traininginstance:
a: 6x-15y=-3
b: 3x+4y=lO
with the teacheradvice to apply mult(b,2). The
problem is that the rule formedby PET will have a
2 on the RHS (for the operation)but no 2 on the
LHS. In this case, we say that the rule is not
predictiveof the operator.
An augmentationof the instancelanguageis
needed to relatethe 2 on the PHS with some term
on the LHS. In this case, the additional
knowledgeneeded is the 3-ary predicateproductof,
specificallyproductof(2,3,6).Now the rule to
cover the traininginstancecan be formed (after
perturbationand augmentation):
{term(a,G*x),term(b,3*x),
productof(2,3,6))
=> mult(a,2)
Augmentationis similarto selectingbackground
knowledge. One problemwith both approachesis
determininghow much backgroundknowledgeto
incorporate. Incorporating
too little kncwledge,
which resultsin an over-generalized
rule, can be
detectedby an associationchain violationor' in
PET, by a non-predictiverule. However,detecting
when too much knowledgehas been pulled in is
difficult. In this case, the rule formedwill be
over-specialized.We overcomethis problem (to a
large extent)by perturbation. Vere reliessolely
on forminga disjunctionof rules (eachoverly
specialized)for the correctgeneralization.
Vere allows only one conceptin the background
knowledge. This furthersimplifiesthe task of
knowinghaw much knowledgeto pull in. However,
as the complexityof problemdomainsincrease,
more backgroundknowledgemust be broughtto bear.
Our augmentationaddressessome of the problemsof
managingthis knowledge.
8. Examplesof SystemPerformance
This sectiondiscusseshighlightsfrom PET's
episodiclearningfor problemsolvingin the
domain of linearequations.
8.1. Examplel-Learning Combine
The rulebaseis initiallyempty and, as PET
learns, rules are added, generalized,and
supplanted. PET requestsadvicewheneverthe
currentrules do not apply to the problemstate.
The teacherpresentsthe traininginstance:
a: 2x+3y=5
b: 2x+4y=6
Conceptsin the augmentationlanguageform a
second-ordersearchspace (figure7-1) for
generalizingan operation. The space consistsof
a (partial)list of conceptsthat a studentmight
rely on for understandingrelationsbetween
numbers. When a predictiverule cannotbe found
in the first-ordersearch space then PET tries to
form a rule using the augmentationas well.
Conceptsare pulled from the list and added to a
developingrule. If the conceptmakes the rule
predictive,then it is retained. Otherwise,it is
removedand anotherconceptis tried. If no
predictiverule can be found then PET ignoresthe
traininginstance.
with the advice sub(a,b). PET appliesthe
operatorwhich yields:
a: 2x+3y=5
b: 2x-2x+4y-3y=6-5
PET must understandwhy an operatoris useful
before a rule is formed. The operatorfailed to
simplifythe equations(in fact the number of
terms in the equationswent from six to nine) and
did not enable any other rules (sincethe rulebase
is empty). PET cannot form a rule and waits for
somethingunderstandable
to happen.
The teachernow suggeststhat combinex(b)be
applied,yielding:
sumof(L,M,N)
(sum of L and M is N)
productof(L,M,N)(productof L and M is N)
squareof(~,~) (squareof M is N)
a: 2x+3y=5
b: Ox+4y-3y=6-5
Figure7-l: augmentationsearchspace
The number of terms is reduced,so PET
hypothesizesa rule. Perturbationtests each term
in the equationsto determinewhich are essential
and which can be generalized. PET forms the rule:
Vere [12]has also addressedthe problemof
learningin the presenceof "background
information."For example,learninga general
rule for a straightin a poker hand requires
knowledgeof the next number in sequence. This is
consideredbackgroundto the knowledgein the
poker domain. Vere describesan 'association
chain"which links togethereach term in a rule.
If a term in the rule is not linked in the chain
kern'hpos (N)*xl,
term(b,neg(M)*x))
=> combinex(b)
which means:
194
a: 2x+3y=5
b: 2x+4y=6
given a problemstate,whenever
equationb containsan x-termwith a
positivecoefficientand an x-termwith a
negativecoefficient,then combinethe two
terms.
The operatorenablesthe partially-learned
rule
for combinex(b),so PET employsperturbationto
form the rule:
PET is unable to apply currentknowledge(i.e.
the rule for combinex(b))to the currentproblem
state so the teachersuggestscombiney(b) which
yields:
{term(a,2*x),term(b,2*x),
term(b,pos(N)*'y)
1 => sub(a,b)
The rule is assigneda score of 1, looselylinking
it with the rules for combineand deletezero.
a: 2x+3y=5
b: Ox+ly=6-5
Presentedwith furthertraininginstances,PET
inducesthe rule:
Stage 1 learningproducesthe rule:
{term b,pos
(N) 9)
(term(egn(Ll),nonzero(N)*var(X)),
term(eqn(L2),nonzero(M)*var(X)),
term(egn(L2),nonzero(O)*var(Y)))
=> =Jweqmu
,eqnW) 1
I
term(b,neg(M)*y))
=> c&iney(b)
This rule cannotbe generalizedwith the current
rulelistand is simplyadded.
8.3. Example3-Learning Multiply
The teacherpresentsPET with an exampleof
multiplywith the traininginstance:
Learningrules for the operatorsccrmbinec(b)
and deletezero are similarand will be assumed
to be completed.
a: 3x+4y=7
b: 6x+2y=8
Stage 2 learningof the combineoperators
involvesrelatingthem to episodes,or sequences
of rules. Since combinesimplifiesa problem
state immediately,it is given a score of zero.
