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Decision procedures for recursive data structures with integer constraints
 In International Joint Conference on Automated Reasoning, volume 3097 of LNCS
, 2004
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Complexity and Uniformity of Elimination in Presburger Arithmetic
 UNIVERSITAT PASSAU
, 1997
"... The decision complexity of Presburger Arithmetic PA and its variants has received much attention in the literature. We investigate the complexity of quantifier elimination procedures for PA  a topic that is even more relevant for applications. First we show that the the author's triply exponential ..."
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Cited by 13 (3 self)
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The decision complexity of Presburger Arithmetic PA and its variants has received much attention in the literature. We investigate the complexity of quantifier elimination procedures for PA  a topic that is even more relevant for applications. First we show that the the author's triply exponential upper bound is essentially tight. This fact seems to preclude practical applications. By weakening the concept of quantifier elimination slightly to bounded quantifier elimination, we show, however, that the upper and lower bound for quantifier elimination in PA can be lowered by exactly one exponential. Moreover we gain uniformity in the coefficients, a property that we prove to be impossible for complete quantifier elimination in PA. Thus we have tight upper and lower complexity bounds for elimination theory in PA and uniform PA. The results are inspired by experimental implementations of bounded quantifier elimination that have solved nontrivial application problems e.g. in parametric i...
Verifying and reflecting quantifier elimination for Presburger arithmetic
 LOGIC FOR PROGRAMMING, ARTIFICIAL INTELLIGENCE, AND REASONING
, 2005
"... We present an implementation and verification in higherorder logic of Cooper’s quantifier elimination for Presburger arithmetic. Reflection, i.e. the direct execution in ML, yields a speedup of a factor of 200 over an LCFstyle implementation and performs as well as a decision procedure handcode ..."
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Cited by 11 (7 self)
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We present an implementation and verification in higherorder logic of Cooper’s quantifier elimination for Presburger arithmetic. Reflection, i.e. the direct execution in ML, yields a speedup of a factor of 200 over an LCFstyle implementation and performs as well as a decision procedure handcoded in ML.
Knowledge Representation and Classical Logic
"... Mathematical logicians had developed the art of formalizing declarative knowledge long before the advent of the computer age. But they were interested primarily in formalizing mathematics. Because of the important role of nonmathematical knowledge in AI, their emphasis was too narrow from the perspe ..."
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Cited by 10 (4 self)
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Mathematical logicians had developed the art of formalizing declarative knowledge long before the advent of the computer age. But they were interested primarily in formalizing mathematics. Because of the important role of nonmathematical knowledge in AI, their emphasis was too narrow from the perspective of knowledge representation, their formal languages were not sufficiently expressive. On the other hand, most logicians were not concerned about the possibility of automated reasoning; from the perspective of knowledge representation, they were often too generous in the choice of syntactic constructs. In spite of these differences, classical mathematical logic has exerted significant influence on knowledge representation research, and it is appropriate to begin this handbook with a discussion of the relationship between these fields. The language of classical logic that is most widely used in the theory of knowledge representation is the language of firstorder (predicate) formulas. These are the formulas that John McCarthy proposed to use for representing declarative knowledge in his advice taker paper [176], and Alan Robinson proposed to prove automatically using resolution [236]. Propositional logic is, of course, the most important subset of firstorder logic; recent
Verifying mixed realinteger quantifier elimination
 IJCAR 2006, LNCS 4130
, 2006
"... We present a formally verified quantifier elimination procedure for the first order theory over linear mixed realinteger arithmetics in higherorder logic based on a work by Weispfenning. To this end we provide two verified quantifier elimination procedures: for Presburger arithmitics and for lin ..."
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Cited by 8 (5 self)
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We present a formally verified quantifier elimination procedure for the first order theory over linear mixed realinteger arithmetics in higherorder logic based on a work by Weispfenning. To this end we provide two verified quantifier elimination procedures: for Presburger arithmitics and for linear real arithmetics.
Proof synthesis and reflection for linear arithmetic. Submitted
, 2006
"... This article presents detailed implementations of quantifier elimination for both integer and real linear arithmetic for theorem provers. The underlying algorithms are those by Cooper (for Z) and by Ferrante and Rackoff (for R). Both algorithms are realized in two entirely different ways: once in ta ..."
