By Dr. rer. nat. H. Ehrig, cand. math. K.-D. Kiermeier, Dipl.-Math. H.-J. Kreowski, Dipl.-Math. W. Kühnel (auth.)

ISBN-10: 3322966445

ISBN-13: 9783322966445

ISBN-10: 3519020548

ISBN-13: 9783519020547

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2 Reduction and Minimization: Given an automaton A we now are going to consider the problem of constructing an automaton A' equivalent to A with minimal number of states. e. M(A) (s) = M(A) (s'). Equivalence of states defines an equivalence relation on S, called Nerode-equivalence. Problem 1: Is it possible to factorize the set of states by the Nerode-equivalence in order to get an equivalent automaton A' with state object morphic image of A? S/= S which is a homo- Minimization: By definition the cardinality of the behavior E(A) is less than or equal to the cardinality of the states of A Problem 2: Is there an automaton A' state set equal or isomorphic to E(A)?

M(A) (s) = M(A) (s'). Equivalence of states defines an equivalence relation on S, called Nerode-equivalence. Problem 1: Is it possible to factorize the set of states by the Nerode-equivalence in order to get an equivalent automaton A' with state object morphic image of A? S/= S which is a homo- Minimization: By definition the cardinality of the behavior E(A) is less than or equal to the cardinality of the states of A Problem 2: Is there an automaton A' state set equal or isomorphic to E(A)? equivalent to A with Solution of the problems: Factorizing S by the Nerodeequivalence, which is the equivalence relation caused by the machine function M(A), we get a surjective function e: S ~ S/= and an injective supplement m: Sf:, ~ <1+,0> satisfying mo e = M(A).

2). • Applications of this theorem will be given in the following chapters 5, 7 and 9. 2 we have remarked that minimal systems in some sense have minimal cardinality of states. Now we want to be more precise: We assume that there is a cardinality function card assigning to each system a cardinal number, for example the cardinality of states or the dimension of vectorspaces. Moreover given a reduction f:S ~ S1 we assume card(S1) ~ card(S) and if card(S) is equal to card(S1) and finite we assume that f is already an isomorphism.

### Universal Theory of Automata: A Categorical Approach by Dr. rer. nat. H. Ehrig, cand. math. K.-D. Kiermeier, Dipl.-Math. H.-J. Kreowski, Dipl.-Math. W. Kühnel (auth.)

by Donald

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