This paper discusses the fundamental combinatorial question of whether or not, for a given string α, there exists a morphism σ such that σ is unambiguous with respect to α, i.e. there exists no other morphism τ satisfying τ(α) = σ(α). While Freydenberger et al. [Int. J. Found. Comput. Sci. 17 (2006) 601–628] characterise those strings for which there exists an unambiguous nonerasing morphism σ, little is known about the unambiguity of erasing morphisms, i.e. morphisms that map symbols onto the empty string. The present paper demonstrates that, in contrast to the main result by Freydenberger et al., the existence of an unambiguous erasing morphism for a given string can essentially depend on the size of the target alphabet of the morphism. In addition to this, those strings for which there exists an erasing morphism over an infinite target alphabet are characterised, complexity issues are discussed and some sufficient conditions for the (non-)existence of unambiguous erasing morphisms are given.

We show that some natural refinements of the Straubing and Brzozowskihierarchies correspond (via the so called leaf-languages) step by step tosimilar refinements of the polynomial-time hierarchy. This extends a result of Burtschik and Vollmer on relationship between the Straubing and thepolynomial hierarchies. In particular, this applies to the Boolean hierarchyand the plus-hierarchy.

Based on the formal framework of reaction systems by Ehrenfeucht and Rozenberg[Fund. Inform. 75 (2007) 263–280], reaction automata (RAs)have been introduced by Okubo et al. [Theoret. Comput. Sci.429 (2012) 247–257], as language acceptors with multiset rewritingmechanism. In this paper, we continue the investigation of RAs with a focus on the twomanners of rule application: maximally parallel and sequential. Considering restrictionson the workspace and the λ-input mode, we introduce the correspondingvariants of RAs and investigate their computation powers. In order to explore Turingmachines (TMs) that correspond to RAs, we also introduce a new variant of TMs withrestricted workspace, called s(n)-restricted TMs. Themain results include the following: (i) for a language L and a functions(n), L is accepted by ans(n)-bounded RA with λ-input mode insequential manner if and only if L is accepted by alog s(n)-bounded one-way TM; (ii) if a languageL is accepted by a linear-bounded RA in sequential manner, thenL is also accepted by a P automaton [Csuhaj−Varju and Vaszil, vol. 2597of Lect. Notes Comput. Sci. Springer (2003) 219–233.] in sequentialmanner; (iii) the class of languages accepted by linear-bounded RAs in maximally parallelmanner is incomparable to the class of languages accepted by RAs in sequential manner.

Parikh matrices have become a useful tool for investigation of subword structure of words. Several generalizations of this concept have been considered. Based on the concept of formal power series, we describe a general framework covering most of these generalizations. In addition, we provide a new characterization of binary amiable words – words having a common Parikh matrix.

This paper deals with rational base number systems for p-adic numbers. We mainly focus on the system proposed by Akiyama et al. in 2008, but we also show that this system is in some sense isomorphic to some other rational base number systems by means of finite transducers. We identify the numbers with finite and eventually periodic representations and we also determine the number of representations of a given p-adic number.

We study five extensions of the polymorphically typed lambda-calculus (system F) by type constructs intended to model fixed-points of monotone operators. Building on work by Geuvers concerning the relation between term rewrite systems for least pre-fixed-points and greatest post-fixed-points of positive type schemes (i.e., non-nested positive inductive and coinductive types) and so-called retract types, we show that there are reduction-preserving embeddings even between systems of monotone (co)inductive types and non-inter leav ing positive fixed-point types (which are essentially those retract types). The reduction relation considered is β- and η-reduction for system FF plus either (full) primitive recursion on the inductive types or (full) primitive corecursion on the coinductive types or an extremely simple rule for the fixed-point types. Monotonicity is not confined to the syntactic restriction on type formation of having only positive occurrences of the type variable α in ρ for the inductive type µαρ or the coinductive type ναρ. Instead of that only a “monotonicity witness” which is a term of type ∀α∀β.(α → β) → ρ → ρ[α:=β] is required. This term may already use (co)recursion such that our monotone (co)inductive types may even be “interleaved” and not only nested.

We first prove an extremal property of the infinite Fibonacci* word f: the family of the palindromic prefixes {hn | n ≥ 6} of f is not only a circular code but “almost” a comma-free one (see Prop. 12 in Sect. 4). We also extend to a more general situation the notion of a necklace introduced for the study of trinucleotides codes on the genetic alphabet, and we present a hierarchy relating two important classes of codes, the comma-free codes and the circular ones.

In recent work we have proposed a novel approach to define idealized type systems for object-oriented languages, based on abstract compilation ofprograms into Horn formulas which are interpreted w.r.t. the coinductive (that is, the greatest) Herbrand model. In this paper we investigate how this approach can be applied also inthe presence of imperative features. This is made possible by considering a natural translation of Static Single Assignment intermediate form programs into Horn formulas, where φ functionscorrespond to union types.

This paper proposes an operational semantics for value recursion in the context of monadic metalanguages. Our technique for combining value recursion with computational effects works uniformly for all monads. The operational nature of our approach is related to the implementation of recursion in Scheme and its monadic version proposed by Friedman and Sabry, but it defines a different semantics and does not rely on assignments. When contrasted to the axiomatic approach proposed by Erkök and Launchbury, our semantics for the continuation monad invalidates one of the axioms, adding to the evidence that this axiom is problematic in the presence of continuations.

