By B. I. Plotkin
The booklet is dedicated to the research of algebraic constitution. The emphasis is at the algebraic nature of genuine automation, which seems as a usual three-sorted algebraic constitution, that permits for a wealthy algebraic conception. according to a common classification place, fuzzy and stochastic automata are outlined. the ultimate bankruptcy is dedicated to a database automata version. Database is outlined as an algebraic constitution and this permits us to contemplate theoretical difficulties of databases.
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Additional info for Algebraic Structures in Automata and Database Theory
Image T form y 1 under the mapping u j i s the set o f a l l elements o f the =aoy, y e r . Since the automaton 9 i s c y c l i c w i t h the generating element a, then t h i s set c o i n c i d e s w i t h the set A; s i m i l a r l y , image T 26 under the mapping p 3 i s B. Hence, (/i . 2 proves the P r o p o s i t i o n . Let t h e mapping 1/1 o f t h e semigroup T t o t h e s e t B be g i v e n . A 1 n a t u r a l automaton r e a l i z i n g such mapping i s the automaton (r ,r,B) w i t h the operations ° and * introduced as f o l l o w s : x«y=xy; x*y=(xy) ; XeT , yer.
E. the automata with the mapping f:A®r -* B according t o the l ( a , y ) ' = a * y . Then (e ,e , v ) i s a homomorphism i n output s i g n a l s o f A the 3 Atm (A,D (A,D). Proof. Let us d e f i n e rule: from i corresponding semigroup automata. I t s u n i c i t y i s v e r i f i e d immedia- t e l y i n a s i m i l a r way, as i t was done i n the previous p r o p o s i t i o n s . 30 The given p r o p o s i t i o n i m p l i e s t h a t the o p e r a t i o n o f any automaton 3 (A, T, B) can be modeled by the o p e r a t i o n o f Atm (A, D , t h e automaton w i t h o u t extraneous output s i g n a l s .
Thus, d e f i n i n g o f the automaton (A,X,B) i s e q u i v a l e n t t o t h a t o f — the r e p r e s e n t a t i o n f:X * S(A,B), w h i l e d e f i n i n g o f the semigroup automaton (A,r,B) i s e q u i v a l e n t t o t h a t o f the homomorphism f:T We w i l l c a l l t h i s homomorphism the automaton representation S(A,B). o f the semi- group r . An a b s o l u t e l y pure automaton (A,X,B) i s c a l l e d an exact if the associated mapping X —* S(A,B) i s i n j e c t i v e . B) i s an exact one, i f the homomorphism f:T ~* S(A,B) i s a monomorphism o f semigroups, i .