knowledge representation method based on predicate calculus in an ...

cagLl~ 82, J. Horecl~ (e~.) North.Holland Publishln~ Cor,~my © Academia, 1982

KNOWLEDGE REPRESENTATION METHOD B A S E D ON pREDICATE CALCULUS IN AN I N T E L L I G E N T CAI S Y S T E M Barbara Belier R e g i o n a l Computer C e n t e r T e c h n i c a l U n i v e r s i t y o f Pozna~ Poland

The k n o w l e d g e r e p r e s e n t a t i o n m e t h o d is i n t r o d u c e d to be applied i n the ICAI system to teach p r o g ~ m i n ~ language° K n o w l e d g e about syntax and semanties of that language is represented b y a set of axioms w ~ i t t e n i n the predicate calculus language, T h e d i r e c t e d g r a p h of concepts is m e n t i o n e d as a m e t h o d to represent a n i n s t r ~ c t l o n a l structure of the d o m a i n knowledge. The pros[ p r o c e d u r e to answ e r student's questions is described. INTRODUCTION The early 70S have brouF~at the Intelligent CAI / I C A I / systems [3~4~o I n t h e s e systems all course m a t e r i a l is represented i n d e p e n d e n t l y of teachin~ procedures. T h e goal of ICAI r e s e a r c h is to obtain a n indiv i d u a l i z a t i o n of inst1~aotion by p r o v i d i n ~ a n ability of a n s w e r l n ~ studentts questions as w e l l as genel~atin~ remedial o o m m e n t s ~ p r o b l e m s and advioes~ aeco1~Lin~ly to current studentPs respormos and his abilities a n d preferences i n general. ICAI researchers have f i r s t l y f o c u s e d their i n v e s t i g a t i o n s on representation of the subject m a t t e r . M o s t l y s e m a n t i c nets h a v e b e e n used as r e p r e s e n t a t i o n of static [~J and p r o c e d u r a l [~p~] d o m a i n k n o w l e d Ge. D a t a b a s e of a n i n s t z ~ c t l o ~ L l system includes also a r e p r e s e n t a tion of c u r r l c u l u m t o o r ~ n l z e a n instz~uotlon of n e w k n o w l e d g e [~pY~ It is h e l p f u l i n seleotln~ material to be presented to the student. Th£s paper presents the knowledge representation method based on predicate calculus i n an intelligent C AT system~ w h i c h is a p p l i e d to teach the p r o g r a m m i ~ l a n g u a g e [J]o T h e p r o t o t y p e of the p r e s e n t e d a p p r o a c h w~s a n a p p l i c a t i o n of predloate calculus to d e s c r i b e pro~ramsp w r i t t e n i n ALGOL~ to l~ove their oozTe©tness~ i n t r o d u c e d by ~stall [2]. T h e ~ a o w l e d g e about s y n t u and semantlos of" the progr m m n i ~ l a n 6 ~ a ~ e has b e e n r e p r e s e n t e d i n the f o r m of a set of first oz~er logic axloms~ Them are given some notes how to construct the predicate calculus lan~age and axioms desor~bin~ the subject matter. The directed graph of concepts as a method to represent an instx~ctiona~ structure of tile domain knowledge has been introduced aXong with the runner of ~enemtin~ instructional te~ts to the student° Then the procedure answeri~ studGnt~s queat~ons has been desoribedo

