Testable Languages Research Articles

From words to pictures: Row-column combinations and Chomsky-Schützenberger theorem

The row-column combination RCC maps two (word) languages over the same alphabet onto the set of rectangular arrays, i.e., pictures, such that each row/column is a word of the first/second language. The resulting array is thus a crossword of the component words. Depending on the family of the components, different picture (2D) language families are obtained: e.g., the well-known tiling-system recognizable languages are the alphabetic projection of the crossword of local (regular) languages. We investigate the effect of the RCC operation especially when the components are context-free, also with application of an alphabetic projection. The resulting 2D families are compared with others defined in the past. The classical characterization of context-free languages, known as Chomsky-Schützenberger theorem, is extended to the crosswords in this way: the projection of a context-free crossword is equivalent to the projection of the intersection of a 2D Dyck language and the crossword of strictly locally testable language. The definition of 2D Dyck language relies on a new more flexible so-called Cartesian RCC operation on Dyck languages. The proof involves the version of the Chomsky-Schützenberger theorem that is non-erasing and uses a grammar-independent alphabet.

Tropical representations and identities of the stylic monoid

We exhibit a faithful representation of the stylic monoid of every finite rank as a monoid of upper unitriangular matrices over the tropical semiring. Thus, we show that the stylic monoid of finite rank n generates the pseudovariety varvec{{mathcal {J}}}_n, which corresponds to the class of all piecewise testable languages of height n, in the framework of Eilenberg’s correspondence. From this, we obtain the equational theory of the stylic monoids of finite rank, show that they are finitely based if and only if n le 3, and that their identity checking problem is decidable in linearithmic time. We also establish connections between the stylic monoids and other plactic-like monoids, and solve the finite basis problem for the stylic monoid with involution.

Sublinear-Time Language Recognition and Decision by One-Dimensional Cellular Automata

After an apparent hiatus of roughly 30 years, we revisit a seemingly neglected subject in the theory of (one-dimensional) cellular automata: sublinear-time computation. The model considered is that of ACAs, which are language acceptors whose acceptance condition depends on the states of all cells in the automaton. We prove a time hierarchy theorem for sublinear-time ACA classes, analyze their intersection with the regular languages, and, finally, establish strict inclusions in the parallel computation classes [Formula: see text] and (uniform) [Formula: see text]. As an addendum, we introduce and investigate the concept of a decider ACA (DACA) as a candidate for a decider counterpart to (acceptor) ACAs. We show the class of languages decidable in constant time by DACAs equals the locally testable languages, and we also determine [Formula: see text] as the (tight) time complexity threshold for DACAs up to which no advantage compared to constant time is possible.

Two-Sided Strictly Locally Testable Languages

A two-sided extension of strictly locally testable languages is presented. In order to determine membership within a two-sided strictly locally testable language, the input must be scanned from both ends simultaneously, whereby it is synchronously checked that the factors read are correlated with respect to a given binary relation. The class of two-sided strictly locally testable languages is shown to be a proper subclass of the even linear languages that is incomparable to the regular languages with respect to inclusion. Furthermore, closure properties of the class of two-sided strictly locally testable languages and decision problems are studied. Finally, it is shown that two-sided strictly k-testable languages are learnable in the limit from positive data.

On Rough Approximations of Languages under Infinite Index Indiscernibility Relations

In the paper [13] Păun, Polkowski and Skowron introduce several indiscernibility relations among strings that are infinite index equivalence or tolerance relations, and study lower and upper rough approximations of languages defined by them. In this paper we develop a further study of some of these indiscernibility relations among strings. We characterize the classes defined by them, and the rough approximations of general and context free languages under them. We also compare some of the rough approximations these relations produce to the ones given by the congruences defining testable, reverse testable, locally testable, piecewise testable and commutative languages. Those yield languages belonging to that families. Next, we modify some of the relations to obtain congruences, and study the families of languages the rough approximations under them give rise to. One of these modificated relations turns out to be the k-abelian congruence, that was defined by J. Karhumäki in [7], in the context of combinatorics on words. We show that it defines a pseudo-principal +-variety, a term defined in [9]. Our results in that work are then applied to determine when a given language has a best upper approximation in that family. Finally, we make some comments on the accuracy of the rough approximations obtained in each case.

Synthesis of quantifier-free DNF sentences from inconsistent samples of strings with EF games and SAT

We present a method to extract knowledge in terms of quantifier-free sentences in disjunctive normal form from noisy samples of classified strings. We show that the problem to find such a sentence is NP-complete, and our approach for solving it is based on a reduction to the Boolean satisfiability problem. Moreover, our method bounds the number of disjuncts and the maximum number of literals per clause since sentences with few clauses and few literals per clause are easier to interpret. As the logic we are considering defines exactly the class of locally threshold testable (LTT) languages, our results can be useful in grammatical inference when the goal is to find a model of an LTT language from a sample of strings. We also use results of the Ehrenfeucht–Fraïssé game over strings in order to handle consistent and inconsistent samples of strings.

