International Journal of Foundations of Computer Science

We study FAD-languages, which are regular languages defined by finite sets of forbidden factors, together with their “canonical” recognizing automata. We are mainly interested in the possible asymptotic orders of growth for such languages. We analyze certain simplifications of sets of forbidden factors and show that they “almost” preserve the canonical automata. Using this result and structural properties of canonical automata, we describe an algorithm that effectively lists all canonical automata having a sink strong component isomorphic to a given digraph, or reports that no such automata exist. This algorithm can be used, in particular, to prove the existence of a FAD-language over a given alphabet with a given exponential growth rate. On the other hand, we give an example showing that the algorithm cannot prove non-existence of a FAD-language having a given growth rate. Finally, we provide some examples of canonical automata with a nontrivial condensation graph and of FAD-languages with a “complex” order of growth.

Weighted automata model quantitative aspects of systems like the consumption of resources during executions. Traditionally, the weights are assumed to form the algebraic structure of a semiring, but recently also other weight computations like average have been considered. Here, we investigate quantitative context-free languages over very general weight structures incorporating all semirings, average computations, lattices. In our main result, we derive the Chomsky-Schützenberger Theorem for such quantitative context-free languages, showing that each arises as the image of the intersection of a Dyck language and a recognizable language under a suitable morphism. Moreover, we show that quantitative context-free languages are expressively equivalent to a model of weighted pushdown automata. This generalizes results previously known only for semirings. We also investigate under which conditions quantitative context-free languages assume only finitely many values.

The standard regular expressions over finite alphabets have been widely accepted as the most basic formalism to query graph databases. However, the major drawback of this approach is that it ignores the presence of data. In this paper we study the so called regular expressions with binding (REWB), that is, regular expressions equipped with variables to store data within a well defined scope. In particular, we study the complexity of the query evaluation of REWB queries over graph databases.

We define a model of advised computation by finite automata where the advice is provided on a separate tape. We consider several variants of the model where the advice is deterministic or randomized, the input tape head is allowed real-time, one-way, or two-way access, and the automaton is classical or quantum. We prove several separation results among these variants, demonstrate an infinite hierarchy of language classes recognized by automata with increasing advice lengths, and establish the relationships between this and the previously studied ways of providing advice to finite automata.

We give a Kleene-type operational characterization of Muller context-free languages (MCFLs) of well-ordered and scattered words.

A two-dimensional code of pictures is defined as a set X ⊆ Σ** such that any picture over Σ is tilable in at most one way with pictures in X. It is proved that in general it is undecidable whether a finite set of picture is a code. The subclass of prefix codes is introduced and it is proved that it is decidable whether a finite set of pictures is a prefix code. Further a polynomial time decoding algorithm for finite prefix codes is given. Maximality and completeness of finite prefix codes are studied.

The Dual Post Correspondence Problem asks whether, for a given word α, there exists a pair of distinct morphisms σ,τ, one of which needs to be non-periodic, such that σ(α) = τ(α) is satisfied. This problem is important for the research on equality sets, which are a vital concept in the theory of computation, as it helps to identify words that are in trivial equality sets only.

Little is known about the Dual PCP for words α over larger than binary alphabets, especially for so-called ratio-primitive examples. In the present paper, we address this question in a way that simplifies the usual method, which means that we can reduce the intricacy of the word equations involved in dealing with the Dual PCP. Our approach yields large sets of words for which there exists a solution to the Dual PCP as well as examples of words over arbitrary alphabets for which such a solution does not exist.

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) = U_i≥0 ♠ⁱ(w) of a word w.

We consider the so-called measure once finite quantum automata model introduced by Moore and Crutchfield in 2000. We show that given a language recognized by such a device and a linear context-free language, it is decidable whether or not they have a nonempty intersection. This extends a result of Blondel et al. which can be interpreted as solving the problem with the free monoid in place of the family of linear context-free languages.

Multi-pushdown systems are formal models of multi-threaded programs. As they are Turing powerful in their full generality, several decidable subclasses, constituting under-approximations of the original system, have been studied in the recent years. Ordered Multi-Pushdown Systems (OMPDSs) impose an order on the stacks and limit pop actions to the lowest non-empty stack. The control state reachability for OMPDSs is 2-ETIME-COMPLETE. We propose a restriction on OMPDSs, called Adjacent OMPDSs (AOMPDS), where values may be pushed only on the lowest non-empty stack or one of its two neighbours. We describe EXPTIME decision procedures for reachability and LTL model-checking and establish matching lower bounds. We demonstrate the utility of this model as an algorithmic tool via optimal reductions from other models.

In the literature, many bijections between (labeled) Motzkin paths and various other combinatorial objects are studied. We consider abelian (un)bordered words and show the connection with irreducible symmetric Motzkin paths and paths in ℤ not returning to the origin. This study can be extended to abelian unbordered words over an arbitrary alphabet and we derive expressions to compute the number of these words. In particular, over a 3-letter alphabet, the connection with paths in the triangular lattice is made. Finally, we characterize the lengths of the abelian unbordered factors occurring in the Thue–Morse word using some kind of automatic theorem-proving provided by a logical characterization of the k-automatic sequences.

We extend the concept of factorization on finite words to ω-rational languages and show how to compute them. We define a normal form for Büchi automata and introduce a universal automaton for Büchi automata in normal form. We prove that, for every ω-rational language, this Büchi automaton, based on factorization, is canonical and that it is the smallest automaton that contains the morphic image of every equivalent Büchi automaton in normal form.

Almost all the decision questions concerning the resource requirements of a computational device are undecidable. Here we want to understand the exact boundary that separates the undecidable from the decidable cases of such problems by considering the time complexity of very simple devices that include NFAs (1-way and 2-way), PDAs and PDAs augmented with counters - and their unambiguous restrictions. We consider several variations - based on whether the bound holds exactly or as an upper-bound and show decidability as well as undecidability results. In the case of decidable problems, we also attempt to determine more precisely the complexity class to which the problem belongs.

Stateless variants of deterministic one-way multi-head finite automata with pebbles, that is, automata where the heads can drop, sense, and pick up pebbles, are studied. The relation between heads and pebbles is investigated, and a proper double hierarchy concerning these two resources is obtained. Moreover, it is shown that a conversion of an arbitrary automaton to a stateless automaton can always be achieved at the cost of additional heads and/or pebbles. On the other hand, there are languages where one head cannot be traded for any number of additional pebbles and vice versa. Finally, the emptiness problem and related problems are shown to be undecidable even for the ‘simplest’ model, namely, for stateless one-way finite automata with two heads and one pebble.

In this paper we are interested in the study of the combinatorial aspects related to the extension of the Burrows-Wheeler transform to a multiset of words. Such study involves the notion of suffixes and conjugates of words and is based on two different order relations, denoted by <_lex and ≺_ω, that, even if strictly connected, are quite different from the computational point of view. In particular, we introduce a method that only uses the <_lex sorting among suffixes of a multiset of words in order to sort their conjugates according to ≺_ω-order. In this study an important role is played by Lyndon words. This strategy could be used in applications specially in the field of Bioinformatics, where for instance the advent of “next-generation” DNA sequencing technologies has meant that huge collections of DNA sequences are now commonplace.