13

14

B. BEGIER

PREDICATE CALCULUS APPLIED TO REPRESENT EN01~EDGE ABOUT THE PROGRAMMING LANGUAGE The followin~ criteria may b e r e f e r r e d to knowledge representation methods for ICAI systems [.1,7]: - a b i l i t y to e x p r e s s a l a r g e set of c o n c e p t s of t h e d o m a i n b e i n ~ • taught p - f a c i l i t y of c o d l n ~ t h e s e c o n c e p t s a n d r e l a t i o n s a m o n ~ them~ - e a s y w a y to t ~ a n s f o r m t h e f o r m a l n o t a t i o n i n t o t h e n a t u r ~ l l a n guage formp - e f f i o l e n o y of i n f o r m a t i o n r e t r i e v a l d u r i ~ the p r o c e s s of a n s w e r i n ~ u s e r ' s q u e r y a n d p r o v i n g the c o r r e c t n e s s of h i s a n s w e r ~ - a b i l i t y of a u t o m a t e d d e d u c t i o n a p p l i c a t i o n i n the q u e s t i o n answerln~ process. L e t us c o n s i d e ~ a s u b s e t of a n A L G O L - l l k e progx~ammin~ l a n ~ u a g e ~ c o n t a i n i n ~ s i m p l e a r i t h m e t i c and l o g i c a l e x p r e s s i o n s ~ i n s t r u c t i o n of substitution and conditional and o ~ instructions. We assume that e a c h i n s t r u c t i o n h a s b e e n w r i t t e n i n a s e p a r a t e l i n e of p r o g r a m . T h e p r e d i c a t e c a l c u l u s l a n g u a g e d e v e l o p e d to r e p r e s e n t k n o w l e d g e a b o u t the progI~ammlng l a n 6 ~ a g e c o n t a i n s : - n a m e s of sets~ c a l l e d s o r t s of o b j e c t s ~ r e p r e s e n t i n g elements of s y n t a x a n d s e m a n t i c s of a p r o ~ T a m m l n ~ l a n ~ u a g e p - funotions~ transformln~ obJeotsp - predioates~ representin~ relations between objects. S o m e s o r t s j f u n c t i o n s a n d p r e d i c a t e s a r e i n t r o d u c e d to r e p r e s e n t s y n t a x of t h e p r o g ~ a m m l n g l a n ~ u a g e . 0 t h e r s r e p r e s e n t its s e m a n t i c s . N o t a t i o n. T h e o r d i n a r y p r e d i c a t e c a l c u l u s n o t a t i o n h a s b e e n u s e d . S o m e m o d i f i c a t i o n s i m p r o v e t h e r e a d a b i l i t y of s t a t e m e n t s : - u n q u a n t i f i e d vat-fables a r e g e n e r ~ l l y qua~t~.~le~. ~ - two-places predicates are written in an infix manner~ - binary arithmetic functions are written in an infix ma~n~rp - parenthesis are used in the ordinary meanin~ - c l a u s e s a r e s e p a r a t e d by dots. P r o g r a m s y n t a x. T h e p r o g r a m s y n t a x h a s b e e n d e s c r i b e d b y a set of c l a u s e s w ~ i t t e n i n t h e p r e d i c a t e c a l c u l u s l a n g u a g e . S o r t s of o b j e c t s / e x a m p l e s / ~ i d e n t i f i e r ~ n u m b e r ~ e x p r e s s l o n ~ a r l t h m e t i o e x p r e s s i o n ~ l o g i c a l e x p r e s s i o n ~ label~ i n s t r u c t i o n ~ p r o g r a m line. Functions transform some dod_~ w a x w a - w - w a it: w a ~ w a - - - w l pod_~ id x w y ~ in sko; e t ~ in ifl: wl × wi ----in

e x p r e s s i o n s _ i n t o te~ms~ b y e x a m p l e : where: ~a - arithmetic expression~ wl - l o g i c a l e x p r e s s i o n ~ id - i d e n t i f i e r ~ w y - e x p r e s s i o n ~ et - label~ in - instruction~ ~ri - p r o g r a m line. F i r s t of t h e s e f u n c t i o n s c o n s t r u c t s a n e x p r e s s i o n ~ w h i c h r e p r e s e n t s a n o p e r a t i o n of a d d i t i o n ~ the s e c o n d one g i v e s as a r e s u l t an exp r e s s i o n r e p r e s e n t i n g the "less than" r e l a t i o n a n d t h e o t h e r s c o n s t r ~ e t a p p r o p r i a t e l y the s u b s t i t u t i o n i n s t r ~ c t i o n ~ the ~ instruct i o n a n d the c o n d i t i o n a l i n s t r u c t i o n . S o m e p ~ e d l c a t e s h a v e b e e n i n t r o d u c e d to r e p r e s e n t the s y n t a x relations between syntax obJeots~ like followin~: wins ~wix in p e t ~ et ~ w i ~ s ~ wl x wi F i r s t of t h e s e p r e d i c a t e s i n d i c a t e s the l o c a t i o n of a n i n s t r u c t i o n