Synthesis of a DNF formula from a sample of strings using Ehrenfeucht–Fraïssé games

In order to define a DNF version of first-order sentences over strings in which atomic sentences represent substring properties of strings, we use results of the Ehrenfeucht–Fraïssé game over strings. Then, given a sample of strings and the number of disjunctive clauses, we investigate the problem of finding a DNF formula that is consistent with the sample. We show that this problem is NP-complete, and we solve it by a translation into Boolean satisfiability. We also present an extension of this problem that is robust concerning noisy samples. We solve the generalized version by a codification into the maximum satisfiability problem. As first-order logic over strings defines exactly the class of locally threshold testable (LTT) languages, our results can be useful in the grammatical inference framework when the goal is to find a model of a LTT language from a sample of strings.

Left and right negatively orderable semigroups and a one-sided version of Simon’s theorem

We study the classes \(\mathrm {LNO}\) and \(\mathrm {RNO}\) of left and right negatively orderable semigroups, arising as natural one-sided generalizations of negatively orderable semigroups (\(\mathrm {NO}\)). Negatively ordered monoids are well-known, for instance, from an equivalent formulation by Straubing and Therien, of a celebrated theorem by I. Simon on piecewise testable languages. The main aim of this paper is to prove a one-sided version of Simon’s theorem for \(\mathrm {LNO}\) (\(\mathrm {RNO}\)). Analogues of some well-known results about negatively orderable semigroups are established in these cases. A characterisation of right negatively orderable semigroups as semigroups of certain type of decreasing mappings on partially ordered sets is obtained. We present sets of quasi-identities defining the quasivarieties \(\mathrm {LNO}\) and \(\mathrm {RNO}\) and show that these classes cannot be defined by finite sets of quasi-identities. We prove \(\mathrm {NO}=\mathrm {LNO}\cap \mathrm {RNO}\) and describe \(\mathrm {LNO}\vee \mathrm {RNO}\). As a one-sided version of Simon’s theorem, the pseudovariety generated by all finite semigroups in \(\mathrm {LNO}\) (\(\mathrm {RNO}\)) is determined and an Eilenberg-type correspondence between this pseudovariety and a variety of languages is established.

Separability by piecewise testable languages is PTime-complete

Piecewise testable languages form the first level of the Straubing–Thérien hierarchy. The membership problem for this level is decidable and testing if the language of a DFA is piecewise testable is NL-complete. So far, this question has not been addressed for NFAs in the literature. We fill in this gap and show that it is PSpace-complete. The main interest of this paper is, however, the lower-bound complexity of separability of regular languages by piecewise testable languages. Two regular languages are separable by a piecewise testable language if the piecewise testable language includes one of them and is disjoint from the other. For languages represented by NFAs, separability by piecewise testable languages is decidable in PTime. We show that it is PTime-hard and that it remains PTime-hard even if the input automata are minimal DFAs. As a result, it is unlikely that separability of regular languages by piecewise testable languages can be solved in a restricted space or effectively parallelized.

On Boolean combinations forming piecewise testable languages

A regular language is k-piecewise testable (k-PT) if it is a Boolean combination of languages of the form La1a2…an=Σ⁎a1Σ⁎a2Σ⁎⋯Σ⁎anΣ⁎, where ai∈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 Lu, 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.

Local testability from words to traces, a suitable definition

The family of locally testable languages has been extensively studied. We propose an extension of the notion of local testability from the free monoid to the free partially commutative monoid (the so called trace monoid). We show that to formalize the notion of locality in traces is conceptually difficult, and we introduce a new kind of factor which takes into account these difficulties. Thus we define a locally testable trace language as a union of classes of a new equivalence relation of a finite index, which is proved to be a congruence using some combinatorics on traces. Then we give a set-theoretic characterization of the family of locally testable trace languages, in terms of some sets called here quasi-ideals, as a generalization of ideals in words. Finally, analyzing the extreme cases of the free monoid and the free commutative monoid, we prove that this new family becomes exactly that of locally testables languages in the case of words.

Finitely based monoids

We present a method for proving that a semigroup is finitely based (FB) and find some new sufficient conditions under which a monoid is FB. As an application, we find a class of finite aperiodic monoids where the finite basis property behaves in a complicated way with respect to the lattice operations but can be recognized by a simple algorithm. The method results in a short proof of the theorem of E. Lee that every monoid that satisfies $$xt_1xyt_2y \approx xt_1yxt_2y$$ and $$xyt_1xt_2y \approx yxt_1xt_2y$$ is FB. Also, the method gives an alternative proof of the theorem of F. Blanchet-Sadri that a pseudovariety of $$n$$ -testable languages is FB if and only if $$n \le 3$$ .

On Separation 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.

