On the index of Simon's congruence for piecewise testability

Karandikar, Prateek; Kufleitner, Manfred; Schnoebelen, Philippe

doi:10.1016/j.ipl.2014.11.008

Cited by 38 publications

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“…The accepting states then represent the ∼ k -classes forming the language L. The ∼ k -canonical DFA can be effectively constructed. Moreover, although the precise size of the ∼ k -canonical DFA is not known, see the estimations in [15], we show that the tight upper bound on its depth is k+n k − 1, where n is the cardinality of the alphabet (Theorem 31). This paper is an extended version of paper [23] presented at the DLT 2015 conference, containing full proofs and updated with the latest results and open problems.…”

mentioning

confidence: 86%

“…To justify our choice of a 2-PT language, we point out that if we considered a 3-PT language, then the size of the ∼ 3 -canonical DFA over a binary alphabet would contain 68 states [15] and it would not be possible to present it here in a reasonable form. It is an open question whether it is possible to avoid the use of the ∼ k -canonical DFA.…”

Section: Examplementioning

confidence: 99%

“…A closely related problem to the question on the depth of the ∼ k -canonical DFA over an n element alphabet is the question on the size (number of states) of the ∼ k -canonical DFA. Although this problem has been investigated in the literature, see Karandikar, Kufleitner and Schnoebelen [15], the precise answer is still open.…”

Section: Exponential Gap Between K-pt and The Depth Of Minimal Dfasmentioning

confidence: 99%

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On Boolean combinations forming piecewise testable languages

Masopust

Thomazo

2017

Theoretical Computer Science

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A regular language is k-piecewise testable (k-PT) if it is a Boolean combination of languages of the form L a 1 a , where a i ∈ Σ and 0 ≤ n ≤ k. Given a finite automaton A , if the language L(A ) is piecewise testable, we want to express it as a Boolean combination of languages of the above form. The idea is as follows. If the language is k-PT, then there exists a congruence ∼ k of finite index such that L(A ) is a finite union of ∼ k -classes. Every such class is characterized by an intersection of languages of the from L u , for |u| ≤ k, and their complements. To represent the ∼ k -classes, we make use of the ∼ k -canonical DFA. We identify the states of the ∼ k -canonical DFA whose union forms the language L(A ) and use them to construct the required Boolean combination. We study the computational and descriptional complexity of related problems.1. Check whether the regular language L is piecewise testable. 2. If so, compute the minimal k ≥ 0 for which L is k-piecewise testable. 3. Compute the finite number of representatives of the equivalence classes that form the union of the language L, express them as above and form their union.We study the computational and descriptional complexity of this approach, provide an overview of related results, and formulate several open problems.The complexity of the first step has been studied in the literature. Simon [26] proved that PT languages are exactly those regular languages whose syntactic monoid is J -trivial, which gives decidability. Stern [28] showed that the problem is decidable in polynomial time for languages represented by DFAs and Cho and Huynh [5] proved NLcompleteness for DFAs. Later, Trahtman [31] showed that the problem is solvable in time quadratic with respect to the number of states of the DFA and linear with respect to the size of the alphabet, and Klíma and Polák [19] gave an algorithm that is quadratic in the size of the input alphabet and linear in the number of states of the DFA. For languages represented by NFAs, the problem is PSPACE-complete [11].The second step gives rise to the k-piecewise testability problem formulated as follows:The problem is trivially decidable for any k because there are only finitely many k-PT languages over the alphabet of A . We investigate and overview the computational complexity of this problem. The upper bound complexity for DFAs has been independently solved in [10,18,23]. The co-NP upper bound on the k-piecewise testability problem for DFAs first appeared in [10] without proof, formulated in terms of separability. 3 In this paper, we recall (without proof) the result of [18] showing that the problem is co-NP-complete for DFAs if k ≥ 4. We then focus on the complexity of the problem for k < 4. In particular, for the input given as the minimal DFA, the problem is trivial for k = 0, belongs to AC 0 for k = 1 (Theorem 6), and is NL-complete for k = 2, 3 (Theorems 13 and 18). For NFAs, we show that the problem is PSPACE-complete for any k ≥ 0 (Theorem 20).There is an interesting observation by Klíma and Polák [19] ...

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confidence: 86%

Section: Examplementioning

confidence: 99%

Section: Exponential Gap Between K-pt and The Depth Of Minimal Dfasmentioning

confidence: 99%

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On Boolean combinations forming piecewise testable languages

Masopust

Thomazo

2017

Theoretical Computer Science

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“…Binomial coefficients of words have been extensively studied [19]: u x denotes the number of occurrences of x as a subword, i.e., a subsequence, of u. They have been successfully used in several applications: p-adic topology [6], non-commutative extension of Mahler's theorem on interpolation series [24], formal language theory [13], Parikh matrices, and a generalization of Sierpiński's triangle [18].…”

Section: Introductionmentioning

confidence: 99%

Computing the k-binomial Complexity of the Thue–Morse Word

Lejeune

Leroy

Rigo

2019

Developments in Language Theory

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Two words are k-binomially equivalent whenever they share the same subwords, i.e., subsequences, of length at most k with the same multiplicities. This is a refinement of both abelian equivalence and the Simon congruence. The k-binomial complexity of an infinite word x maps the integer n to the number of classes in the quotient, by this kbinomial equivalence relation, of the set of factors of length n occurring in x. This complexity measure has not been investigated very much. In this paper, we characterize the k-binomial complexity of the Thue-Morse word. The result is striking, compared to more familiar complexity functions. Although the Thue-Morse word is aperiodic, its k-binomial complexity eventually takes only two values. In this paper, we first obtain general results about the number of occurrences of subwords appearing in iterates of the form Ψ ℓ (w) for an arbitrary morphism Ψ. We also thoroughly describe the factors of the Thue-Morse word by introducing a relevant new equivalence relation.

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“…Formally, a language L is k-piecewise testable if x ∈ L and x ∼ k y implies that y ∈ L, where x ∼ k y if and only if x and y have the same scattered subwords of length at most k. It is easy to see that ∼ k is a congruence, the so-called Simon's congruence, with finite index. Some estimations of this index can be found in [3] and [4]. Furthermore, in [4] the word problem for the syntactic monoids of the varieties of k-piecewise testable languages are analyzed and a normal form of the words is presented for k = 2 and k = 3.…”

Section: Introductionmentioning

confidence: 99%

Normal forms under Simon’s congruence

Pach

2017

Semigroup Forum

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Simon's congruence, denoted by ∼ k , relates the words having the same subwords of length at most k. In this paper a normal form is presented for the equivalence classes of ∼ k . The length of this normal form is the shortest possible. Moreover, a canonical solution of the equation pwq ∼ k r is also shown (the words p, q, r are parameters), which can be viewed as a generalization of giving a normal form for ∼ k . In this paper, there can be found an algorithm with which the canonical solution can be determined in O((L + n)n k ) time, where L denotes the length of the word pqr and n is the size of the alphabet.

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On the index of Simon's congruence for piecewise testability

Abstract: Simon's congruence, denoted ∼ n , relates words having the same subwords of length up to n. We show that, over a k-letter alphabet, the number of words modulo ∼ n is in 2 Θ(n k−1 log n) .

Cited by 38 publications

References 9 publications

On Boolean combinations forming piecewise testable languages

On Boolean combinations forming piecewise testable languages

Computing the k-binomial Complexity of the Thue–Morse Word

Normal forms under Simon’s congruence

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