15

KNOWLEDGE REPRESENTATION METHOD IN AN ICAI SYSTEM

i n a g i v e n p r o ~ z ~ m l l n e r t h e s e c o n d one a s s i ~ s a l a b e l at t h e b e 6~i~ning of a p r o g m a m l i n e a n d t h e t h i r d one d e t e r m i n e s t h e d i r e c t s u c c e s s i o n of t w o progx~um l i n e s i n a s e q u e n c e . Example,. T h e s y n t a x of a p r o ~ a m containing three followin~ substi~ u t i one : J = L L= L+ 2 EK = J + L is d e s c r i b e d l i k e tb~tz wl w i n s pod(J~LJ . w 2 b n a s wl. w 2 w i n s p o d ( L p d o d ( L , 2~ . E pet W3o w ~ b n a s W2o w3 wln____sspod K ~ d o d ( J , L ) ) . P r o g r a m s e m a n t i c s° S e m a n t i c s of a n a l ~ o r i t h m i c lang u a g e is r e p r e s e n t e d b y a set of a x i o m s ~ r i t % e n i n t h e p r e d i c a t e calculus l~age. N e w s o r t s of obJects~ f u n c t i o n s a n d p r e d i c a t e s a r e to be i n t r O d u c e d . S o r t s of o b ~ e e t s z v a l u e ~ s t a t e V a l u e set c o n t a i n s v a l u e s of ax-lthmeti0 a n d l o g i c a l e x p r e s s l o n s ~ a r ~ys~ s u b r o u t i n e s or p r o c e d u r e s etCo T h e p a r t i c u l a r k i n d of v a l u e is a l o c a t i o n i n a p r o ~ n a m ~ r e p r e s e n t e d b y a progr~um llne. A s t a t e • is a s s i E n e d to a progx~un l i n e and it is d e t e r m i n e d a l s o by the ca. l u a t i o n of v a r i a b l e s . Functions evaluate expressions • owar: w y X s t - - ' ~ T and s e q u e n c e of s t a t e s : haS: st ~ s t

where:

wy - expression~ st ~ state~ w~ - ~lueo

P r e d i c a t e s a s s i g n s t a t e s to i o o a t i o r ~ i n a p r o g r a m : stwe c-st ~wi stzm ~st ×wi Pirs$,c~" t h e m a s s o c i a t e s a s t a t e to a p r o g r a m l i n e b e f o r e a n e x e c u t i o n of a n i n s t r u c t i o n f r o m this lineo T h e s e c o n d o n e i n d i c a t e o 9 that the c o n t r o l p ~ s s e e ~o a n i n s t r u c t i o n ~ r i % t e n i n a ~ i v e n p r o g r a m l i n e i n a ~ i v e n state. S o r t s o~ ob~eets~ f u n c t i o n s a n d p r e d i c a t e s a r e the b ~ i s o~ a ~ r ~ m m a r of ~ h e p r e d i c a t e c a l ~ u l u s l a n g ~ a ~ e ~ w h i c h e x p r e u ~ i o n s a r e ~ s e d %o r e p m e s e n t k n o w l ~ d ~ e a b o u t %he prosx'anuruin~ l a n G u ~ e o A x i o m s w r i t t e n ~ n this l a n g u a g e descx~ibe a ~ h m e % l o 1~les~ p r o p e P ~ ties of e x ~ r e s s i o n s ~ s e m a n % i c s of i n s t r u c t i o n s / p r i n c l p l e ~ of e x e = o u t i o n of i n s t r ~ c t i o n s / ~ m e a n i n g of p r o g r a m s e ~ n e n % c or b l o c k s etOo Ez~unple~ A ~ a x i o m d e ~ o r i b i n ~ tution: w l ~ i n ~ p o d (J~X] s s t w e wl A w 2 bna____~sw l

semantics

of a ~ i n s ~ m u o $ i o n

of s u b s t i -

T h e p r e m i s e s of this a x i o m ! n e l u d e ~ % h o l o c a t i o n el" the ~ u b s t l t u t l o n J = X i n t h e p r o ~ T a m i~ne~ ~ ~ %he ~ s s i ~ m ~ of the s % ~ % e b e f o r e a n e x e c u t i o n o.~. %~.~is i n s t r u e ~ i o n ~ a n d a f a o t ~ t h a t t h e p r o g r a m l i n e w 2 d i r e c t l y f o l l o ~ s " w l o T h e o o n c l ~ s l o ~ says~ % . h ~ a s~;ate n e x t of s as %he s t a t e b e f o r e a n e x e r t i o n of a n i n s t r u c t i o n w r i t t e n in ~ 2 a n d 8 v a l u e of t h e v a r i a b l e J i n the s t a t e n e x t of s is a value of a n exp~-ession X i n the ~ t a t e s ~nd ~ ~ l u e of a n y v a r i a b l e Y ~ J d o e s n ' t e h a n ~ e d u r i n g %he % r a r ~ f e r f r o m t h e s t a t e s to the n e x t o f ~o A l ~ r ~ e s u b s e ~ o£ F O R T R A N h a s b e e n d e $ ~ r l b e d i n t h i s m a n n e r [~] ° it

