210 research outputs found
On the formalization of termination techniques based on multiset orderings
Multiset orderings are a key ingredient in certain termination techniques like the recursive path ordering and a variant of size-change termination. In order to integrate these techniques in a certifier for termination proofs, we have added them to the Isabelle Formalization of Rewriting. To this end, it was required to extend the existing formalization on multiset orderings towards a generalized multiset ordering. Afterwards, the soundness proofs of both techniques have been established, although only after fixing some definitions. Concerning efficiency, it is known that the search for suitable parameters for both techniques is NP-hard. We show that checking the correct application of the techniques-where all parameters are provided-is also NP-hard, since the problem of deciding the generalized multiset ordering is NP-hard. © René Thiemann, Guillaume Allais, and JulianNagele
CoLoR: a Coq library on well-founded rewrite relations and its application to the automated verification of termination certificates
Termination is an important property of programs; notably required for
programs formulated in proof assistants. It is a very active subject of
research in the Turing-complete formalism of term rewriting systems, where many
methods and tools have been developed over the years to address this problem.
Ensuring reliability of those tools is therefore an important issue. In this
paper we present a library formalizing important results of the theory of
well-founded (rewrite) relations in the proof assistant Coq. We also present
its application to the automated verification of termination certificates, as
produced by termination tools
Proof Pearl: a Formal Proof of Higman’s Lemma in ACL2
Higman’s lemma is an important result in infinitary combinatorics, which
has been formalized in several theorem provers. In this paper we present a formalization
and proof of Higman’s Lemma in the ACL2 theorem prover. Our formalization
is based on a proof by Murthy and Russell, where the key termination argument
is justified by the multiset relation induced by a well-founded relation. To our
knowledge, this is the first mechanization of this proof.Ministerio de Ciencia e Innovación MTM2009-13842-C02-0
Confluence by Decreasing Diagrams -- Formalized
This paper presents a formalization of decreasing diagrams in the theorem
prover Isabelle. It discusses mechanical proofs showing that any locally
decreasing abstract rewrite system is confluent. The valley and the conversion
version of decreasing diagrams are considered.Comment: 17 pages; valley and conversion version; RTA 201
Certification of Complexity Proofs using CeTA
Nowadays certification is widely employed by automated termination tools for term rewriting, where certifiers support most available techniques. In complexity analysis, the situation is quite different. Although tools support certification in principle, current certifiers implement only the most basic technique,
namely, suitably tamed versions of reduction orders. As a consequence, only a small fraction of the proofs generated by state-of-the-art complexity tools can be certified. To improve upon this situation, we formalized a framework for the certification
of modular complexity proofs and incorporated it into CeTA. We
report on this extension and present the newly supported techniques
(match-bounds, weak dependency pairs, dependency tuples, usable rules, and usable replacement maps), resulting in a significant increase in the number of certifiable complexity proofs. During our work we detected conflicts in theoretical results as well as bugs in
existing complexity tools
AC-KBO Revisited
Equational theories that contain axioms expressing associativity and
commutativity (AC) of certain operators are ubiquitous. Theorem proving methods
in such theories rely on well-founded orders that are compatible with the AC
axioms. In this paper we consider various definitions of AC-compatible
Knuth-Bendix orders. The orders of Steinbach and of Korovin and Voronkov are
revisited. The former is enhanced to a more powerful version, and we modify the
latter to amend its lack of monotonicity on non-ground terms. We further
present new complexity results. An extension reflecting the recent proposal of
subterm coefficients in standard Knuth-Bendix orders is also given. The various
orders are compared on problems in termination and completion.Comment: 31 pages, To appear in Theory and Practice of Logic Programming
(TPLP) special issue for the 12th International Symposium on Functional and
Logic Programming (FLOPS 2014
Formal proofs about rewriting using ACL2
We present an application of the ACL2 theorem prover to reason about rewrite systems
theory. We describe the formalization and representation aspects of our work using the firstorder,
quantifier-free logic of ACL2 and we sketch some of the main points of the proof effort.
First, we present a formalization of abstract reduction systems and then we show how this
abstraction can be instantiated to establish results about term rewriting. The main theorems
we mechanically proved are Newman’s lemma (for abstract reductions) and Knuth–Bendix
critical pair theorem (for term rewriting).Ministerio de Educación y Ciencia TIC2000-1368-CO3-0
AC Dependency Pairs Revisited
Rewriting modulo AC, i.e., associativity and/or commutativity of certain symbols, is among the most frequently used extensions of term rewriting by equational theories. In this paper we present a generalization of the dependency pair framework for termination analysis to rewriting modulo AC. It subsumes existing variants of AC dependency pairs, admits standard dependency graph analyses, and in particular enjoys the minimality property in the standard sense. As a direct benefit, important termination techniques are easily extended; we describe usable rules and the subterm criterion for AC termination, which properly generalize the non-AC versions.
We also perform these extensions within IsaFoR - the Isabelle formalization of rewriting - and thereby provide the first formalization of AC dependency pairs. Consequently, our certifier CeTA now supports checking proofs of AC termination
A Formally Verified Prover for the ALC Description Logic
The Ontology Web Language (OWL) is a language used for
the Semantic Web. OWL is based on Description Logics (DLs), a family
of logical formalisms for representing and reasoning about conceptual
and terminological knowledge. Among these, the logic ALC is a ground
DL used in many practical cases. Moreover, the Semantic Web appears
as a new field for the application of formal methods, that could be used
to increase its reliability. A starting point could be the formal verification
of satisfiability provers for DLs. In this paper, we present the PVS
specification of a prover for ALC , as well as the proofs of its termination,
soundness and completeness. We also present the formalization of
the well–foundedness of the multiset relation induced by a well–founded
relation. This result has been used to prove the termination and the
completeness of the ALC prover.Ministerio de Educación y Ciencia TIN2004–0388
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