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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 non-trivial application problems e.g. in parametric i...

We present an implementation and verification in higher-order logic of Cooper’s quantifier elimination for Presburger arithmetic. Reflection, i.e. the direct execution in ML, yields a speed-up of a factor of 200 over an LCF-style implementation and performs as well as a decision procedure hand-coded in ML.

We present a formally verified quantifier elimination procedure for the first order theory over linear mixed real-integer arithmetics in higher-order 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.

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 proof-producing functional program, and once by reflection, i.e. by computations inside the logic rather than in the meta-language. 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

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 second-order 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...

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].

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 first-order (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 first-order logic; recent