tul~n8 out ~ h a t form%11as

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16

'B. BEGIER

f o r m into n a % ~ r a l l a ~ e~os£~. A x l e s are d i v i d e d into simple sentences. T r a n s l a t i ~ rales are applied to simple sentences. E a c h object has a n d m e i n n a t u r ~ l l ~ g e . Also an appropriate ~%ural l a n g u a g e e x p r e s s i o n is selected f o r e a o h f u n o t i o n . E a a h p r e d i cate cOrTesponds w i t h a v e r b p h r a s e i n n a t u r a l language. T h e p r o p e r t r a n s l a t i o n rules f o r f u n c t i o n s and objects sure a p p l i e d w i t h r e f e rence te arguments of a p r e d l o a t e . ~ l a t i o n ~ l e s f o r P o l i s h language have b e e n r e p o r t e d i n [I] as well as their a ~ p l i c a t i o n to all axioms d e s c r l b i n ~ FORTRAN. D I R E C T E D G R A P H AS A D ~ r H O D OF I N S T R U C T I O N A L S T R U C T U R E R E P R E S E N T A T I O N A n a s s u m p t i o n is done that all k n o w l e d g e to be tau6ht can be divided to i n s t r u c t i o n a l u n i t s . T h u s the first step to oonstr~ot a n i n s t r u c tional s t r u c t u r e r e p r e s e n t a t i o n is to select such units /concepts/. E a c h of them has a n a m e a n d a t least one sentence can be told about it / u n r e a l concepts are not a l l o w e d / . S o m e i n t r o d u c t o r y concepts are a s s u m e d t o be k n o w n t o the student. T h e next step is lations could be: /a/ Concept / b / Concept ~ oncept Concept / e / Concept / f / Concept

to s p e c i f y all relations b e t w e e n c o n c e p t s . T h e s e reX X X X X X

is a part of Y, has a property~ r e p r e s e n t e d b y Z~ is a r e a s o n O r a J u s t i f i c a t i o n of T, b e l o n g s to the object class r e p r e s e n t e d by K, is a n a l t e r n a t i v e of A~ is equivalent to W at least i n some circumstances.

E a c h r e l a t i o n corresponds w i t h a graph, w h i c h nodes r e p r e s e n t concepts f r o m a n i n t r o d u c e d set of c o n c e p t s . T h e c o m p o s i t i o n of all obtained 6~aphs rerults i n a f i n a l g r a p h ~ w h l c h r e p r e s e n t s a n i n s t r u c tional structure of the subject m a t t e r . B e c a u s e of the d i f f e r e n t int e r p r e t a t i o n of the p a r t i c u l a r arches of this ~ T a p h / w h i c h are des c r i b e d b y v a r i o u s r e l a t i o n s h i p s / the "superior-inferior" r e l a t i o n is i n t r o d u c e d as the u n i v e r s a l one w h i c h r e p r e s e n t s every r e l a t i o n b e t w e e n c o n c e p t s . T h u s the d l r e s t e d ~ p h h a s b e e n o b t a i n e d , ~ i t h arrows directed to the s u p e r i o r concepts. A set of axioms is a s s o c i a t e d w i t h each node of the concept graph. Also some other i n f o r m a t i o n may b e a s s o c i a t e d w i t h it. A N S W E R I N G STUDENT'S

QUESTIONS

The f o l l o w i n ~ problems h a v e to be solved| - choice and s p e c i f y i n ~ of classes of u s e r ' s q u e r i e s , w h i c h c a n be a n s w e r e d by the ICAI system~ - reco@~nition of a m a i n subject of the query, t r a n s l a t i o n of the query f r o m n a t u r a l language to the predicate calculus l a n g u a g e f o r m u l a , - a p p l i c a t i o n of the a u t o m a t e d t h e o r e m - p r o v i n ~ techniques to retrieve a n answer, - g e n e r a t i n ~ of a n a n s w e r i n n a t u r a ~ language form. T h r e e classes of queries have b e e n qonsidered: / I / D e c i s i o n queries of g e n e r ~ l \ f o r m i n n a t u r a l l a n g u a g e < sentence> 7 where: / e x i s t i n ~ i n P o l i s h / determines that a q u e s t i o n belongs to this class 9 ~ e n t e n o e > - i n d i c a t i v e sentence,

