197 research outputs found
New results on rewrite-based satisfiability procedures
Program analysis and verification require decision procedures to reason on
theories of data structures. Many problems can be reduced to the satisfiability
of sets of ground literals in theory T. If a sound and complete inference
system for first-order logic is guaranteed to terminate on T-satisfiability
problems, any theorem-proving strategy with that system and a fair search plan
is a T-satisfiability procedure. We prove termination of a rewrite-based
first-order engine on the theories of records, integer offsets, integer offsets
modulo and lists. We give a modularity theorem stating sufficient conditions
for termination on a combinations of theories, given termination on each. The
above theories, as well as others, satisfy these conditions. We introduce
several sets of benchmarks on these theories and their combinations, including
both parametric synthetic benchmarks to test scalability, and real-world
problems to test performances on huge sets of literals. We compare the
rewrite-based theorem prover E with the validity checkers CVC and CVC Lite.
Contrary to the folklore that a general-purpose prover cannot compete with
reasoners with built-in theories, the experiments are overall favorable to the
theorem prover, showing that not only the rewriting approach is elegant and
conceptually simple, but has important practical implications.Comment: To appear in the ACM Transactions on Computational Logic, 49 page
Basic paramodulation
We introduce a class of restrictions for the ordered paramodulation and superposition calculi (inspired by the {\em basic\/} strategy for narrowing), in which paramodulation inferences are forbidden at terms introduced by substitutions from previous inference steps. In addition we introduce restrictions based on term selection rules and redex orderings, which are general criteria for delimiting the terms which are available for inferences. These refinements are compatible with standard ordering restrictions and are complete without paramodulation into variables or using functional reflexivity axioms. We prove refutational completeness in the context of deletion rules, such as simplification by rewriting (demodulation) and subsumption, and of techniques for eliminating redundant inferences
The Vampire and the FOOL
This paper presents new features recently implemented in the theorem prover
Vampire, namely support for first-order logic with a first class boolean sort
(FOOL) and polymorphic arrays. In addition to having a first class boolean
sort, FOOL also contains if-then-else and let-in expressions. We argue that
presented extensions facilitate reasoning-based program analysis, both by
increasing the expressivity of first-order reasoners and by gains in
efficiency
Abstract Canonical Inference
An abstract framework of canonical inference is used to explore how different
proof orderings induce different variants of saturation and completeness.
Notions like completion, paramodulation, saturation, redundancy elimination,
and rewrite-system reduction are connected to proof orderings. Fairness of
deductive mechanisms is defined in terms of proof orderings, distinguishing
between (ordinary) "fairness," which yields completeness, and "uniform
fairness," which yields saturation.Comment: 28 pages, no figures, to appear in ACM Trans. on Computational Logi
Equality elimination for the inverse method and extension procedures
We demonstrate how to handle equality in the inverse method using equality elimination. In the equality elimination method, proofs consist of two parts. In the first part we try to solve equations obtaining so called solution clauses. Solution clauses are obtained by a very refined strategy — basic superposition with selection function. In the second part, we perform the usual sequent proof search by the inverse method. Our approach is called equality elimination because we eliminate all occurrences of equality in the first part of the proof. Unlike the previous approach proposed by Maslov, our method uses most general substitutions, orderin
Rewrite-based equational theorem proving with selection and simplification
We present various refutationally complete calculi for first-order clauses with equality that allow for arbitrary selection of negative atoms in clauses. Refutation completeness is established via the use of well-founded orderings on clauses for defining a Herbrand model for a consistent set of clauses. We also formulate an abstract notion of redundancy and show that the deletion of redundant clauses during the theorem proving process preserves refutation completeness. It is often possible to compute the closure of nontrivial sets of clauses under application of non-redundant inferences. The refutation of goals for such complete sets of clauses is simpler than for arbitrary sets of clauses, in particular one can restrict attention to proofs that have support from the goals without compromising refutation completeness. Additional syntactic properties allow to restrict the search space even further, as we demonstrate for so-called quasi-Horn clauses. The results in this paper contain as special cases or generalize many known results about Knuth-Bendix-like completion procedures (for equations, Horn clauses, and Horn clauses over built-in Booleans), completion of first-order clauses by clausal rewriting, and inductive theorem proving for Horn clauses
Set of support, demodulation, paramodulation: a historical perspective
This article is a tribute to the scientific legacy of automated reasoning pioneer and JAR founder Lawrence T. (Larry) Wos. Larry's main technical contributions were the set-of-support strategy for resolution theorem proving, and the demodulation and paramodulation inference rules for building equality into resolution. Starting from the original definitions of these concepts in Larry's papers, this survey traces their evolution, unearthing the often forgotten trails that connect Larry's original definitions to those that became standard in the field
Defining the meaning of TPTP formatted proofs
International audienceThe TPTP library is one of the leading problem libraries in the automated theorem proving community. Over time, support was added for problems beyond those in first-order clausal form. TPTP has also been augmented with support for various proof formats output by theorem provers. Such proofs can also be maintained in the TSTP proof library. In this paper we propose an extension of this framework to support the semantic specification of the inference rules used in proofs
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