Quotient Complexity of Ideal Languages

The state complexity of a regular language is the number of states in a minimal deterministic finite automaton accepting the language. The syntactic complexity of a regular language is the cardinality of its syntactic semigroup. The syntactic complexity of a subclass of regular languages is the worst-case syntactic complexity taken as a function of the state complexity n of languages in that class. We prove that n n−1 , n n−1 + n − 1, and n n−2 + (n − 2)2 n−2 + 1 are tight upper bounds on the syntactic complexities of right ideals and prefix-closed languages, left ideals and suffix-closed languages, and two-sided ideals and factor-closed languages, respectively. Moreover, we show that the transition semigroups meeting the upper bounds for all three types of ideals are unique, and the numbers of generators (4, 5, and 6, respectively) cannot be reduced.

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“…The languages L and L have the same syntactic semigroup, but one is a left ideal while the other is not. The following remark was proved in [4]:…”

Section: Lemma 2 If D N Is the Quotient Dfa Of A Left Ideal All The mentioning

confidence: 95%

“…The state complexity of operations on the classes of ideal languages was studied by Brzozowski, Jirásková and Li [4]. The same problem for the classes of prefix-, suffix-, factor-, and subword-closed languages was studied by Han and K. Salomaa [17], Han, K. Salomaa, and Wood [18], and Brzozowski, Jirásková and Zou [5].…”

Section: Introductionmentioning

confidence: 99%

Syntactic Complexity of Regular Ideals

Brzozowski

Szykuła

2017

Theory Comput Syst

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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%

“…If n = 1, then either KL = ∅, or KL = KΣ * . In the second case the bound is m for both star-free and regular languages [4,18].…”

Section: Productmentioning

confidence: 99%

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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“…Reversal It was proved in [10] that the reverse has complexity 2 n−1 , and in [17] that the number of atoms is the same as the complexity of the reverse. 4.…”

mentioning

confidence: 99%

Complexity of Right-Ideal, Prefix-Closed, and Prefix-Free Regular Languages

Brzozowski¹,

Sinnamon²

2017

Acta Cybern

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A language L over an alphabet Σ is prefix-convex if, for any words x, y, z ∈ Σ * , whenever x and xyz are in L, then so is xy. Prefix-convex languages include right-ideal, prefix-closed, and prefix-free languages as special cases. We examine complexity properties of these special prefix-convex languages. In particular, we study the quotient/state complexity of boolean operations, product (concatenation), star, and reversal, the size of the syntactic semigroup, and the quotient complexity of atoms. For binary operations we use arguments with different alphabets when appropriate; this leads to higher tight upper bounds than those obtained with equal alphabets. We exhibit right-ideal, prefix-closed, and prefix-free languages that meet the complexity bounds for all the measures listed above.

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Quotient Complexity of Ideal Languages

Cited by 35 publications

References 19 publications

Syntactic Complexity of Regular Ideals

Syntactic Complexity of Regular Ideals

Large Aperiodic Semigroups

Complexity of Right-Ideal, Prefix-Closed, and Prefix-Free Regular Languages

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