Syntactic Semigroup Research Articles

A hierarchy of shift equivalent sofic shifts

We define new subclasses of the class of irreducible sofic shifts. These classes form an infinite hierarchy where the lowest class is the class of almost finite type shifts introduced by B. Marcus. We give effective characterizations of these classes with the syntactic semigroups of the shifts. We prove that these classes define invariants for shift equivalence (and thus for conjugacy). Finally, we extend the result to the case of reducible sofic shifts.

Nesting Until and Since in Linear Temporal Logic

We provide an effective characterization of the “until-since hierarchy” of linear temporal logic over finite models (strings), that is, we show how to compute for a given temporal property of strings the minimal nesting depth in “until” and “since” required to express it. This settles the most prominent classification problem for linear temporal logic. Our characterization of the individual levels of the “until-since hierarchy” is algebraic: for each n, we present a decidable class of finite semigroups and show that a temporal property is expressible with nesting depth at most n if and only if the syntactic semigroup of the formal language associated with the property belongs to the class provided. The core of our algebraic characterization is a new description of substitution in linear temporal logic in terms of block products of finite semigroups.

Bilateral locally testable languages

We give an algebraic characterization of a new variety of languages that will be called bilateral locally testable languages and denoted as BLT. Given k>0, the membership of a word x to a BLT (k-BT) language can be decided by means of exploring the segments of length k of x, as well as considering the order of appearance of those segments when we scan the prefixes and the suffixes of x. In this paper, we also characterize the syntactic semigroup of BLT languages in terms of the equations of the variety they belong to, as well as in terms of the join of two previously studied varieties.

Syntactic semigroups and graph algebras

We describe all directed graphs with graph algebras isomorphic to syntactic semigroups of languages.

Right and left locally testable languages

The aim of this paper is to give a definition as well as an algebraic characterization of two new varieties of languages that will be referred to as right (respectively left) locally testable languages and denoted as RLT (resp. LLT). Both families strictly contain the class of locally testable languages. Given k>0, the membership of a word x to a RLT ( k-RT) language can be decided by means of exploring the prefix and suffix of length k−1 of x and the segments of length k, as well as considering the order of appearance of those segments when we scan the prefixes of x. Membership of x to a k-LT can be decided in a similar way, but we have to change the word “prefixes” for “suffixes” in the above description. In this paper we also show that S is a syntactic semigroup of a k-RT (resp. k-LT) language if and only if the local subsemigroups of S are idempotents and right (resp. left) repetition free.

Strongly locally testable semigroups with commuting idempotents and related languages

If we consider words over the alphabet which is the set of all elements of a semigroup S , then such a word determines an element of S : the product of the letters of the word. S is strongly locally testable if whenever two words over the alphabet S have the same factors of a fixed length k , then the products of the letters of these words are equal. We had previously proved [19] that the syntactic semigroup of a rational language L is strongly locally testable if and only if L is both locally and piecewise testable. We characterize in this paper the variety of strongly locally testable semigroups with commuting idempotents and, using the theory of implicit operations on a variety of semigroups, we derive an elementary combinatorial description of the related variety of languages.

Over testable languages

In this paper we give a combinatorial description of the languages that are both locally and piecewise testable. We call such languages over testable. The locally testable semigroups were discovered in the study of finite automata. The definition of locally testable semigroup is similar to that of locally testable language. If we consider words over the alphabet which is the set of all elements of a semigroup S, then such a word determines an element of S: the product of the letters of the word. A semigroup S is locally testable if whenever two words over the alphabet S have the same factors of a fixed length k and the same prefix and suffix of length k − 1, then the products of the letters of these words are equal. A natural extension of locally testable semigroups is obtained by dropping the condition about the prefix and suffix. We call such semigroups strongly locally testable. We prove in this paper that the family of strongly locally testable semigroups is exactly the intersection of the variety of locally idempotent and commutative semigroups and that of J -trivial semigroups. In particular, a language is over testable if and only if its syntactic semigroup is strongly locally testable. We use this algebraic characterization to derive a simple family of generators for the variety of over testable languages.

On the expressive power of temporal logic

We study the expressive power of linear propositional temporal logic interpreted on finite sequences or words. We first give a transparent proof of the fact that a formal language is expressible in this logic if and only if its syntactic semigroup is finite and aperiodic. This gives an effective algorithm to decide whether a given rational language is expressible. Our main result states a similar condition for the “restricted” temporal logic (RTL), obtained by discarding the “until” operator. A formal language is RTL-expressible if and only if its syntactic semigroup is finite and satisfies a certain simple algebraic condition. This leads to a polynomial time algorithm to check whether the formal language accepted by an n-state deterministic automaton is RTL-expressible.

