Abstract We give an axiomatic description of parallel, synchronous algorithms. Our main result is that every such algorithm can be simulated, step for step, by an abstract state machine with a background that provides for multisets. \Lambda

A first-order sentence θ of vocabulary σ ∪ {S} is successor-invariant in the finite if for every finite σ-structure M and successor relations S1 and S2 on M, (M, S1) | = θ ⇐ ⇒ (M, S2) | = θ. In this paper I give an example of a non-first-order definable class of finite structures which is, however, defined by a successor-invariant first-order sentence. This strengthens a corresponding result for order-invariance in the finite, due to Y. Gurevich. 1

The inductive operators nio (due to Arvind and Biswas) and c-ifp (due to Gire and Hoang) allow for a nondeterministic choice of tuples at each stage in the inductive construction of a relation. We consider the extensions of rst-order logic with each of these operators, presenting a formal semantics for each, in which formulae denote sets of relations. We derive normal forms for these formulae and prove that the operators have equal expressive power. Finally, we show that, by using an appropriate notion of satisfaction for nondeterministic formulae, essentially any computational complexity class de ned in terms of nondeterministic Turing machines operating within polynomial time bounds can be expressed in terms of nondeterministic xedpoint formulae.

Hilbert’s ε-calculus is based on an extension of the language of predicate logic by a term-forming operator εx. Two fundamental results about the ε-calculus, the first and second epsilon theorem, play a rôle similar to that which the cut-elimination theorem plays in sequent calculus. In particular, Herbrand’s Theorem is a consequence of the epsilon theorems. The paper investigates the epsilon theorems and the complexity of the elimination procedure underlying their proof, as well as the length of Herbrand disjunctions of existential theorems obtained by this elimination procedure.

Abstract We provide machine-independent characterizations of some complexity classes, over an arbitrary structure, in the model of computation proposed by L. Blum, M. Shub and S. Smale. We show that the levels of the polynomial hierarchy correspond to safe recursion with predicative minimization. The levels of the digital polynomial hierarchy correspond to safe recursion with digital predicative minimization. Also, we show that polynomial alternating time corresponds to safe recursion with predicative substitutions and that digital polynomial alternating time corresponds to safe recursion with digital predicative substitutions. 1

Gire and Hoang introduce a xed-point logic with a `symmetric ' choice operator that makes a nondeterministic choice from a de nable set of tuples at each stage in the inductive construction of a relation, as long as the set of tuples is an automorphism class of the structure. We present a clean de nition of the syntax and semantics of this logic and investigate its expressive power. We extend the logic of Gire and Hoang with parameterized and nested xed points and rst-order combinations of xed points. We show that the ability to supply parameters to xed points strictly increases the power of the logic. Our logic can express the graph isomorphism problem and we show that, on almost all structures, it captures P , the class of problems decidable in polynomial time by a deterministic Turing machine with an oracle for graph isomorphism.

Specification of system requirements is often involved with ambiguity and nondeterminism. Formal methods tend to mitigate ambiguity but nondeterminism remains as an inherent part of specification.

Abstract. This paper studies the applicability of Hilbert’s ε-calculus in automated reasoning, in particular its incorporation in free-variable tableaux is investigated. Based on this study a careful comparison between δε-tableaux [1] and δ-tableaux [8] becomes possible. Finally, in the spirit of [8], a new liberalization of the δ-rule [3] is defined. This liberalization allows a non-elementary speed-up over δ-tableaux. 1

Abstract. Hilbert’s ε-calculus is based on an extension of the language of predicate logic by a term-forming operator εx. Two fundamental results about the ε-calculus, the first and second epsilon theorem, play a rôle similar to that which the cut-elimination theorem plays in sequent calculus. In particular, Herbrand’s Theorem is a consequence of the epsilon theorems. The paper investigates the epsilon theorems and the complexity of the elimination procedure underlying their proof, as well as the length of Herbrand disjunctions of existential theorems obtained by this elimination procedure. Keywords: Hilbert’s ε-calculus, epsilon theorems, Herbrand’s theorem, proof complexity 1.