KNOWLEDGEREPRESENTATION lVlETHODIN AN ICAI SYSTEM

17

w h i c h r e q u i r e a n a n s w e r i n the f o r m .Y~s, or "No". / 2 / Objective quez-lee of ~enez~al f o z ~ i n n a t u z ~ l l a n s u a g e W h l o h ~senex~l name> i n t h e q u e r y , A(X) -. f o r m u l a o b t a i n e d i r o n t h e t z ~ n s l a t e d q u e r y p which describes some p r o p e ~ t i e s of X. /3/

Problem

/a/

~

quex-ies of

general

form

in

natural

language

?

w h l o h can be t ~ s f o .treed into the form: Which Z eatlsfies: Z~ ~clause> ? /b/ What is i m p l i e d b y ~elause> ? w h i c h c a n b e t z ~ n s f o ~ m e d into the formz W h i c h Z satisfies = ~ o l a u s e ~ Z ? I n the above p r o b l e m queries: Z - the clause to be f o u n d t ~olause~ - clause obta/ned f r o m the tx~nslate~ query. P ~ o b l e m queries ~ e q u l r e a n a n s w e r i n the f o z ~ of a sentenCOo A n analysis Of a userts quex-y should f l x t h e m a i n subject of it i n the terms of a subset of c o n o e p t s ~ r e p r e s e n t e d on the concept g ~ p h . A d o f i D / t l o n of a c c e p t a b l e l a n g u a g e of u s e r ' s queries involves the f o r m of t r a n s l a t i o n rules f r o m n~tux~tl lan~u.age into the p r e d l o a t e calculus formula. It is w o r t h n o t l o l n ~ that: - querles i n the n a t u r a l lar~ua~e £ o r m have the threefold nature~ it means they can be counted into the three m e n t i o n e d a b o v e olasses~ - queries f r a g m e n t s i n the f o r m of i n d i c a t i v e clauses are built f r o m expressed i n n a t u r a l language p ~ e d l o a t e s ~ i n t r o d u e e ~ i n the p r e sent ed f or~alization~ - i n respect of q u a n t i t y of expressions the language of user's queries is comparable w i t h the l a a 1 & ~ e obtained i n the process of t r a n s l a t i o n of p ~ e d / o a t e calculus axioms into n a t u r a l lan~ua~e~ - lan@uaGe o~ user's queries and the predloate calculus l ~ a ~ e have a c o m m o n base of b a s l e concepts b e c a u s e the sorts of ob~eots~ functions and pFedicates i n t r o d u c e d i n the predloate calculus language correspond w i t h some s p e c i f i e d n a t u r a l language expressions. It has b e e n submitted that q u e s t l o n - a n s w e r i n g p r o b l e m m ~ y b e s o l v e d with a n a p p l i c a t i o n o~ a u t o ~ t e ~ theorem-pyo~,in~ teohnlques~ n a m e l y on the b a s e o[ the r e s o l u t i o n principle [8] ,This method~ b a s e d only on the s~rnteac o f olauses~ doesn't r e q u i r e to control the proo~ procedure b y the u s e r , T h e r e s o l u t i o n prlnelple requires to convert all formulas into the S k o l e m conjunctive form. Thus each f o r m u l a b e c o mes a set of olauses~ each of them b e i ~ E a d l s j u n o t i o n of literals, The q u e s t i o n - a n s ~ e r i n g procedure f o r d e o l s l o n queries tries to pro~e that a n e g a t i o n of f o r m u l a ~ob%alned a f t e r t r a n s l a t i o n of the que r ~ is false, I£ it's so~ a n a n s w e r is "Yes". If a proo~ procedure a p p l i e d t o the f o r m u l a i n its a~firmati~/e ~ o r ~ provides a sucoess~ a n a n s w e r is "NO".Some questions m a y b e u n s o l v a b l e i n the lack of kno~le~e. T h e proof p r o c e d u r e £ o r objective queries examines a £ o r m u l a ~ ~ x (w (x} ~ A (x))

