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24
Theorem Proving in Cancellative Abelian Monoids
, 1996
"... We describe a refined superposition calculus for cancellative abelian monoids. They encompass not only abelian groups, but also such ubiquitous structures as the natural numbers or multisets. Both the AC axioms and the cancellation law are difficult for a general purpose superposition theorem prover ..."
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We describe a refined superposition calculus for cancellative abelian monoids. They encompass not only abelian groups, but also such ubiquitous structures as the natural numbers or multisets. Both the AC axioms and the cancellation law are difficult for a general purpose superposition theorem prover, as they create many variants of clauses which contain sums. Our calculus requires neither explicit inferences with the theory clauses for cancellative abelian monoids nor extended equations or clauses. Improved ordering constraints allow us to restrict to inferences that involve the maximal term of the maximal sum in the maximal literal. Furthermore, the search space is reduced drastically by certain variable elimination techniques. Keywords Automated Theorem Proving, FirstOrder Logic, Superposition, Cancellative Abelian Monoids, Associativity, Commutativity, Variable Elimination, Term Rewriting. 1 Introduction To be useful in applications such as program verification and synthesis, a...
Orderings, ACTheories and Symbolic Constraint Solving (Extended Abstract)
 In Tenth Annual IEEE Symposium on Logic in Computer Science
, 1995
"... We design combination techniques for symbolic constraint solving in the presence of associative and commutative (AC) function symbols. This yields an algorithm for solving ACRPO constraints (where ACRPO is the ACcompatible total reduction ordering of [16]), which was a missing ingredient for autom ..."
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Cited by 7 (4 self)
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We design combination techniques for symbolic constraint solving in the presence of associative and commutative (AC) function symbols. This yields an algorithm for solving ACRPO constraints (where ACRPO is the ACcompatible total reduction ordering of [16]), which was a missing ingredient for automated deduction strategies with ACconstraint inheritance [15, 19]. As in the ACunification case (actually the ACunification algorithm of [9] is an instance of ours), for this purpose we first study the pure case, i.e. we show how to solve ACordering constraints built over a single AC function symbol and variables. Since ACRPO is an interpretationbased ordering, our algorithm also requires the combination of algorithms for solving interpreted constraints and noninterpreted constraints.
On The Use Of Constraints In Automated Deduction
, 1995
"... . This paper presents three approaches dealing with constraints in automated deduction. Each of them illustrates a different point. The expression of strategies using constraints is shown through the example of a completion process using ordered and basic strategies. The schematization of complex un ..."
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Cited by 7 (1 self)
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. This paper presents three approaches dealing with constraints in automated deduction. Each of them illustrates a different point. The expression of strategies using constraints is shown through the example of a completion process using ordered and basic strategies. The schematization of complex unification problems through constraints is illustrated by the example of an equational theorem prover with associativity and commutativity axioms. The incorporation of builtin theories in a deduction process is done for a narrowing process which solves queries in theories defined by rewrite rules with builtin constraints. Advantages of using constraints in automated deduction are emphasized and new challenging problems in this area are pointed out. 1 Motivations Constraints have been introduced in automated deduction since about 1990, although one could find similar ideas in theory resolution [32] and in higherorder resolution [16]. The idea is to distinguish two levels of deduction and t...
Reducing ACTermination to Termination
 Proc. 23rd MFCS, LNCS 1450
, 1997
"... We present a new technique for proving ACtermination. We show that if certain conditions are met, ACtermination can be reduced to termination, i. e., termination of a TRS S modulo an ACtheory can be inferred from termination of another TRS R with no ACtheory involved. This is a new perspective a ..."
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We present a new technique for proving ACtermination. We show that if certain conditions are met, ACtermination can be reduced to termination, i. e., termination of a TRS S modulo an ACtheory can be inferred from termination of another TRS R with no ACtheory involved. This is a new perspective and opens new possibilities to deal with ACtermination. 1 Introduction Termination of term rewriting systems (TRS's) is crucial for the use of rewriting in proofs and computations, and many theories have been developed in this field, especially for the case where function symbols do not obey any particular law or property. However, many interesting and useful systems have operators which are associative and commutative (AC), and most techniques developed for proving termination of TRS's do not carry over to rewriting modulo equational theories so that the theory developed to study termination of TRS's needs to be adapted to the equational case. Along these lines, a lot of work has been done...
Proving AssociativeCommutative Termination Using RPOcompatible Orderings
 in Proc. Automated Deduction in Classical and NonClassical Logics, LNAI 1761
, 2000
"... Developing path orderings for associativecommutative (AC) rewrite systems has been quite a challenge at least for a decade. Compatibility with the recursive path ordering (RPO) schemes is desirable, and this property helps in orienting the commonly encountered distributivity axiom as desired. For a ..."
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Developing path orderings for associativecommutative (AC) rewrite systems has been quite a challenge at least for a decade. Compatibility with the recursive path ordering (RPO) schemes is desirable, and this property helps in orienting the commonly encountered distributivity axiom as desired. For applications in theorem proving and constraint solving, a total ordering on ground terms involving AC operators is often required. It is shown how the main solutions proposed so far ([7],[13]) with the desired properties can be viewed as arising from a common framework. A general scheme that works for nonground (general) terms also is proposed. The proposed definition allows flexibility (using different abstractions) in the way the candidates of a term with respect to an associativecommutative function symbol are compared, thus leading to at least two distinct orderings on terms (from the same precedence relation on function symbols).
