Generalized synthesis problem of finite Mealy-automata Everywhere in this paragraph X and Y will denote finite

nonempty sets, with Y = <y^,...,ym >. As in the introduction it was shown any finite automaton mapping a : X* Y* can be given by a complete regular i-vector a . In [5 ] we have

defined the left side e-free derivations of ^-vectors with respect to a word p 6 X* as follows; the left side e-free derivation of a = [a^,...,am ] with respect to the word p

e

X* is the 5,-vector £Dp (a) = [^D^a-j),-- '£Dp(am ^ ' where

i,Dp (a± ) = <q 6 X+ I pq

e

ai> (i=1,...,m) . It can be easily verified that if a is complete and regular then 5D (a ) is also complete and regular for all p 6 X * . Moreover, the regularity of a implies that the set of all different left side e-free derivations of a is a finite set.

In [5] was proved the following

THEOREM 2. Let a ; X* ->■ Y* be a finite automaton

mapping and let a = [a^,...,a ] be the corresponding complete regular %-vector. Then a can be induced by the finite cyclic reduced Mealy-automaton Vt = (A,X,Y,6,A) , with

A = <^Dp(a) | p € X*>, in which a = (a) is the generator state and the functions <5 and X are defined for all „D (a) 6 A

x. p — and x 6 X by

<$(£Dp(a),x) = £Dp x (a) and X(^Dp (a),x) = yk <=> x 6 £D~(ak ) (1 < k < m) . More precisely a = a holds.

cl

Now we consider a finite set <ou : X* -* Y* | i e I>

of finite automaton mappings and verify that a finite cyclic reduced Mealy-automaton can be constructed which induces every mapping ou (i

e

I) . For all i

e

I let = [a^,...

* **'aim] he the complete regular £-vector corresponding

to ai - By the previous theorem it is enough to prove that there is a complete regular 2-vector b = [b^,...,bm ] such that every a^ (i e I) can be obtained as a left side e-free derivation of b with respect to some word p e X* . This is, that will be proved in the following theorem.

THEOREM 3. Let <a^ = [a^^,...,a^m ] | i 6 I> be a finite set of complete regular 9,-vectors over X. Then there exists a complete regular 9,-vector b = [b^,...,bm ] over X such that for all i e I

— la . holds with some p 6 X * .

P r o o f : Let n be the less natural number for which card(I) < card(Xn ), where Xn denotes the set of all words p 6 X* with |pI = n. Moreover, let <L^ | i 6 I> be a set of pairwise disjunct languages such that

Z L. = Xn . iei 1

Finally, let b' = [bij , . . . ,b^] be an 2,-vector over X for which

m

j ft k => b' fib' = 0 (1 < j,k < m) and £ b' = X+...+X

3 K j=1 3

We form the left side linear combination of 2,-vectors au (i 6 I) with languages (i 6 I) as follows:

Z

iei

L. a.

l— i [ £ iei

L . a . .

l i1 I

iei

r

where L ^ a ^ O - j - m) means the catenation of languages L. and a. .

i iD and

iei

L . a . .

i ID means the sum of languages L. a. . (i 6 I) .

l l]

Let b be the ß-vector given by b = £ L . a . + b ' .

iei 1 1

We shall prove that b satisfies the demanded requirements.

It is obvious that b is regular since the catenation and the sum of regular languages is regular. Let us assume that j ^ k but p 6 b. fl b, holds for some p 6 X+ . Since j f k implies

D K

that b' Db' = 0, the length of p must be greater than n D K

and

p 6 £ L.a. fl £ h±aik . iei ^{i ID} iei

But |pI > n implies that p = qr (q,r 6 X ) with |q| = n and -taking into account that <L^ | i 6 I> is a partition of Xn ,

p = qr 6 £ L.a.. fl £ L.a., iei 1 13 iei 1 1

implies that there exists a unique i 6 I such that p = qr e L.a. . fl L.a.k .

Since L. E Xn it follows that l

q 6 L . and r 6 a.. (1 a.. ,

^ l I] lk

which contradicts to the completeness of the Ä-vector a^.

