Results 1  10
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12
Abstract state machines capture parallel algorithms
 ACM Transactions on Computational Logic
, 2003
"... 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 ..."
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Cited by 58 (23 self)
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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
Successorinvariance in the finite
 In Proceedings of the 18th IEEE Symposium on Logic in Computer Science (LICS’03
, 2003
"... A firstorder sentence θ of vocabulary σ ∪ {S} is successorinvariant in the finite if for every finite σstructure M and successor relations S1 and S2 on M, (M, S1)  = θ ⇐ ⇒ (M, S2)  = θ. In this paper I give an example of a nonfirstorder definable class of finite structures which is, however, ..."
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Cited by 4 (2 self)
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A firstorder sentence θ of vocabulary σ ∪ {S} is successorinvariant in the finite if for every finite σstructure M and successor relations S1 and S2 on M, (M, S1)  = θ ⇐ ⇒ (M, S2)  = θ. In this paper I give an example of a nonfirstorder definable class of finite structures which is, however, defined by a successorinvariant firstorder sentence. This strengthens a corresponding result for orderinvariance in the finite, due to Y. Gurevich. 1
FixedPoint Logics with Nondeterministic Choice
 Journal of Logic and Computation
"... The inductive operators nio (due to Arvind and Biswas) and cifp (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 rstorder logic with each of these operators, presenting a formal semant ..."
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Cited by 3 (3 self)
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The inductive operators nio (due to Arvind and Biswas) and cifp (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 rstorder 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.
The epsilon calculus and Herbrand Complexity
 STUDIA LOGICA
, 2006
"... Hilbert’s εcalculus is based on an extension of the language of predicate logic by a termforming operator εx. Two fundamental results about the εcalculus, the first and second epsilon theorem, play a rôle similar to that which the cutelimination theorem plays in sequent calculus. In particular ..."
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Cited by 2 (0 self)
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Hilbert’s εcalculus is based on an extension of the language of predicate logic by a termforming operator εx. Two fundamental results about the εcalculus, the first and second epsilon theorem, play a rôle similar to that which the cutelimination 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.
Tailoring Recursion to Characterize NonDeterministic Complexity Classes Over Arbitrary Structures
 in "3rd IFIP International Conference on Theoretical Computer Science  TCS’2004
, 2004
"... Abstract We provide machineindependent 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. Th ..."
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Abstract We provide machineindependent 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
A FixedPoint Logic with Symmetric Choice
"... Gire and Hoang introduce a xedpoint 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 clea ..."
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Gire and Hoang introduce a xedpoint 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 rstorder 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.
Making Nondeterminism Explicit in Z
"... 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. ..."
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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.
SUCCESSORINVARIANT FIRSTORDER LOGIC ON FINITE STRUCTURES
"... Abstract. We consider successorinvariant firstorder logic (FO + succ)inv, consisting of sentences Φ involving an “auxiliary ” binary relation S such that (A, S1)  = Φ ⇐⇒ (A, S2)  = Φ for all finite structures A and successor relations S1, S2 on A. A successorinvariant sentence Φ has a welldefin ..."
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Abstract. We consider successorinvariant firstorder logic (FO + succ)inv, consisting of sentences Φ involving an “auxiliary ” binary relation S such that (A, S1)  = Φ ⇐⇒ (A, S2)  = Φ for all finite structures A and successor relations S1, S2 on A. A successorinvariant sentence Φ has a welldefined semantics on finite structures A with no given successor relation: one simply evaluates Φ on (A, S) for an arbitrary choice of successor relation S. In this article, we prove that (FO + succ)inv is more expressive on finite structures than firstorder logic without a successor relation. This extends similar results for orderinvariant logic [7] and epsiloninvariant logic [10]. §1. Introduction. Let σ and τ be disjoint relational vocabularies, and let C be an isomorphismclosed class of τstructures, which we call “auxiliary” structures. A sentence Φ in the firstorder language of σ ∪ τ is Cinvariant if A  = Φ ⇐ ⇒ B  = Φ for all (σ ∪τ)structures A and B such that Aσ = Bσ and Aτ, Bτ ∈ C where Aσ denotes the σreduct of A (and likewise: Aτ, Bσ, Bτ).
Epsilon, Delta, and SpeedUps WORK in PROGRESS
"... Abstract. This paper studies the applicability of Hilbert’s εcalculus in automated reasoning, in particular its incorporation in freevariable tableaux is investigated. Based on this study a careful comparison between δεtableaux [1] and δtableaux [8] becomes possible. Finally, in the spirit of [8 ..."
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Abstract. This paper studies the applicability of Hilbert’s εcalculus in automated reasoning, in particular its incorporation in freevariable tableaux is investigated. Based on this study a careful comparison between δεtableaux [1] and δtableaux [8] becomes possible. Finally, in the spirit of [8], a new liberalization of the δrule [3] is defined. This liberalization allows a nonelementary speedup over δtableaux. 1