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Cited by 6 (5 self)
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This article presents detailed implementations of quantifier elimination for both integer and real linear arithmetic for theorem provers. The underlying algorithms are those by Cooper (for Z) and by Ferrante and Rackoff (for R). Both algorithms are realized in two entirely different ways: once in tactic style, i.e. by a proofproducing functional program, and once by reflection, i.e. by computations inside the logic rather than in the metalanguage. Both formalizations are highly generic because they make only minimal assumptions w.r.t. the underlying logical system and theorem prover. An implementation in Isabelle/HOL shows that the reflective approach is between one and two orders of magnitude faster. 1
Implementing WS1S via Finite Automata
 In Automata Implementation, WIA '96, Proceedings, volume 1260 of LNCS
, 1997
"... It has long been known that WS1S is decidable through the use of finite automata. However, since the worst case running time has been proven to grow extremely quickly, few have explored the implementation of the algorithm. In this paper we describe some of the points of interest that have come up ..."
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Cited by 6 (0 self)
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It has long been known that WS1S is decidable through the use of finite automata. However, since the worst case running time has been proven to grow extremely quickly, few have explored the implementation of the algorithm. In this paper we describe some of the points of interest that have come up while coding and running the algorithm. These points include the data structures used as well as the special properties of the automata, which we can exploit to perform minimization very quickly in certain cases. We also present some data that enable us to gain insight into how the algorithm performs in the average case, both on random inputs ans on inputs that come from the use of Presburger Arithmetic (which can be converted to WS1S) in compiler optimization. 1 Introduction 1.1 Definitions 1.1.1 WS1S The language L S1S is the secondorder predicate calculus ranging over the natural numbers, with variables x 1 ; X 1 ; x 2 ; X 2 ; : : : (to represent numbers and sets of numbers), r...
On the Complexity of the Theories of Weak Direct Products
, 1974
"... Let N be the set of nonnegative integers and let < N ,+> be the weak direct product of < N,+> with itself. Mostowski[ 9 ] shows that the theory of < N ,+> is decidable, but his decision procedure isn't elementary recursire. We present here a more efficient procedure which operates 2 cn within spa ..."
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Cited by 2 (0 self)
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Let N be the set of nonnegative integers and let < N ,+> be the weak direct product of < N,+> with itself. Mostowski[ 9 ] shows that the theory of < N ,+> is decidable, but his decision procedure isn't elementary recursire. We present here a more efficient procedure which operates 2 cn within space 2 2 . As corollaries we obtain the same upper bound for the theory of finite abelJan groups, the theory of finitely generated abelian groups, and the theory of the structure < N ,' > of positive integers under multiplication. Fischer and Rabin have shown that the theory of <N ,+> 2 dn requires time 2 on nondeterministic Turing machines [5].
On the Satisfiability of Modular Arithmetic Formulae
"... Abstract. Modular arithmetic is the underlying integral computation model in conventional programming languages. In this paper, we discuss the satisfiability problem of propositional formulae in modular arithmetic can be obtained by solving alternationfree Presburger arithmetic, it is easy to see t ..."
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Cited by 1 (0 self)
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Abstract. Modular arithmetic is the underlying integral computation model in conventional programming languages. In this paper, we discuss the satisfiability problem of propositional formulae in modular arithmetic can be obtained by solving alternationfree Presburger arithmetic, it is easy to see that the problem is in fact NPcomplete. Further, we give an efficient reduction to integer programming with the number of constraints and variables linear in the length of the given linear modular arithmetic formula. For nonlinear modular arithmetic formulae, an additional factor of ω is needed. With the advent of efficient integer programming packages, our algorithm could be useful to software verification in practice. over the finite ring Z2 ω. Although an upper bound of 22O(n4) 1
INTEGRATION OF DECISION PROCEDURES INTO HIGHORDER INTERACTIVE PROVERS
, 2006
"... An efficient proof assistant uses a wide range of decision procedures, including automatic verification of validity of arithmetical formulas with linear terms. Since the final product of a proof assistant is a formalized and verified proof, it prompts an additional task of building proofs of formula ..."
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An efficient proof assistant uses a wide range of decision procedures, including automatic verification of validity of arithmetical formulas with linear terms. Since the final product of a proof assistant is a formalized and verified proof, it prompts an additional task of building proofs of formulas, which validity is established by such a decision procedure. We present an implementation of several decision procedures for arithmetical formulas with linear terms in the MetaPRL proof assistant in a way that provides formal proofs of formulas found valid by those procedures. We also present an implementation of a theorem prover for the logic of justified common knowledge S4 J n introduced in [Artemov, 2004]. This system captures the notion of justified common knowledge, which is free of some of the deficiencies of the usual common knowledge operator, and is yet sufficient for the analysis of epistemic problems where common knowledge has been traditionally applied. In particular, S4 J n enjoys cutelimination, which introduces the possibility of automatic proof search in the logic of common