18

B. BEGIER

w h i c h is supposed to be f a l s e . T h e proof p r r o e d u r e tries to r e t r i e v e a counter-example, if i% exists, w h i c h w i l l be s u b s t i t u t e d in the place of X o I% has b e e n a s s u m e d that p r o b l e m queries are i n the i m p l i c a t i o n f o r m a f t e r the t r a n s l a t i o n proeess. Q u e s t i o n - a n s w e r i n ~ p r o c e d u r e f o r this class of queries has b e e n reduced to such onetwhich tries to retrieve a n a n s w e r f r o m one axiom. For the f i r s t subclass of p r o b l e m queries a search is made f o r a n a x i o m i n the i m p l i c a t i o n form, ~ h i c h c o n c l u s i o n embodies the c o n c l u s i o n of the f o r m u l a o b t a i n e d f r o m the tx~ansformed query. T h e premises of this a x i o m are a n answer° The proof procedure a p p l i e d to a n s w e r a q u e s t i o n of the second subclass tries to f i n d a n axiom~ w h i c h premises are i m p l i e d by premises of the f o r m u l a obtained f ~ o m the transformed query. The c o n c l u s i o n of this a x i o m is an answer. T h e p r o p e r way to r e d u c e a n u m b e r of clauses t a k i n g a part i n the r e s o l u t i o n process is to constr,~ct a n i n i t i a l active set of clauses as a set c o n t a i n i n g only clauses of axioms c o n c e r n i n g concepts reco~LIzed i n the query and clauses of f o r m u l a obtained f r o m the query. The t r a n s l a t i o n r u l e s , a p p l i e d to t r a n s f o r m axioms f r o m the predicate calculus language into the n a t u r a l language expressions t c a n be u s e d also to translate the r e t r i e v e d a n s w e r to the n a t u r a l language° CONCLUSIONS D e s c r i b e d above ~ 1 o ~ l e ~ e r e p r e s e n t a t i o m m e t h o d b a s e d on predicate calculus has b e e n applied to the large subset of F O R T R A N 1900 [I] . I% has b e e n sho~n that this m e t h o d satisfies c r i t e r i a r e q u i r e d in the ICAI system. A n a p p l i c a t i o n of the p r e d i c a t e calculus language to d e s c r i b e k n o w l e d g e about the p r o g r a m m i n g language provides the a u t o m a t e d a n s w e r i n g of student's q u e s t i o n s , w h l c h is the m a i n advantage of this method. T~Lis m e t h o d is a p p l i c a b l e to those domains of ~ o w l e d g e , w h i c h can be r e p r e s e n t e d b y the set of first or~er logic axioms, w i t h regard to their f o r m a l i z e d nature. REFERENCES [J] B e g i e r B. ~ M e t h o d of r e p r e s e n t a t i o n of subject m a t t e r in the c o m p u t e r system to teach p r o ~ a m m l n g lan~ua~e~ Ph.D. Thesis / i n Polish/, Reg. Comp. Center, Tech. Univ. of P o z n a ~ /1980/. [2] B u r s t a l l R.M.~ Formal d e s c r i p t i o n of p r o g r a m str~cture and sem a n t i c s i n first order logic, Math. Intell. 5 / 1 9 6 9 / 79-98 /3] C a r b e n e l l J.R.~ AI i n CAI: a n a r t i f i c i a l i n t e l l i ~ e n c e a p p r o a c h to c o m p u t e r - a s s i s t e d instruction, IEEE Trans. on M a n - M a c h i n e Systems 11 / 1 9 7 0 / 190-202 [4] C l a n c e y W.J,, B e ~ e t J.S., C o h e n J.R. ~ A p p l l c a t i o n s - o r i e n t e d AI research= education, Rep. STAN-CS-79-749~ Dep. of Comp. So., Stanford Univ. / J u l y 1979/ [5] C o m p u t e r - a i d e d i n s t r u c t i o n system to t ~ c h the p r o ~ a m m l n g language F O R T R A ~ Rep. of Re~. Compo Centerp T e o h . U n l v . P o z n a d /1980/ [6] G o l d s t e i n I.P. ~ T h e genetic graph: a r e p r e s e n t a t i o n f o r the eve. l u t i o n of p r o c e d u r a l Enowledge, I n t . J . M a n - M a c h i n e Studies 11 / 1 9 7 9 / 51-77 [~] L a u b s c h J.H., Some t h o ~ h % about r e p r e s e n ¢ i n g k n o w l e d g e in ins t r u c t i o n a l systems~ F o u r t h Int. J o i n t Conf. on Art. Intel~.,

~ilisi

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[8] L o v e l a n d D.W.~ A u t o m a t e d theorem proving= / N o r t h Holland~ Amsterdam, 19~8 /

a losical basis