41 research outputs found
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
Automated deduction with built-in theories: completeness results and constraint solving techniques
Postprint (published version
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
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
Studies in the completeness and efficiency of theorem-proving by resolution
Inference systems Τ and search strategies E for T are distinguished from proof procedures β = (T,E)
The completeness of procedures is studied by studying
separately the completeness of inference systems and of
search strategies. Completeness proofs for resolution
systems are obtained by the construction of semantic
trees. These systems include minimal α-restricted
binary resolution, minimal α-restricted M-clash resolution
and maximal pseudo-clash resolution. Certain refinements
of hyper-resolution systems with equality axioms are
shown to be complete and equivalent to refinements of
the pararmodulation method for dealing with equality.
The completeness and efficiency of search strategies
for theorem-proving problems is studied in sufficient
generality to include the case of search strategies for
path-search problems in graphs. The notion of theorem-proving problem is defined abstractly so as to be dual to
that of and" or tree. Special attention is given to
resolution problems and to search strategies which generate
simpler before more complex proofs.
For efficiency, a proof procedure (T,E) requires
an efficient search strategy E as well as an inference
system T which admits both simple proofs and relatively
few redundant and irrelevant derivations. The theory
of efficient proof procedures outlined here is applied
to proving the increased efficiency of the usual method
for deleting tautologies and subsumed clauses. Counter-examples
are exhibited for both the completeness and
efficiency of alternative methods for deleting subsumed
clauses.
The efficiency of resolution procedures is improved
by replacing the single operation of resolving a clash
by the two operations of generating factors of clauses
and of resolving a clash of factors. Several factoring
methods are investigated for completeness. Of these the
m-factoring method is shown to be always more efficient
than the Wos-Robinson method
Systèmes Canoniques Abstraits : Application à la Déduction Naturelle et à la Complétion
Canonical Systems and Inference (ACSI) were introduced by N. Dershowitz and C. Kirchner to formalize the intuitive notion of good proof and good inference appearing typically in first-order logic or in Knuth-Bendix like completion procedures. Since this abstract framework, based on proof orderings, is intended to be generic, it is of fundamental interest to show its adequacy to represent the main systems of interest. It is here done for minimal propositional natural deduction, and for the standard (Knuth-Bendix) completion, closing an open question. For the first proof system, a generalisation of the ACSI is needed. We provide here a conservative one, in the sense that all results of the original framework still hold. For the second one, two proof representations, proof terms and proofs by replacement, are compared to built a proof ordering that provides an instantiation adapted to the ACSI framework