The currentrulelist(withscores)is:
No rule in the kncwledgebase applies,so PET
requestsadvice. mult(a,2)is suggestedwhich
yields:
0 -- {term(b,pos(N)*x),
term(b,neg(M)*x)}
=> combinex(b)
O{termbps (NJ9) I
term(b,neg(M)*y))
=> combiney(b)
0 -- bmb,posW
1I
term(b,neg(M))
=> combinec(b)
0 -- (term(b,O*x))
=> deletezero
0 -- (term(b,O))=> deletezero
a: 6x+8y=14
b: 6x+2y=8
Now sub(a,b)appliesso the rule for mult(a,2)
can be learned. Mult(a,2)is given a score of 2
(onemore than the score of the rule enabled).
After perturbationPET forms the rule:
{term(a,3*x),term(b,6*X)r
term(b,pos(N)%))=> mult(a,2)
With furthertraininginstancesfor the combine
operators,PET forms the rules:
0--{term(egn(L),int(N)*x),
term(egn(L),int(M)*x))=>ccPnbinex(egn(L))
O-hmn(eqn(L) ,int(N)
%I,
term(egn(L)lint(M)%)]=>ccmbiney(egn(L))
0-(term(eqn(L),int(N)),
term(egn(L),int(M))
=> combinec(egn(L))
0-(term(egn(L),O*var(X))}
=> deletezero(egn(L))
0--(term(egn(L),O))
=> deletezero(egn(L))
At this point PET realizesthat it has
over-generalized
since the rule is non-predictive
(the2 on the RHS does not occur on the LHS). PET
augmentsthe instancedescriptionand forms the
candidaterule:
(term(a,3*x),term(b,6*x),term(b,pos(N)%),
productof(2,3,6)}
=> mult(a,2)
After additionalexamples,PET forms the
correctrule:
2
8.2. Examplea--LearningSubtract
Now that the ccmbineoperatorsare partially
learned,PET can begin to learn subtract. Since
our episodesare looselypackaged,there is not a
problemwith furthergeneralizingthe rules for
the combineoperatorsonce subtractis affiliated
with them. Learningfor each operatorcan proceed
independently.
-- (term(a,pos(K)*x),term(b,pos(L)*x)
I
termbpos 04 9) r
productof(pos(N)
,pos(K)
,pos(L)
1
=> mult(a,pos(N))
which supplantsthe more specificrule in the
rulebase.
8.4. Example4-Learning "CrossMultiply"
The teacherpresentsthe traininginstance:
Naw the traininginstanceabove which PET had
to ignorecan be understood. The instancewith
teacheradvicesub(a,b)is:
a: 2xt6y=8
b: 3x+4y=7
Since no rule is enabled,the teacheradvice to
awly mult(a,3)yields:
'deletezero
(b) could also be suggested,but we
continuewith a combineoperatorfor continuity.
a: 6x+18y=24
b: 3x+ 4y=7
195
Since mult(b,2)is enabled,mult(a,3)can be
learned. After perturbation,PET acguiresthe
rule:
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(term(a,2*x),term(b,3*x),term(b,pos(N)~)]
=> mult(a,3)
This rule is given a score of 3 since it enablesa
rule with score 2. The rule will be generalized
(withsubsequenttraininginstances)to:
3 -- {term(egn(I),nonzero(N)*var(X)),
term(egn(J),nonzero(M)*var(X)),
term(eqn(J),nonzero(L)*var(Y))]
=> mult(egn(I),nonzero(M))
9. Limitations and Extensions
As with most learningprogramswe requirethat
the conceptto be learnedbe representablein our
generalization
language. In additionPET has to
be suppliedwith some coarsenotion of when an
operatorhas been effectivein simplifyingthe
current state. Furthermorewe assume that the
teachergives only appropriateadviceand there is
no "noise."
Extensionsthat we are consideringare:
- Learningfrom negativeinstancesas well
as positiveones.
- Improvingthe use of augmentationby
introducingstructuredconcepts. These
would permit climbingtree
generalizations
for this second-order
knowledge. Another improvementwould be
allowingmultipleconceptsto be pulled
into a rule from the augmentationsearch
space. This requiresa requisitechange
in the test for predictiveness.
- Applyingthe theory to learning
operatorsin other domains. Integration
problemshave been attempted [8]. We
would like to try our approachin the
calculusproblemdomain.
10. Conclusions
A system,PET, has been describedwhich learns
sequencesof rules,or episodes,for problem
solving. The learningis incrementaland
thorough. The system learnswhen and why
operatorsare applied. AlthoughPET startswith
an extremelygeneraland coarse notion of why an
operatorshouldbe appliedit's representation
beccmes increasinglyfine and completeas it forms
rules from examples. PET detectswhen rules are
non-predictive
and augmentsthe generalization
languagewith higher level concepts. Due to the
power of perturbation,our systemcan learn
episodeswith minimalteacherinteraction. The
episodesare segmentedinto discrete,re-usable
segments,each accomplishing
a recognizable
simplification
of the problem state. The approach
is shown effectivein the dcnnain
of solving
simultaneouslinear equations. We suspectthe
techniquewill also work for solvingproblemsin
symbolicintegrationand differentialequations
196