Set theory in first-order logic: clauses for Go¨del's axioms
Journal of Automated Reasoning
The automation of syllogistic I. Syllogistic normal forms
Journal of Symbolic Computation
Towards a computation system based on set theory
Theoretical Computer Science
Foundations of Equational Logic Programming
Foundations of Equational Logic Programming
Introduction to algorithms
Computable set theory
Expressing infinity without foundation
Journal of Symbolic Logic
Renaming a Set of Clauses as a Horn Set
Journal of the ACM (JACM)
Proving termination with multiset orderings
Communications of the ACM
EUROCAL '85 Research Contributions from the European Conference on Computer Algebra-Volume 2
Embedding Finite Sets in a Logic Programming Language
ELP '92 Proceedings of the Third International Workshop on Extensions of Logic Programming
Decidability results for sets with atoms
ACM Transactions on Computational Logic (TOCL)
Theory-specific automated reasoning
A 25-year perspective on logic programming
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Four weak theories of pure sets are axiomatically characterized. A decision method is given for checking sentences of the form @?"y"1...@?"y"n@?"x"p, where n varies over natural numbers and p over unquantified matrices, for provability in each theory. Dually, the method can be used to check @?"y"1...@?"y"n@?"x"@?"p for satisfiability. The completeness proof is fully constructive: this means that given a satisfiable constraint of the form @?"y"1...@?"y"n@?"x"@?"p, a computable model of the axioms which also fulfills the constraint can be synthesized. In this sense, we have a way of automatically generating a concrete representation of the abstract data-type ''set'' under varying axioms. The problem is also addressed of how to determine @? x p; i.e., how to find a @z fulfilling p(@x"1,...,@x"n,@z) in a computable model M of one of our theories, as a function of input M-sets @x"1,...,@x"n. A partial solution to this problem is supplied, which works when @?"y"1...@?"y"n@?"x"p is a theorem and M meets a suitable condition which happens to be satisfied by those models that are produced by our automatic synthesis algorithm. A stronger condition on M is also characterized that makes @? x p computable in all cases (at worst through a blind search method). Examples showing the expressive power of the @?*@?-constraints and the usefulness of @?-expressions in set computations are included. Envisaged extensions of the proposed methods to axiomatic set theories antithetic to the classical ones are briefly hinted at.