Drawing on an analogy with temporal fixpoint logic, we relate the arithmetic fixpoint definable sets to the winning positions of certain games, namely games whose winning conditions lie in the difference hierarchy over $\Sigma^0_2$. This both provides a simple characterization of the fixpoint hierarchy, and refines existing results on the power of the game quantifier in descriptive set theory. We raise the problem of transfinite fixpoint hierarchies.

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].

In our main result, we establish a formal connection between Lindström quantifiers with respect to regular languages and the double semidirect product of finite monoids with a distinguished set of generators. We use this correspondence to characterize the expressive power of Lindström quantifiers associated with a class of regular languages.

We prove that for every countable ordinal α one cannot decidewhether a given infinitary rational relation is in the Borel class ${\bf \Sigma_{\alpha}^0}$ (respectively ${\bf \Pi_{\alpha}^0}$ ). Furthermoreone cannotdecide whether a given infinitary rational relation is a Borel set or a ${\bf \Sigma_{1}^1}$ -complete set. We prove some recursive analogues to theseproperties. In particular one cannot decide whether an infinitary rational relation is anarithmetical set.We then deduce from the proof ofthese results some other ones, like: one cannot decide whether thecomplement ofan infinitary rational relation is also an infinitary rational relation.

Motivated by some examples from functional programming, we propose a generalization of the notion of trace to symmetric premonoidal categories and of Conway operators to Freyd categories. We show that in a Freyd category, these notions are equivalent, generalizing a well-known theorem relating traces and Conway operators in Cartesian categories.

Processes in Place/Transition (P/T) nets are defined inductively by a peculiar numbering of place occurrences. Along with an associative sequential composition called catenation and a neutral process, a monoid of processes is obtained. The power algebra of this monoid contains all process languages with appropriate operations on them. Hence the problems of analysis and synthesis, analogous to those in the formal languages and automata theory, arise. Here, the analysis problem is: for a given P/T net with an initial marking find the set of all processes the net may evoke. The synthesis problem is: given a process language L decide if there exists a marked net whose evolutions (represented by processes) are collected in L and, in the positive case, find such net and its initial marking. The problems are posed and given a general solution.

The notion of pseudovarieties of homomorphisms onto finite monoids was recently introduced by Straubing as an algebraic characterization for certain classes of regular languages. In this paper we provide a mechanism of equational description of these pseudovarieties based on an appropriate generalization of the notion of implicit operations. We show that the resulting metric monoids of implicit operations coincide with the standard ones, the only difference being the actual interpretation of pseudoidentities. As an example, an equational characterization of the pseudovariety corresponding to the class of regular languages in AC0 is given.

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].

The aim of this paper is to study the threshold behavior for the satisfiability property of a random k-XOR-CNF formula or equivalently for the consistency of a random Boolean linear system with k variables per equation. For k ≥ 3 we show the existence of a sharp threshold for the satisfiability of a random k-XOR-CNF formula, whereas there are smooth thresholds for k=1 and k=2.

Algebraic systems of equations define functions using recursion where parameter passing is permitted. This generalizes the notion of a rational system of equations where parameter passing is prohibited. It has been known for some time that algebraic systems in Greibach Normal Form have unique solutions. This paper presents a categorical approach to algebraic systems of equations which generalizes the traditional approach in two ways i) we define algebraic equations for locally finitely presentable categories rather than just Set; and ii) we define algebraic equations to allow right-hand sides which need not consist of finite terms. We show these generalized algebraic systems of equations have unique solutions by replacing the traditional metric-theoretic arguments with coalgebraic arguments.

We show that the size of a Las Vegas automaton and the size of a complete, minimal deterministic automaton accepting a regular language are polynomially related. More precisely, we show that if a regular language L is accepted by a Las Vegas automaton having r states such that the probability for a definite answer to occur is at least p, then r ≥ np, where n is the number of the states of the minimal deterministic automaton accepting L. Earlier this result has been obtained in [2] by using a reduction to one-way Las Vegas communication protocols, but here we give a direct proof based on information theory.

