The Rabin index of a rational language of infinite words given by a parity automaton with n states is computable in time O(n2c) where c is the cardinality of the alphabet. The number of values used by a parity acceptance condition is always greater than the Rabin index and conversely, the acceptance condition of a parity automaton can always be replaced by an equivalent acceptance condition whose number of used values is exactly the Rabin index. This new acceptance condition can also be computed in time O(n2c).

We give an explicit criterionfor unavoidability of word sets. We characterize extendible, finitelyand infinitely as well, elements in them. We furnish a reasonable upperbound and an exponential lower bound on the maximum leghth of words in a reduced unavoidable set of agiven cardinality.

Ordered binary decision diagrams (OBDDs) and several more general BDD models have turned out to be representations of Boolean functions which are useful in applications like verification, timing analysis, test pattern generation or combinatorial optimization.The hidden weighted bit function (HWB) is of particular interest, since it seems to be the simplest function with exponential OBDD size. The complexity of this function with respect to different circuit models, formulas, and various BDD models is discussed.

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.

In this paper, the identification of stochastic regular languages is addressed. For this purpose, we propose a class of algorithms which allow for the identification of the structure of the minimal stochastic automaton generating the language. It is shown that the time needed grows only linearly with the size of the sample set and a measure of the complexity of the task is provided. Experimentally, our implementation proves very fast for application purposes.

The notion of solvability in the call-by-value λ-calculus is defined and completely characterized, both from an operational and a logical point of view. The operational characterization is given through a reduction machine, performing the classical β-reduction, according to an innermost strategy. In fact, it turns out that the call-by-value reduction rule is too weak for capturing the solvability property of terms. The logical characterization is given through an intersection type assignment system, assigning types of a given shape to all and only the call-by-value solvable terms.

Ordered binary decision diagrams are an important data structure for the representation of Boolean functions. Typically, the underlying variable ordering is used as an optimization parameter. When finite state machines are represented by OBDDs the state encoding can be used as an additional optimization parameter. In this paper, we analyze the influence of the state encoding on the OBDD-representations of counter-type finite state machines. In particular, we prove lower bounds, derive exact sizes for important encodings and construct a worst-case encoding which leads to exponential-size OBDDs.

Using an inductive definition of normal terms of thetheory of Cartesian ClosedCategories with a given graph of distinguished morphisms, we give a reduction freeproof of the decidability of this theory. This inductive definition enables us toshow via functional completeness that extensions of such a theory by new constants (“indeterminates”) are conservative.

In this paper we give a simple proof that the alternation-depth hierarchy of the μ-calculus for binary trees is strict. The witnesses for this strictness are the automata that determine whether there is a winning strategy for the parity game played on a tree.

A simplified proof is given for the following result due to Lisovik: it is undecidable for two given ε–free finitesubstitutions, whether they are equivalent on the regular languageb{0,1}* c.

We study D0L power series over commutative semirings. We show that a sequence(cn )n≥0 of nonzero elements of a field A is the coefficientsequence of a D0L power series if and only if there exist a positive integerk and integers βi for 1 ≤ i ≤ k such that $c_{n+k}=c_{n+k-1}^{\beta_1}c_{n+k-2}^{\beta_2}\ldots c_n^{\beta_k}$ for alln ≥ 0. As a consequence we solve the equivalence problem of D0L powerseries over computable fields.

We provide an elementary proof of the fixpoint alternation hierarchy in arithmetic, which in turn allows us to simplify the proof of the modal mu-calculus alternation hierarchy. We further show that the alternation hierarchy on the binary tree is strict, resolving a problem of Niwiński.

We define a kind of cellular automaton called a hexagonal partitioned cellular automaton (HPCA), and study logical universality of a reversible HPCA. We give a specific 64-state reversible HPCA H1, and show that a Fredkin gate can be embedded in this cellular space. Since a Fredkin gate is known to be a universal logic element, logical universality of H1 is concluded. Although the number of states of H1 is greater than those of the previous models of reversible CAs having universality, the size of the configuration realizing a Fredkin gate is greatly reduced, and its local transition function is still simple. Comparison with the previous models, and open problems related to these model are also discussed.

We show that some classical P-complete problems can be solved efficiently in average NC. The probabilistic model we consider is the sample space of input descriptions of the problem with the underlying distribution being the uniform one. We present parallel algorithms that use a polynomial number of processors and have expected time upper bounded by (e ln 4 + o(1))log n, asymptotically with high probability, where n is the instance size.

Under the assumption that the Polynomial-TimeHierarchy does not collapsewe show for a regular language L:the unbalanced polynomial-time leaf language class determined by L equals  iff L is existentially but notquantifierfree definable in FO[<, min, max, +1, −1].Furthermore, no suchclass lies properly between NP and co-1-NP or NP⊕co-NP. The proofs rely on a result of Pin and Weilcharacterizing the automata of existentially first-order definable languages.

In this paper we show that neither the set of all valid equationsbetween shuffle expressions nor the set of schemas of valid equations isrecursively enumerable. Thus, neither of the sets can be recursivelygenerated by any axiom system.

The notion of a bisimulation relation is of basic importance in many areas of computation theory and logic. Of late, it has come to take a particular significance in work on the formal analysis and verification of hybrid control systems, where system properties are expressible by formulas of the modal μ-calculus or weaker temporal logics. Our purpose here is to give an analysis of the concept of bisimulation, starting with the observation that the zig-zag conditions are suggestive of some form of continuity. We give a topological characterization of bisimularity for preorders, and then use the topology as a route to examining the algebraic semantics for the µ-calculus, developed in recent work of Kwiatkowska et al., and its relation to the standard set-theoretic semantics. In our setting, μ-calculus sentences evaluate as clopen sets of an Alexandroff topology, rather than as clopens of a (compact, Hausdorff) Stone topology, as arises in the Stone space representation of Boolean algebras (with operators). The paper concludes by applying the topological characterization to obtain the decidability of μ-calculus properties for a class of first-order definable hybrid dynamical systems, slightly extending and considerably simplifying the proof of a recent result of Lafferriere et al.

If we consider words over the alphabet which is the set of all elements of a semigroup S, then such a word determines an element of S: the product of the letters of the word. S is strongly locally testable if whenever two words over the alphabet S have the same factors of a fixed length k, then the products of the letters of these words are equal. We had previously proved [19] that the syntactic semigroup of a rational language L is strongly locally testable if and only if L is both locally and piecewise testable. We characterize in this paper the variety of strongly locally testable semigroups with commuting idempotents and, using the theory of implicit operations on a variety of semigroups, we derive an elementary combinatorial description of the related variety of languages.

A deterministic automaton recognizing a givenω-regular languageis constructed from an ω-regular expressionwith the help of derivatives.The construction is related to Safra's algorithm, in about the same way as the classicalderivative method is related to the subset construction.

Proof systems with sequents of the form U ⊢ Φ for proving validity of a propositional modal μ-calculus formula Φ over a set U of states in a given model usually handle fixed-point formulae through unfolding, thus allowing such formulae to reappear in a proof. Tagging is a technique originated by Winskel for annotating fixed-point formulae with information about the proof states at which these are unfolded. This information is used later in the proof to avoid unnecessary unfolding, without having to investigate the history of the proof. Depending on whether tags are used for acceptance or for rejection of a branch in the proof tree, we refer to “positive” or “negative” tagging, respectively. In their simplest form, tags consist of the sets U at which fixed-point formulae are unfolded. In this paper, we generalise results of earlier work by Andersen et al. which, in the case of least fixed-point formulae, are applicable to singleton U sets only.