On the other hand,

m m m

Z b . = Z ( E L , a . . + b') = Z L. Z a . . + X + ... + Xn =

j=1 3 j=l iei 1

13

3 iei 1 j=1

13

= Z L . X+ + X + ... + Xn = X + ...+Xn+XnX+ = X+ .

iei 1

Thus we have got that b is a complete regular £-vector. To verify that every a^ (i 6 I) can be obtained as a left side e-free derivation of b, let p be any word from for

arbitrarily fixed i 6 I. Then, taking into account that |p| = n implies that for all j(=1,...,m) .D (b') = 0 , and for

X, p j

a

arbitrary languages L and L

rD-(L.£)

- i v L >b +

A Z Ő (L)„D_ (L)

q £ r qr=p ^

holds, where

e if q 0 L, V L >

= <

. 0 if

q 0

l, ^<

we have got that

V ^ 1 - [ ÍDP ‘Jj. Lia i1 + b i>.... iDp Liaim + b;»l

-iei

t £ T £Dp (Lia i1 ] ' ’ • • ' . L £Dp (Liaim) J “

iei iei

= [ a ...,a. ] = a . . □ L 1 1 un J — 1

Constructing the finite cyclic reduced Mealy-automaton inducing the automaton mapping determined by the £-vector b according to the Theorem 2 it will induces every finite automaton mapping of the given set <ct^: x* -*■ Y* | i e I>.

BODNARCUK, V. G. , CucTenu ypaBHeHMH b a/ireöpe coőbiTMÍÍ, WypH. BMHMCJi . naTen. m MaTen. ÓM3., 3 (1963), 1077-1088.

GÉCSEG, F. and PEÁK, I., Algebraic Theory of Automata, Akadémiai Kiadó, Budapest, 1972.

PEÁK, I., Bevezetés az automaták elméletébe.I I . /Introduction to the Theory of Automata, Vol.II, in Hungarian/ Tankönyvkiadó, Budapest, 1978.

PUSKÁS, CS., Matrix equations in the algebra of languages with applications to automata, Papers on Automata Theory,

IV., K. Marx Univ. of Economics, Dept, of Math., Budapest, 1982, No. DM 82-1, 77-89.

PUSKÁS, CS., A common method for analysis of finite deterministic and non-deterministic automata, Papers on Automata Theory, V., K. Marx Univ. of Economics, Dept, of Math., Budapest, 1983, No. DM 83-3, 77-90.

IM Y C S ' 8 4

ARTIFICIAL

INTELLIGENCE •

IM J'C S' 8 4

sonance of special ternary relations on data base components. The consonance of data base is expecially important in the case of active data base, i.e. when it includes some procedures of knowledge generation.

The verification of active data base consonance may be realized using ordinary, fuzzy or linguistic re­

lations between components.

INTRODUCTION

The procedure of data base designing suggests the definition of informational model of the problem environment to a precision of the names of the object's types and to a precision of the specifi­

cation of links between them. The name of the object's type defines several objects of this type distinguished by their attributes. The structuration of data base is based on fixation of links between the pairs of objects. Omitting the problems of data base designing we'll point out several important distinctions between data base organization and knowledge base organization. At first, knowledge simulation and knowledge representation have special forms, which reflect knowledge-dependent inner structure of every knowledge type.

At second, knowledge structuration suggests much more strong inter­

knowledge relations and in general case they can't be presented only by binary relations.

The inner knowledge representation is difficult to discuss without problem environment description. But as far as general structura­

tion problem is concerning, the problem-environment independent areas exist. One of such areas is the knowledge base integrity pro­

blem which is unsolvable without elimination of contradictions in­

side of the knowledge set, i.e. without transformation of the know­

ledge base into consonance state.

TYPES OP RELATIONS

By analogy with data base we define knowledge base as interconnec­

ted set of knowledge pieces. We consider knowledge base as further development of data base from the point of view of introduction of new relations on data types. As a result fragments of declarative and procedural knowledge arise and they produce semantic network.

In formal way we may present the knowledge base as quintuple (T, L, R, G, P), where T - a set of terms; L,R and G - local, regional and global relations; P - a set of knowledge base management proce­

In document MAGYAR TUDOMÁNYOS AKADÉMIA SZÁMÍTÁSTECHNIKAI ÉS AUTOMATIZÁLÁSI KUTATÓ INTÉZETE (Pldal 148-157)