Language Theoretic Properties Research Articles

On the word problem for weakly compressible monoids

We study the language-theoretic properties of the word problem, in the sense of Duncan & Gilman, of weakly compressible monoids, as defined by Adian & Oganesian. We show that if C is a reversal-closed super-, as defined by Greibach, then M has word problem in C if and only if its compressed left monoid L(M) has word problem in C. As a special case, we may take C to be the class of context-free or indexed languages. As a corollary, we find many new classes of monoids with decidable rational subset membership problem. Finally, we show that it is decidable whether a one-relation monoid containing a non-trivial idempotent has context-free word problem. This answers a generalization of a question first asked by Zhang in 1992.

On the word problem for special monoids

A monoid is called special if it admits a presentation in which all defining relations are of the form w = 1. Every group is special, but not every monoid is special. In this article, we describe the language-theoretic properties of the word problem, in the sense of Duncan and Gilman, for special monoids in terms of their group of units. We prove that a special monoid has context-free word problem if and only if its group of units is virtually free, giving a full generalisation of the Muller-Schupp theorem. This fully answers, for the class of special monoids, a question posed by Duncan and Gilman (Math Proc Camb Philos Soc 136:513–524, 2004). We describe the congruence classes of words in a special monoid, and prove that these have the same language-theoretic properties as the word problem. This answers a question first posed by Zhang (Math Proc Camb Philos Soc 112:495–505, 1992). As a corollary, we prove that it is decidable (in polynomial time) whether a special one-relation monoid has context-free word problem. This completely answers another question from 1992, also posed by Zhang.

Language theoretic properties of regular DAG languages

We study sets of directed acyclic graphs, called regular DAG languages, which are accepted by a recently introduced type of DAG automata motivated by current developments in natural language processing. We prove (or disprove) closure properties, establish pumping lemmata, characterize finite regular DAG languages, and show that “unfolding” turns regular DAG languages into regular tree languages, which implies a linear growth property and the regularity of the path languages of regular DAG languages. Further, we give polynomial decision algorithms for the emptiness and finiteness problems, and show that deterministic DAG automata can be minimized and tested for equivalence in polynomial time.

Pseudoknot-generating operation

A pseudoknot is a crucial intra-molecular structure formed primarily in RNA strands and closely related to important biological processes. This motivates us to define an operation that generates all pseudoknots from a given sequence and consider algorithmic and language theoretic properties of the operation. We design an efficient algorithm that decides whether or not a given string is a pseudoknot of a regular language L. Our algorithm runs in linear time if L is given by a deterministic finite automaton. We study closure and decision properties of the pseudoknot-generating operation. For DNA encoding applications, pseudoknot structures are undesirable. We give polynomial-time algorithms that check whether or not a regular language L contains a pseudoknot or a pseudoknot generated by some string of L. Furthermore, we show that the corresponding questions for context-free languages are undecidable.

Node replacement graph grammars with dynamic node relabeling

The paper introduces node replacement graph grammars with dynamic node relabeling as an extension of the eNCE families of graph grammars. Initiated by NLC grammars in which graph embedding is based on node labels only, node distinction of the embedded graph extended NLC grammars to NCE grammars and dynamic edge relabeling further extended NCE grammars to eNCE grammars. Now, dynamic node relabeling, the node counterpart of the dynamic edge relabeling, is introduced as the next extension, which we call neNCE. The paper analyzes basic language-theoretical properties of the neNCE system.

INNER PALINDROMIC CLOSURE

We introduce the inner palindromic closure as a new operation ♠, which consists in expanding a factor u to the left or right by a non-empty word v such that vu or uv, respectively, is a palindrome of minimal length. We investigate several language theoretic properties of the iterated inner palindromic closure ♠*(w) = Ui≥0 ♠i(w) of a word w.