Algebraic Characterization of the Class of Languages Recognized by Measure Only Quantum Automata

We study a model of one-way quantum automaton where only measurement operations are allowed (MON-1QFA). We give an algebraic characterization of LMO(Σ), showing that the syntactic monoids of the languages in LMO(Σ) are exactly the J-trivial literally idempotent syntactic monoids, where J is the Green's relation determined by two-sided ideals. We also prove that LMO(Σ) coincides with the literal variety of literally idempotent piecewise testable languages. This allows us to prove the existence of a polynomial-time algorithm for deciding whether a regular language belongs to LMO(Σ).

Numéro 10/2013. Réponses

Among the variety of representation theorems for context-free languages, the Chomsky–Schützenberger theorem is unique in that it consists of Dyck languages, regular languages, and simple operations. In the present paper, we obtain some characterizations of and representation theorems for languages in the Chomsky hierarchy by using insertion systems, strictly locally testable languages, and morphisms in the framework of the Chomsky–Schützenberger theorem. For instance, each context-free language L can be represented in the form L=h(L(γ)∩R), where γ is an insertion system of weight (1,1), R is a strictly 2-testable language, and h is a morphism. In contrast, if R is instead a strictly 1-testable language, the class of languages generated is a proper subclass of context-free languages and incomparable with the class of regular languages.

FROM REGULAR TO STRICTLY LOCALLY TESTABLE LANGUAGES

A classical result (often credited to Y. Medvedev) states that every language recognized by a finite automaton is the homomorphic image of a local language, over a much larger so-called local alphabet, namely the alphabet of the edges of the transition graph. Local languages are characterized by the value k = 2 of the sliding window width in the McNaughton and Papert's infinite hierarchy of strictly locally testable languages (k-slt). We generalize Medvedev's result in a new direction, studying the relationship between the width and the alphabetic ratio telling how much larger the local alphabet is. We prove that every regular language is the image of a k-slt language on an alphabet of doubled size, where the width logarithmically depends on the automaton size, and we exhibit regular languages for which any smaller alphabetic ratio is insufficient. More generally, we express the trade-off between alphabetic ratio and width as a mathematical relation derived from a careful encoding of the states. At last we mention directions for theoretical development and application.

Morphic characterizations with insertion systems controlled by a context of length one

Among the variety of representation theorems for context-free languages, the Chomsky–Schützenberger theorem is unique in that it consists of Dyck languages, regular languages, and simple operations. In the present paper, we obtain some characterizations of and representation theorems for languages in the Chomsky hierarchy by using insertion systems, strictly locally testable languages, and morphisms in the framework of the Chomsky–Schützenberger theorem. For instance, each context-free language L can be represented in the form L=h(L(γ)∩R), where γ is an insertion system of weight (1,1), R is a strictly 2-testable language, and h is a morphism. In contrast, if R is instead a strictly 1-testable language, the class of languages generated is a proper subclass of context-free languages and incomparable with the class of regular languages.

Language theoretical properties of hairpin formations

Hairpin formations arise in biochemical processes and play an important role in DNA-computing. We study language theoretical properties of hairpin formations and our new results concern the hairpin completion Hκ(L1,L2) of two regular languages L1 and L2 and the iterated hairpin lengthening HLκ∗(L) of any language L.Assume that L1 and L2 belong to a certain variety of regular languages which satisfies a mild closure property (being closed by a restricted concatenation), then either Hκ(L1,L2) is not regular or it belongs to the same variety as L1 and L2. This result applies, in particular, to the class of first-order definable languages (which is the class of aperiodic or star-free languages) and it applies to the class of first-order definable languages in two variables with predicates < and successor +1 (i.e., languages in the variety LDA).Furthermore, we solve an open question from Manea et al. [15]: we prove that regular languages are closed under iterated hairpin lengthening. (This has been independently shown in Manea et al. [16].) However, the result here is more precise: if L belongs to a class of languages which satisfies a certain closure property with respect to locally testable languages, then HLκ∗(L) belongs to the same class. This applies to the class of aperiodic languages and many other classes. We also show that the variety DA is not closed under the operation L↦HLκ∗(L). For LDA the situation remains open.

On the word problem for syntactic monoids of piecewise testable languages

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.

From Regular to Strictly Locally Testable Languages

A classical result (often credited to Y. Medvedev) states that every language recognized by a finite automaton is the homomorphic image of a local language, over a much larger so-called local alphabet, namely the alphabet of the edges of the transition graph. Local languages are characterized by the value k=2 of the sliding window width in the McNaughton and Papert's infinite hierarchy of strictly locally testable languages (k-slt). We generalize Medvedev's result in a new direction, studying the relationship between the width and the alphabetic ratio telling how much larger the local alphabet is. We prove that every regular language is the image of a k-slt language on an alphabet of doubled size, where the width logarithmically depends on the automaton size, and we exhibit regular languages for which any smaller alphabetic ratio is insufficient. More generally, we express the trade-off between alphabetic ratio and width as a mathematical relation derived from a careful encoding of the states. At last we mention some directions for theoretical development and application.