About Some Properties of Definite, Reverse-Definite and Related Automata

Ginzburg, Abraham

doi:10.1109/pgec.1966.264264

Cited by 32 publications

(16 citation statements)

References 4 publications

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“…Here is an example of the maximal one for n = 100: (12,11,10,10,9,8,8,7,6,5,5,4,3,2); its syntactic semigroup size exceeds 2.1 × 10 160 . Compare this to the previously known largest semigroup of an R-trivial language; its size is 100!…”

Section: ⊓ ⊔mentioning

confidence: 99%

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Large Aperiodic Semigroups

Brzozowski

Szykuła

2014

Implementation and Application of Automata

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Abstract. The syntactic complexity of a regular language is the size of its syntactic semigroup. This semigroup is isomorphic to the transition semigroup of a minimal deterministic finite automaton accepting the language, that is, to the semigroup generated by transformations induced by non-empty words on the set of states of the automaton. In this paper we search for the largest syntactic semigroup of a star-free language having n left quotients; equivalently, we look for the largest transition semigroup of an aperiodic finite automaton with n states. We introduce two new aperiodic transition semigroups. The first is generated by transformations that change only one state; we call such transformations and resulting semigroups unitary. In particular, we study complete unitary semigroups which have a special structure, and we show that each maximal unitary semigroup is complete. For n 4 there exists a complete unitary semigroup that is larger than any aperiodic semigroup known to date. We then present even larger aperiodic semigroups, generated by transformations that map a non-empty subset of states to a single state; we call such transformations and semigroups semiconstant. In particular, we examine semiconstant tree semigroups which have a structure based on full binary trees. The semiconstant tree semigroups are at present the best candidates for largest aperiodic semigroups. We also prove that 2 n − 1 is an upper bound on the state complexity of reversal of star-free languages, and resolve an open problem about a special case of state complexity of concatenation of star-free languages.

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Section: ⊓ ⊔mentioning

confidence: 99%

“…An upper bound of n((n−1)!−(n−3)!) has been shown to hold [12] for definite and generalized definite languages [9], but it is not known whether this bound is tight.…”

Section: Introductionmentioning

confidence: 99%

Large Aperiodic Semigroups

Brzozowski

Szykuła

2014

Implementation and Application of Automata

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“…An alternate description of the p families is the following: [2,11] if, and only if, it is in p 2K , generalized definite [11,22] if, and only if, it is in p 2 , and locally testable [14] if, and only if, it is in P 3 . The original définitions of these families of languages were somewhat different; however, the équivalence of the définitions is easily proved [6], and the present formulation appears more natural.…”

Section: The Depth-one Finite-cofemte Hierarchy [6]mentioning

confidence: 99%

“…We introducé a family cc 11 of languages (the reason for this notation is explained in Section 9) such that, if L e oc lal , the membership of a word x in L can be determined solely by the set of letters appearing in x. Define xa = {aeA; x = «ai?…”

Section: The Y! Hierarchy [2324]mentioning

confidence: 99%