On the properties of language classes defined by bounded reaction automata

Reaction automata are a formal model that has been introduced to investigate the computing powers of interactive behaviors of biochemical reactions (Okubo et al. (2012) [19]). Reaction automata are language acceptors with multiset rewriting mechanism whose basic frameworks are based on reaction systems introduced in Ehrenfeucht and Rozenberg (2007) [8].In this paper we continue the investigation of reaction automata with a focus on the formal language theoretic properties of subclasses of reaction automata, called linear-bounded reaction automata (LRAs) and exponentially-bounded reaction automata (ERAs). Besides LRAs, we newly introduce an extended model (denoted by λ-LRAs) by allowing λ-moves in the accepting process of reaction, and investigate the closure properties of language classes accepted by both LRAs and λ-LRAs. Further, we establish new relationships of language classes accepted by LRAs and by ERAs with the Chomsky hierarchy. The main results include the following: (i)the class of languages accepted by λ-LRAs forms an AFL with additional closure properties,(ii)any recursively enumerable language can be expressed as a homomorphic image of a language accepted by an LRA,(iii)the class of languages accepted by ERAs coincides with the class of context-sensitive 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.

Languages as hyperplanes: grammatical inference with string kernels

Using string kernels, languages can be represented as hyperplanes in a high dimensional feature space. We discuss the language-theoretic properties of this formalism with particular reference to the implicit feature maps defined by string kernels, considering the expressive power of the formalism, its closure properties and its relationship to other formalisms. We present a new family of grammatical inference algorithms based on this idea. We demonstrate that some mildly context-sensitive languages can be represented in this way and that it is possible to efficiently learn these using kernel PCA. We experimentally demonstrate the effectiveness of this approach on some standard examples of context-sensitive languages using small synthetic data sets.

Fragments of First-Order Logic over Infinite Words

We give topological and algebraic characterizations as well as language theoretic descriptions of the following subclasses of first-order logic FO[<] for ω-languages: Σ2, FO2, FO2∩Σ2, and Δ2 (and by duality Π2 and FO2∩Π2). These descriptions extend the respective results for finite words. In particular, we relate the above fragments to language classes of certain (unambiguous) polynomials. An immediate consequence is the decidability of the membership problem of these classes, but this was shown before by Wilke (Classifying Discrete Temporal Properties. Habilitationsschrift, Universitat Kiel, April 1998) and Bojanczyk (Lecture Notes in Computer Science, vol. 4962, pp. 172–185, 2008) and is therefore not our main focus. The paper is about the interplay of algebraic, topological, and language theoretic properties.

PGF: A Portable Run-time Format for Type-theoretical Grammars

Portable Grammar Format (PGF) is a core language for type-theoretical grammars. It is the target language to which grammars written in the high-level formalism Grammatical Framework (GF) are compiled. Low-level and simple, PGF is easy to reason about, so that its language-theoretic properties can be established. It is also easy to write interpreters that perform parsing and generation with PGF grammars, and compilers converting PGF to other formats. This paper gives a concise description of PGF, covering syntax, semantics, and parser generation. It also discusses the technique of embedded grammars, where language processing tasks defined by PGF grammars are integrated in larger systems.

INTRA-MOLECULAR TEMPLATE-GUIDED RECOMBINATION

This paper considers an intra-molecular version of template-guided recombination (TGR), which has been studied in recent papers as an inter-molecular operation. We demonstrate that the intra-molecular TGR operations are distinct from previous intramolecular models for ciliate gene assembly, including the ld and hi operations. We examine the language-theoretic properties of intra-molecular TGR, including closure properties and solutions to language equations.

Algebraic properties of substitution on trajectories

Language operations on trajectories provide a generalization of many common operations such as concatenation, quotient, shuffle and others. A trajectory is a syntactical condition determining positions where an operation is applied. Besides their elegant language-theoretical properties, the operations on trajectories have been used to solve problems in coding theory, bio-informatics and concurrency theory. We focus on algebraic properties of substitution on trajectories. Their characterization in terms of language-theoretical properties of the associated sets of trajectories is given. The transitivity property is of particular interest. Unlike, e.g., shuffle on trajectories, in the case of substitution the transitive closure of a regular set of trajectories is again regular. This result has consequences in the above-mentioned application areas.

Formalizing Mirror Theory

Mirror theory is a theory of (morpho-) syntax introduced in (Brody, 1997). Here I present a formalization of the theory, and study some of its language theoretic properties.

