On the word problem for syntactic monoids of piecewise testable languages

Abstract

Piecewise testable languages are widely studied area in the theory of automata. We analyze the algebraic properties of these languages via their syntactic monoids. In this paper a normal form is presented for 2-and 3-piecewise testable languages and a log-asymptotic estimate is given for the number of words over these monoids.

Testing Simon's congruence

“…Understanding the combinatorial properties of ∼ k is one of the main tools in the study of piecewise testable languages. For example, in the proof of his theorem, Simon already used that (uv) k ∼ k (uv) k u for all words u, v. Upper and lower bounds on the index of ∼ k were given by Kátai-Urbán et al [7] and Karandikar et al [6].…”

Normal forms under Simon’s congruence

“…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.…”

“…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. In [6] a normal form was given when k = 4.…”

On the index of Simon's congruence for piecewise testability

“…Since the question of estimating C k (n) was raised in [2] (and to the best of our knowledge) no progress has been made on the question, until Kátai-Urbán et al proved the following bounds: Theorem 1.1 (Kátai-Urbán et al [10]). For all k > 1,…”

“…The proof of Theorem 1.2 relies on two new reductions that allows us to relate C k (n) with C k−1 instead of relating it with C k (n − 2) as in [10]. The article is organized as follows.…”