498 research outputs found
Proving Finite Satisfiability of Deductive Databases
It is shown how certain refutation methods can be extended into semi-decision
procedures that are complete for both unsatisfiability and finite satisfiability. The proposed extension
is justified by a new characterization of finite satisfiability. This research was motivated
by a database design problem: Deduction rules and integrity constraints in definite databases
have to be finitely satisfiabl
HoCHC: A Refutationally Complete and Semantically Invariant System of Higher-order Logic Modulo Theories
We present a simple resolution proof system for higher-order constrained Horn
clauses (HoCHC) - a system of higher-order logic modulo theories - and prove
its soundness and refutational completeness w.r.t. the standard semantics. As
corollaries, we obtain the compactness theorem and semi-decidability of HoCHC
for semi-decidable background theories, and we prove that HoCHC satisfies a
canonical model property. Moreover a variant of the well-known translation from
higher-order to 1st-order logic is shown to be sound and complete for HoCHC in
standard semantics. We illustrate how to transfer decidability results for
(fragments of) 1st-order logic modulo theories to our higher-order setting,
using as example the Bernays-Schonfinkel-Ramsey fragment of HoCHC modulo a
restricted form of Linear Integer Arithmetic
Combination of convex theories: Modularity, deduction completeness, and explanation
AbstractDecision procedures are key components of theorem provers and constraint satisfaction systems. Their modular combination is of prime interest for building efficient systems, but their effective use is often limited by poor interface capabilities, when such procedures only provide a simple “sat/unsat” answer. In this paper, we develop a framework to design cooperation schemas between such procedures while maintaining modularity of their interfaces. First, we use the framework to specify and prove the correctness of classic combination schemas by Nelson–Oppen and Shostak. Second, we introduce the concept of deduction complete satisfiability procedures, we show how to build them for large classes of theories, then we provide a schema to modularly combine them. Third, we consider the problem of modularly constructing explanations for combinations by re-using available proof-producing procedures for the component theories
Variations on a Theme: A Bibliography on Approaches to Theorem Proving Inspired From Satchmo
This articles is a structured bibliography on theorem provers,
approaches to theorem proving, and theorem proving applications inspired
from Satchmo, the model generation theorem prover developed
in the mid 80es of the 20th century at ECRC, the European Computer-
Industry Research Centre. Note that the bibliography given in this article
is not exhaustive
A resolution principle for clauses with constraints
We introduce a general scheme for handling clauses whose variables are constrained by an underlying constraint theory. In general, constraints can be seen as quantifier restrictions as they filter out the values that can be assigned to the variables of a clause (or an arbitrary formulae with restricted universal or existential quantifier) in any of the models of the constraint theory. We present a resolution principle for clauses with constraints, where unification is replaced by testing constraints for satisfiability over the constraint theory. We show that this constrained resolution is sound and complete in that a set of clauses with constraints is unsatisfiable over the constraint theory if we can deduce a constrained empty clause for each model of the constraint theory, such that the empty clauses constraint is satisfiable in that model. We show also that we cannot require a better result in general, but we discuss certain tractable cases, where we need at most finitely many such empty clauses or even better only one of them as it is known in classical resolution, sorted resolution or resolution with theory unification
First-Order Logic Theorem Proving and Model Building via Approximation and Instantiation
In this paper we consider first-order logic theorem proving and model
building via approximation and instantiation. Given a clause set we propose its
approximation into a simplified clause set where satisfiability is decidable.
The approximation extends the signature and preserves unsatisfiability: if the
simplified clause set is satisfiable in some model, so is the original clause
set in the same model interpreted in the original signature. A refutation
generated by a decision procedure on the simplified clause set can then either
be lifted to a refutation in the original clause set, or it guides a refinement
excluding the previously found unliftable refutation. This way the approach is
refutationally complete. We do not step-wise lift refutations but conflicting
cores, finite unsatisfiable clause sets representing at least one refutation.
The approach is dual to many existing approaches in the literature because our
approximation preserves unsatisfiability
A logic of graph conditions extended with paths
In this paper we tackle the problem of extending the logic of nested graph conditions with paths. This means, for instance, that we may state properties about the existence of paths between some given nodes. As a main contribution, a sound and complete tableau method is defined for reasoning about this kind of properties.Peer ReviewedPostprint (published version
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