Unions of groups of small height

A semigroup S is called a union of groups if each of its elements lies in a (maximal) subgroup of S. It is well known that in a union of groups Green’s relation 3 (defined by a/‘b if and only if SaS=%S) is a congruence whose classes are completely simple subsemigroups of S and such that S/g is a semilattice (i.e. satisfies x1=x, xu=yx) [I]. In this paper we present a construction of all unions of groups S having the following properties: (1) S is the syntactic semigroup of a language of the form C+ = Un,O Cn, where C is a finite prefix code; (2) S/f is the two-element semilattice. Using the terminology of Clifford and Preston the semigroups studied here are ideal extensions of a completely simple semigroup by another, or unions of groups of height two. All the completely simple semigroups satisfying condition (1) above have been obtained in [6], using the concept of team tournament, All unions of groups S satisfying (1) and having a non-trivial group of units are constructible using certain factorizations of B,, the group of integers modulo n [4]. The two construction techniques are combined here to give all the unions of groups satisfying (1) and (2) as transition semigroups of what we call a ‘standard amalgamation’ of the automata of two team tournaments (Theorem 3.2). It is likely that the process given here generalizes to a construction of all unions of groups satisfying (1) provided some convenient representation of the two 2-classes case is found. We conjecture that all these semigroups are chains of length n of completely simple semigroups, and that they are in general of group complexity n, in the sense of J. Rhodes [3] (cf. also Tilson’s Chapter 12 in [2]). The complexity conjecture, at least in the case n = 2, can be verified on examples using the results of [ll]. Directly related to these considerations is the problem of describing the variety I of all languages whose syntactic semigroups are unions of groups (see [2], [lo]). In view of the recent results of J.E. Pin [9], the following question arises naturally: Is

A characterization of strictly locally testable languages and its application to subsemigroups of a free semigroup

A syntactic characterization of strictly locally testable languages is given by means of the concept of constant . If S is a semigroup and X a subset of S , an element c of S is called constant for X if for all p, q, r, s ε S 1 *[ pcq, rcs εX ⇒ pcs ε X ]. The main result of the paper states that a recognizable subset X of a free semigroup A + is strictly locally testable if and only if all the idempotents of the syntactic semigroup S ( X ) of X are constants for X′ = XΦ , where Φ : A + → S ( X ) is the syntactic morphism. By this result some remarkable consequences are derived for recognizable subsemigroups of A + . In particular we prove that if X is a recognizable free subsemigroup of A + and Y = X/X 2 its base then the following conditions are equivalent : (1) X is strictly locally testable. (2) X is locally testable. (3) X is locally parsable and Y is strictly locally testable. (4) X has a bounded synchronization delay and Y is strictly locally testable (5) A positive integer k exists such that all the elements of A + whose length is greater than or equal to k , are constants for X . (6) For all the idempotents e of the syntactic semigroup S ( X ) of X , eS ( X ) e ⊆ e , 0.

On some properties of the syntactic semigroup of a very pure subsemigroup

On some properties of the syntactic semigroup of a very pure subsemigroup

Synchronization and simplification

We describe the notions of synchronization and simplification with respect to a given subsemigroup P of a semigroup S in terms of the syntactic semigroup of P. These notions derive from coding theory, which corresponds to the case where P is a free subsemigroup of a free semigroup; we apply the results to give a unified account of several theorems previously published.

A generalization of finiteness

We explore an analogy between the family β1, of finite/cofinite languages and the family ψ1 of languages whose syntactic monoids are J-trivial. It is shown that (a) J-trivial monoids, (b)L-trivial monoids, (c) R-trivial monoids, and (d) a recently studied family, that we callK, of aperiodic monoids are natural generalization of the families of syntactic semigroups of (a) finite/cofinite languages, (b) definite languages, (c) reverse definite languages, and (d) generalized definite languages, respectively. In the case of alphabets of one and two letters, the languages corresponding to the familyK of monoids are characterized, illustrating the above-mentioned analogy explicitly.

Locally testable languages

This paper studies the locally testable languages (or “events”) introduced by McNaughton and Papert. We characterize these languages by means of their syntactic semigroups and obtain wreath product and direct product decompositions for these semigroups. As a by-product of our study, we find an algebraic characterization of A. Ginzburg's generalized definite languages.