Separating Regular Languages by Locally Testable and Locally Threshold Testable Languages

A separator for two languages is a third language containing the first one and disjoint from the second one. We investigate the following decision problem: given two regular input languages, decide whether there exists a locally testable (resp. a locally threshold testable) separator. In both cases, we design a decision procedure based on the occurrence of special patterns in automata accepting the input languages. We prove that the problem is computationally harder than deciding membership. The correctness proof of the algorithm yields a stronger result, namely a description of a possible separator. Finally, we discuss the same problem for context-free input languages

Splicing representations of strictly locally testable languages

AbstractThe relationship between the family SH of simple splicing languages, which was recently introduced by Mateescu et al. and the family SLT of strictly locally testable languages is clarified by establishing an ascending hierarchy of families {SiH: i⩾ − 1} of splicing languages for which SH = S1H and the union of the families is the family SLT. A procedure is given which, for an arbitrary regular language L, determines whether L is in SLT and, when L is in SLT, specifies constructively the smallest family in the hierarchy to which L belongs. Examples are given of sets of restriction enzymes for which the action on DNA molecules is naturally represented by splicing systems of the types discussed

Locally threshold testable languages of infinite words

Two versions of local threshold testability for languages of infinite words (omega-languages) are compared: It is proved that an omega-language is finitely locally threshold testable iff it is locally threshold testable and belongs to the intersection of the Borel classes Fsigma and Gdelta. As a consequence we obtain a result on the definability of infinite word structures in the signature of the successor function: It is decidable whether a given monadic second order formula has the same set of infinite word models as some first order formula. For biinfinite word models the corresponding problem was raised by Jean Eric Pin. The major tool in the proofs is the analysis of De Bruijn graphs

from regular to strictly locally testable languages

Comment: In Proceedings WORDS 2011, arXiv:1108.341

A decidable characterization of locally testable tree languages

A regular tree language L is locally testable if membership of a tree in L depends only on the presence or absence of some fix set of neighborhoods in the tree. In this paper we show that it is decidable whether a regular tree language is locally testable. The decidability is shown for ranked trees and for unranked unordered trees

Higher-Order Operator Precedence Languages

Floyd's Operator Precedence (OP) languages are a deterministic context-free family having many desirable properties. They are locally and parallely parsable, and languages having a compatible structure are closed under Boolean operations, concatenation and star; they properly include the family of Visibly Pushdown (or Input Driven) languages. OP languages are based on three relations between any two consecutive terminal symbols, which assign syntax structure to words. We extend such relations to k-tuples of consecutive terminal symbols, by using the model of strictly locally testable regular languages of order k at least 3. The new corresponding class of Higher-order Operator Precedence languages (HOP) properly includes the OP languages, and it is still included in the deterministic (also in reverse) context free family. We prove Boolean closure for each subfamily of structurally compatible HOP languages. In each subfamily, the top language is called max-language. We show that such languages are defined by a simple cancellation rule and we prove several properties, in particular that max-languages make an infinite hierarchy ordered by parameter k. HOP languages are a candidate for replacing OP languages in the various applications where they have have been successful though sometimes too restrictive.Comment: In Proceedings AFL 2017, arXiv:1708.0622

Conditional Lindenmayer systems with subregular conditions : the extended case

We study the generative power of extended conditional Lindenmayer systems where the conditions are finite, monoidal, combinational, definite, nilpotent, strictly locally (k)-testable, commutative, circular, suffix-closed, starfree, and union-free regular languages. The results correspond to those obtained for conditional context-free languages

Free profinite R-trivial, locally idempotent and locally commutative semigroups

This paper is concerned with the structure of implicit operations on R intersection with LJ1, the pseudovariety of all R-trivial, locally idempotent and locally commutative semigroups. We give a unique factorization statement, in terms of component projections and idempotent elements, for the implicit operations on R intersection with LJ1. As an application we give a combinatorial description of the languages that are both R-trivial and locally testable. A similar study is conducted for the pseudovariety DA intersection with LJ1 of locally idempotent and locally commutative semigroups in which each regular D-class is a rectangular band.INVOTAN, grant 4/ A/94/PO.PRC-GdR AM