98,961 research outputs found
Using Well-Founded Relations for Proving Operational Termination
[EN] In this paper, we study operational termination, a proof theoretical notion for capturing the termination behavior of computational systems. We prove that operational termination can be characterized at different levels by means of well- founded relations on specific formulas which can be obtained from the considered system. We show how to obtain such well-founded relations from logical models which can be automatically generated using existing tools.Partially supported by the EU (FEDER), Projects TIN2015-69175-C4-1-R, and GV PROMETEOII/2015/013.Lucas Alba, S. (2020). Using Well-Founded Relations for Proving Operational Termination. Journal of Automated Reasoning. 64(2):167-195. https://doi.org/10.1007/s10817-019-09514-2S167195642Alarcón, B., Gutiérrez, R., Lucas, S., Navarro-Marset, R.: Proving termination properties with MU-TERM. 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Termination orders for 3-dimensional rewriting
This paper studies 3-polygraphs as a framework for rewriting on
two-dimensional words. A translation of term rewriting systems into
3-polygraphs with explicit resource management is given, and the respective
computational properties of each system are studied. Finally, a convergent
3-polygraph for the (commutative) theory of Z/2Z-vector spaces is given. In
order to prove these results, it is explained how to craft a class of
termination orders for 3-polygraphs.Comment: 30 pages, 35 figure
Automated Termination Proofs for Logic Programs by Term Rewriting
There are two kinds of approaches for termination analysis of logic programs:
"transformational" and "direct" ones. Direct approaches prove termination
directly on the basis of the logic program. Transformational approaches
transform a logic program into a term rewrite system (TRS) and then analyze
termination of the resulting TRS instead. Thus, transformational approaches
make all methods previously developed for TRSs available for logic programs as
well. However, the applicability of most existing transformations is quite
restricted, as they can only be used for certain subclasses of logic programs.
(Most of them are restricted to well-moded programs.) In this paper we improve
these transformations such that they become applicable for any definite logic
program. To simulate the behavior of logic programs by TRSs, we slightly modify
the notion of rewriting by permitting infinite terms. We show that our
transformation results in TRSs which are indeed suitable for automated
termination analysis. In contrast to most other methods for termination of
logic programs, our technique is also sound for logic programming without occur
check, which is typically used in practice. We implemented our approach in the
termination prover AProVE and successfully evaluated it on a large collection
of examples.Comment: 49 page
The three dimensions of proofs
In this document, we study a 3-polygraphic translation for the proofs of SKS,
a formal system for classical propositional logic. We prove that the free
3-category generated by this 3-polygraph describes the proofs of classical
propositional logic modulo structural bureaucracy. We give a 3-dimensional
generalization of Penrose diagrams and use it to provide several pictures of a
proof. We sketch how local transformations of proofs yield a non contrived
example of 4-dimensional rewriting.Comment: 38 pages, 50 figure
Singular and Plural Functions for Functional Logic Programming
Functional logic programming (FLP) languages use non-terminating and
non-confluent constructor systems (CS's) as programs in order to define
non-strict non-determi-nistic functions. Two semantic alternatives have been
usually considered for parameter passing with this kind of functions: call-time
choice and run-time choice. While the former is the standard choice of modern
FLP languages, the latter lacks some properties---mainly
compositionality---that have prevented its use in practical FLP systems.
Traditionally it has been considered that call-time choice induces a singular
denotational semantics, while run-time choice induces a plural semantics. We
have discovered that this latter identification is wrong when pattern matching
is involved, and thus we propose two novel compositional plural semantics for
CS's that are different from run-time choice.
We study the basic properties of our plural semantics---compositionality,
polarity, monotonicity for substitutions, and a restricted form of the bubbling
property for constructor systems---and the relation between them and to
previous proposals, concluding that these semantics form a hierarchy in the
sense of set inclusion of the set of computed values. We have also identified a
class of programs characterized by a syntactic criterion for which the proposed
plural semantics behave the same, and a program transformation that can be used
to simulate one of them by term rewriting. At the practical level, we study how
to use the expressive capabilities of these semantics for improving the
declarative flavour of programs. We also propose a language which combines
call-time choice and our plural semantics, that we have implemented in Maude.
The resulting interpreter is employed to test several significant examples
showing the capabilities of the combined semantics.
To appear in Theory and Practice of Logic Programming (TPLP)Comment: 53 pages, 5 figure
A general conservative extension theorem in process algebras with inequalities
We prove a general conservative extension theorem for transition system based process theories with easy-to-check and reasonable conditions. The core of this result is another general theorem which gives sufficient conditions for a system of operational rules and an extension of it in order to ensure conservativity, that is, provable transitions from an original term in the extension are the same as in the original system. As a simple corollary of the conservative extension theorem we prove a completeness theorem. We also prove a general theorem giving sufficient conditions to reduce the question of ground confluence modulo some equations for a large term rewriting system associated with an equational process theory to a small term rewriting system under the condition that the large system is a conservative extension of the small one. We provide many applications to show that our results are useful. The applications include (but are not limited to) various real and discrete time settings in ACP, ATP, and CCS and the notions projection, renaming, stage operator, priority, recursion, the silent step, autonomous actions, the empty process, divergence, etc
Inductive-data-type Systems
In a previous work ("Abstract Data Type Systems", TCS 173(2), 1997), the last
two authors presented a combined language made of a (strongly normalizing)
algebraic rewrite system and a typed lambda-calculus enriched by
pattern-matching definitions following a certain format, called the "General
Schema", which generalizes the usual recursor definitions for natural numbers
and similar "basic inductive types". This combined language was shown to be
strongly normalizing. The purpose of this paper is to reformulate and extend
the General Schema in order to make it easily extensible, to capture a more
general class of inductive types, called "strictly positive", and to ease the
strong normalization proof of the resulting system. This result provides a
computation model for the combination of an algebraic specification language
based on abstract data types and of a strongly typed functional language with
strictly positive inductive types.Comment: Theoretical Computer Science (2002
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
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