An ACCompatible KnuthBendix Order
"... We introduce a family of ACcompatible KnuthBendix simplification orders which are ACtotal on ground terms. Our orders preserve attractive features of the original KnuthBendix orders such as existence of a polynomialtime algorithm for comparing terms; computationally e#cient approximations, for ..."
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We introduce a family of ACcompatible KnuthBendix simplification orders which are ACtotal on ground terms. Our orders preserve attractive features of the original KnuthBendix orders such as existence of a polynomialtime algorithm for comparing terms; computationally e#cient approximations, for instance comparing weights of terms; and preference of light terms over heavy ones. This makes these orders especially suited for automated deduction where e#cient algorithms on orders are desirable.
AssociativeCommutative Reduction Orderings via HeadPreserving Interpretations
, 1995
"... We introduce a generic definition of reduction orderings on term algebras containing associativecommutative (hereafter denoted AC) operators. These orderings are compatible with the AC theory hence makes them suitable for use in deduction systems where AC operators are builtin. Furthermore, they ..."
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We introduce a generic definition of reduction orderings on term algebras containing associativecommutative (hereafter denoted AC) operators. These orderings are compatible with the AC theory hence makes them suitable for use in deduction systems where AC operators are builtin. Furthermore, they have the nice property of being total on AC classes of ground terms, a required property for example to avoid failure in ACcompletion, or to insure completeness of ordered strategies in firstorder theorem proving with builtin AC operators. We show that the two definitions already known of such total and ACcompatible orderings [24, 25] are actually instances of our definition. Finally, we find new such orderings which have more properties, first an ordering based on an integer polynomial interpretation, answering positively to a question left open by Narendran and Rusinowitch, and second an ordering which allow to orient the distributivity axiom in the usual way, answering positively to a ...
Combination of Compatible Reduction Orderings that are Total on Ground Terms (Extended Abstract)
 In 12th Ann. IEEE Symp. on Logic in Computer Science
, 1997
"... Franz Baader LuFg Theoretical Computer Science, RWTH Aachen Ahornstraße 55, 52074 Aachen, Germany email: baader@informatik.rwthaachen.de 1 Introduction Reduction orderings that are total on ground terms play an important role in many areas of automated deduction. For example, unfailing completio ..."
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Franz Baader LuFg Theoretical Computer Science, RWTH Aachen Ahornstraße 55, 52074 Aachen, Germany email: baader@informatik.rwthaachen.de 1 Introduction Reduction orderings that are total on ground terms play an important role in many areas of automated deduction. For example, unfailing completion [4]a variant of KnuthBendix completion that avoids failure due to incomparable critical pairspresupposes such an ordering. In addition, using a reduction ordering that is total on ground terms, one can show that any finite set of ground equations has a decidable word problem [13, 20]. It is very easy to obtain such orderings. Indeed, many of the standard methods for constructing reduction orderings yield orderings that are total on ground terms: both KnuthBendix orderings [12] and lexicographic path orderings [10] are total on ground terms if they are based on a total precedence ordering on the set of function symbols. Things become more complex if one is interested in reduction or...
Rewrite Closures for Ground and Cancellative AC Theories
 Conference on Foundations of Software Technology and Theoretical Computer Science, FST&TCS ’2001
, 2001
"... Given a binary relation E [ R on the set of ground terms over some signature, we dene an abstract rewrite closure for E [R. An abstract rewrite closure is a specialized bottomup tree automata and it can be used to eciently decide the reachability relation ! E[E [R . It is constructed using a co ..."
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Given a binary relation E [ R on the set of ground terms over some signature, we dene an abstract rewrite closure for E [R. An abstract rewrite closure is a specialized bottomup tree automata and it can be used to eciently decide the reachability relation ! E[E [R . It is constructed using a completion like procedure. Correctness is established using proof ordering techniques. The procedure is extended, in a modular way, to deal with signatures containing cancellative associative commutative function symbols. 1
Canonized Rewriting and Ground AC Completion Modulo Shostak Theories
, 2001
"... ACcompletion efficiently handles equality modulo associative and commutative function symbols. When the input is ground, the procedure terminates and provides a decision algorithm for the word problem. In this paper, we present a modular extension of ground ACcompletion for deciding formulas in th ..."
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ACcompletion efficiently handles equality modulo associative and commutative function symbols. When the input is ground, the procedure terminates and provides a decision algorithm for the word problem. In this paper, we present a modular extension of ground ACcompletion for deciding formulas in the combination of the theory of equality with userdefined AC symbols, uninterpreted symbols and an arbitrary signature disjoint Shostak theory X. Our algorithm, called AC(X), is obtained by augmenting in a modular way ground ACcompletion with the canonizer and solver present for the theory X. This integration rests on canonized rewriting, a new relation reminiscent to normalized rewriting, which integrates canonizers in rewriting steps. AC(X) is proved sound, complete and terminating, and is implemented to extend the core of the AltErgo theorem prover.