Parallel communicating grammar systems with terminal transmission

We introduce a new variant of PC grammar systems, called PC grammar systems with terminal transmission, PCGSTT for short. We show that right-linear centralized PCGSTT have nice formal language theoretic properties: they are closed under gsm mappings (in particular, under intersection with regular sets and under homomorphisms) and union; a slight variant is, in addition, closed under concatenation and star; their power lies between that of n-parallel grammars introduced by Wood and that of matrix languages of index n, and their relation to equal matrix grammars of degree n is discussed. We show that membership for these language classes is complete for NL. In a second part of the paper, we discuss questions concerning grammatical inference of these systems. More precisely, we show that PCGSTT whose component grammars are terminal distinguishable right-linear, a notion introduced by Radhakrishnan and Nagaraja in [33,34], are identifiable in the limit if certain data communication information is supplied in addition.

Independent parallelism in finite copying parallel rewriting systems

We consider the class of parallel rewriting systems and investigate the interaction between two complexity measures, that in the literature have been called synchronized parallelism and independent parallelism. It is shown that, when the degree of synchronized parallelism is bounded by some constant greater than one, the degree of independent parallelism induces an infinite non-collapsing hierarchy within the family of generated languages. The result is obtained using an original characterization of parallel rewriting systems. Other language-theoretic properties of parallel rewriting systems are proved in this work, that together with our main result provide an answer to some questions that were left open in the literature.

Simple splicing systems

We consider one of the most restrictive classes of splicing ( H) systems, namely based on splicing rules of the form ( a, λ; a, λ), where a is a symbol in a given set and λ is the empty string. They correspond to a special class of “null context splicing systems”, as introduced in Head (1987). A series of language-theoretic properties of languages generated by such systems with finite sets of axioms are investigated.

Prefix-Rewriting on Context-Free Groups

A class of groups that has received much attention is the class of context-free groups. This class of groups can be characterized algebraically as well as through some language-theoretical properties as well as through certain combinatorial properties of presentations. Here we use the fact that a finitely generated group is context-free if and only if it admits a finite complete presentation of a certain form that we call a virtually free presentation.

TWO-DIMENSIONAL FINITE STATE RECOGNIZABILITY

The purpose of this paper is to investigate about a new notion of finite state recognizability for two-dimensional (picture) languages. This notion takes as starting point the characterization of one-dimensional recognizable languages in terms of local languages and projections. Such notion can be extended in a natural way to the two-dimensional case. We first introduce a notion of local picture language and then we define,a recognizable picture language as a projection of a local picture language. The family of recognizable picture languages is denoted by REC. We study some combinatorial and language-theoretic properties of family REC. In particular we prove some closure properties with respect to different kinds of operations. From this, we derive that some natural families of two-dimensional languages (finite languages, regular languages, locally testable languages) are recognizable. Further we give some necessary conditions for recognizability which provides tools to show that certain languages are not recognizable. Although REC shares several properties of recognizable string languages, however, differently from the case of words, we prove here that REC is not closed under complementation and that the emptyness problem is undecidable for this family of languages. Finally, we report some characterizations of family REC by means of machine-based models and logic-based formalisms.

Decidability of Regularity and Related Properties of Ground Normal Form Languages

We study language-theoretical properties of the set of reducible ground terms and its complement - the set of ground normal forms induced by a given rewriting system. As a tool for our analysis we introduce the property of finite irreducibility of a term with respect to a variable and prove it to be decidable. It turns out that this property generalizes numerous interesting properties of the language of ground normal forms. In particular, we show that testing regularity of this language can be reduced to verifying this property. In this way we prove the decidability of the regularity of the set of ground normal forms, the problem mentioned in the list of open problems in rewriting by Dershowitz, Jouannaud, and Klop ( in "Proceedings, 4th Conference on Rewriting Techniques and Applications" ( R. V. Book, Ed.), Springer-Verlag, Berlin/New York, 1991). Also, the decidability of the existence of an equivalent ground term rewriting system and some other results are proved.