Branching programs are a well-established computation model for boolean functions, especially read-once branching programs (BP1s) have been studied intensively. Recently two restricted nondeterministic (parity) BP1 models, called nondeterministic (parity) graph-driven BP1s and well-structured nondeterministic (parity) graph-driven BP1s, have been investigated. The consistency test for a BP-model M is the test whether a given BP is really a BP of model M. Here it is proved that the consistency test is co-NP-complete for nondeterministic (parity) graph-driven BP1s. Moreover, a lower bound technique for nondeterministic graph-driven BP1s is presented. The method generalizes a technique for the well-structured model and is applied in order to answer in the affirmative the open question whether the model of nondeterministic graph-driven BP1s is a proper restriction of nondeterministic BP1s (with respect to polynomial size).

We give a bound for the ω-equivalence problem of polynomially bounded D0L systems which depends only on the size of the underlying alphabet.

It is well known that, given an endofunctor H on a category C , the initial (A+H-)-algebras (if existing), i.e. , the algebras of (wellfounded) H-terms over different variable supplies A, give rise to a monad with substitution as the extension operation (the free monad induced by the functor H). Moss [17] and Aczel, Adámek, Milius and Velebil [12] have shown that a similar monad, which even enjoys the additional special property of having iterations for all guarded substitution rules (complete iterativeness), arises from the inverses of the final (A+H-)-coalgebras (if existing), i.e. , the algebras of non-wellfounded H-terms. We show that, upon an appropriate generalization of the notion of substitution, the same can more generally be said about the initial T'(A,-)-algebras resp. the inverses of the final T'(A,-)-coalgebras for any endobifunctor T' on any category Csuch that the functors T'(-,X) uniformly carry a monad structure.

We describe Wadge degrees of ω-languages recognizable by deterministic Turing machines. In particular, it is shown that the ordinal corresponding to these degrees is ξω where ξ = ω1CK is the first non-recursive ordinal known as the Church–Kleene ordinal. This answers a question raised in [2].

The basic framework of domain μ-calculus was formulated in [39] more than ten years ago.This paper provides an improved formulation of a fragment of the μ-calculus without function space or powerdomain constructions,and studies some open problemsrelated to this μ-calculus such asdecidability and expressive power.A class of language equations is introducedfor encoding μ-formulas in order toderive results related to decidability and expressive power of non-trivial fragments of the domain μ-calculus.The existence and uniqueness of solutions tothis class of language equations constitute an important component of this approach.Our formulation is based on the recent work of Leiss [23], who established a sophisticated framework for solving language equationsusing Boolean automata(a.k.a. alternating automata [12,35]) and a generalized notion of language derivatives.Additionally, the early notion of even-linear grammars is adopted here totreat another fragment of the domain μ-calculus.

We describe the communicating alternating machines and their simulation. We show that, in the case of communicating alternating machines which are bounded, simultaneously, by polynomial time and logarithmic space, the use of three communication levels instead of two does not increase computational power of communicating alternating machines. This resolves an open problem [2] concerning the exact position of machines with three communication levels in the hierarchy.

Episturmian morphisms generalize Sturmian morphisms. They are defined as compositions of exchange morphisms and two particular morphisms L, and R. Epistandard morphisms are the morphisms obtained without considering R. In [14], a general study of these morphims and of conjugacy of morphisms is given. Here, given a decomposition of an Episturmian morphism f over exchange morphisms and {L,R}, we consider two problems: how to compute a decomposition of one conjugate of f; how to compute a list of decompositions of all the conjugates of f when f is epistandard. For each problem, we give several algorithms. Although the proposed methods are fundamently different, we show that some of these lead to the same result. We also give other algorithms, using the same input, to compute for instance the length of the morphism, or its number of conjugates.

We investigate a Gentzen-style proof system for the first-order μ-calculusbased on cyclic proofs, produced by unfolding fixed point formulasand detecting repeated proof goals. Our system uses explicit ordinalvariables and approximations to support a simple semantic inductiondischarge condition which ensures the well-foundedness of inductivereasoning. As the main result of this paper we propose a new syntacticdischarge condition based on traces and establish its equivalencewith the semantic condition. We give an automata-theoretic reformulationof this condition which is more suitable for practical proofs. Fora detailed comparison with previous work we consider two simpler syntacticconditions and show that they are more restrictive than our new condition.