507 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
On Unification Modulo One-Sided Distributivity: Algorithms, Variants and Asymmetry
An algorithm for unification modulo one-sided distributivity is an early
result by Tid\'en and Arnborg. More recently this theory has been of interest
in cryptographic protocol analysis due to the fact that many cryptographic
operators satisfy this property. Unfortunately the algorithm presented in the
paper, although correct, has recently been shown not to be polynomial time
bounded as claimed. In addition, for some instances, there exist most general
unifiers that are exponentially large with respect to the input size. In this
paper we first present a new polynomial time algorithm that solves the decision
problem for a non-trivial subcase, based on a typed theory, of unification
modulo one-sided distributivity. Next we present a new polynomial algorithm
that solves the decision problem for unification modulo one-sided
distributivity. A construction, employing string compression, is used to
achieve the polynomial bound. Lastly, we examine the one-sided distributivity
problem in the new asymmetric unification paradigm. We give the first
asymmetric unification algorithm for one-sided distributivity
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
Formalizing Knuth-Bendix Orders and Knuth-Bendix Completion
We present extensions of our Isabelle Formalization of Rewriting that cover two historically related concepts: the Knuth-Bendix order and the Knuth-Bendix completion procedure.
The former, besides being the first development of its kind in a proof assistant, is based on a generalized version of the Knuth-Bendix order. We compare our version to variants from the literature and show all properties required to certify termination proofs of TRSs.
The latter comprises the formalization of important facts that are related to completion, like Birkhoff\u27s theorem, the critical pair theorem, and a soundness proof of completion, showing that the strict encompassment condition is superfluous for finite runs. As a result, we are able to certify completion proofs
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