A word u defined over an alphabet $\mathcal A$ is c-balanced (c∈$\mathbb N$) if for all pairs of factors v, w of u of the same length and for all letters a∈$\mathcal A$, the difference between the number of letters a in v and w is less or equal to c. In this paper we consider a ternary alphabet $\mathcal A$ = {L, S, M} and a class of substitutions $\varphi_p$ defined by $\varphi_p$(L) = LpS, $\varphi_p$(S) = M, $\varphi_p$(M) = Lp–1S where p> 1. We prove that the fixed point of $\varphi_p$, formally written as $\varphi_p^\infty$(L), is 3-balanced and that its Abelian complexity is bounded above by the value 7, regardless of the value of p. We also show that both these bounds are optimal, i.e. they cannot be improved.

Motivated by the wavelength division multiplexing in all-optical networks, we consider the problem of finding an optimal (with respect to the least possible number of wavelengths) set of ƒ+1 internally node disjoint dipaths connecting all pairs of distinct nodes in the binary r-dimensional hypercube, where 0 ≤ ƒ < r. This system of dipaths constitutes a routing protocol that remains functional in the presence of up to ƒ faults (of nodes and/or links). The problem of constructing such protocols for general networks was mentioned in [1]. We compute precise values of ƒ-wise arc forwarding indexes and give (describe dipaths and color them) nearly optimal all-to-all ƒ-fault tolerant protocols for the hypercube network. Our results generalize corresponding results from [1, 4, 14].

Let TsH be the graph obtained from a given graph H by subdividing each edge s times. Motivated by a problem raised by Igor Pak [Mixing time and long paths in graphs, in Proc. of the 13th annual ACM-SIAM Symposium on Discrete Algorithms (SODA 2002) 321–328], we prove that, for any graph H, there exist graphs G with O(s) edges that are Ramsey with respect to TsH.

We study the decidability of the following problem: given p affine functions ƒ1,...,ƒp over $\mathbb{N}^k$ and two vectors $v_1, v_2\in\mathbb{N}^k$, is v2 reachable from v1 by successive iterations of ƒ1,...,ƒp (in this given order)? We show that this question is decidable for p = 1, 2 and undecidable for some fixed p.

A reversible automaton is a finite automaton in which eachletter induces a partial one-to-one map from the set of states intoitself. We solve the following problem proposed by Pin. Given an alphabet A, does there exist a sequence of languages K n on A which can be accepted by a reversible automaton, and such that the number of states of the minimal automaton of K n is in O(n), while the minimal number of states of a reversible automaton accepting K n is in O(ρ n ) for some ρ > 1? We give such an example with $\rho=\left(\frac{9}{8}\right)^{\frac{1}{12}}$ .

We prove that the cutwidth of the r-dimensional mesh of d-ary trees is of order $\Theta(d^{(r-1)n+1})$, which improves and generalizes previous results.


A shuffle ideal is a language which is a finite union of languages of the form A*a1A*...A*ak where A is a finite alphabet and the ai's are letters. We show how to represent shuffle ideals by special automata and how to compute these representations. We also give a temporal logic characterization of shuffle ideals and we study its expressive power over infinite words. We characterize the complexity of deciding whether a language is a shuffle ideal and we give a new quadratic algorithm for this problem. Finally we also present a characterization by subwords of the minimal automaton of a shuffle ideal and study the complexity of basic operations on shuffle ideals.

We introduce a new model of cellular automaton called a one-dimensional number-conserving partitioned cellular automaton (NC-PCA). An NC-PCA is a system such that a state of a cell is represented by a triple of non-negative integers, and the total (i.e., sum) of integers over the configuration is conserved throughout its evolving (computing) process. It can be thought as a kind of modelization of the physical conservation law of mass (particles) or energy. We also define a reversible version of NC-PCA, and prove that a reversible NC-PCA is computation-universal. It is proved by showing that a reversible two-counter machine, which has been known to be universal, can be simulated by a reversible NC-PCA.

Computing the image of a regular language by the transitive closure of a relation is a central question in regular model checking. In a recent paper Bouajjani et al. [IEEE Comput. Soc. (2001) 399–408] proved that the class of regular languages L – called APC – of the form UjL0,jL1,jL2,j...Lkj,j, where the union is finite and each Li,j is either a single symbol or a language of the form B* with B a subset of the alphabet, is closed under all semi-commutation relations R. Moreover a recursive algorithm on the regular expressions was given to compute R*(L). This paper provides a new approach, based on automata, for the same problem. Our approach produces a simpler and more efficient algorithm which furthermore works for a larger class of regular languages closed under union, intersection, semi-commutation relations and conjugacy. The existence of this new class, PolC, answers the open question proposed in the paper of Bouajjani et al.

Cover automata for finite languages have been much studied a few years ago. It turns out that a simple mathematical structure, namely similarity relations over a finite set of words, is underlying these studies. In the present work, we investigate in detail for themselves the properties of these relations beyond the scope of finite languages. New results with straightforward proofs are obtained in this generalized framework, and previous results concerning cover automata are obtained as immediate consequences.

We prove in this paper that there exists some infinitary rational relations which are analytic but non Borel sets, giving an answer to a question of Simonnet [20].