3,270 research outputs found
Non-termination using Regular Languages
We describe a method for proving non-looping non-termination, that is, of
term rewriting systems that do not admit looping reductions. As certificates of
non-termination, we employ regular (tree) automata.Comment: Published at International Workshop on Termination 201
Proving Looping and Non-Looping Non-Termination by Finite Automata
A new technique is presented to prove non-termination of term rewriting. The
basic idea is to find a non-empty regular language of terms that is closed
under rewriting and does not contain normal forms. It is automated by
representing the language by a tree automaton with a fixed number of states,
and expressing the mentioned requirements in a SAT formula. Satisfiability of
this formula implies non-termination. Our approach succeeds for many examples
where all earlier techniques fail, for instance for the S-rule from combinatory
logic
Classes of Terminating Logic Programs
Termination of logic programs depends critically on the selection rule, i.e.
the rule that determines which atom is selected in each resolution step. In
this article, we classify programs (and queries) according to the selection
rules for which they terminate. This is a survey and unified view on different
approaches in the literature. For each class, we present a sufficient, for most
classes even necessary, criterion for determining that a program is in that
class. We study six classes: a program strongly terminates if it terminates for
all selection rules; a program input terminates if it terminates for selection
rules which only select atoms that are sufficiently instantiated in their input
positions, so that these arguments do not get instantiated any further by the
unification; a program local delay terminates if it terminates for local
selection rules which only select atoms that are bounded w.r.t. an appropriate
level mapping; a program left-terminates if it terminates for the usual
left-to-right selection rule; a program exists-terminates if there exists a
selection rule for which it terminates; finally, a program has bounded
nondeterminism if it only has finitely many refutations. We propose a
semantics-preserving transformation from programs with bounded nondeterminism
into strongly terminating programs. Moreover, by unifying different formalisms
and making appropriate assumptions, we are able to establish a formal hierarchy
between the different classes.Comment: 50 pages. The following mistake was corrected: In figure 5, the first
clause for insert was insert([],X,[X]
Automated verification of termination certificates
In order to increase user confidence, many automated theorem provers provide
certificates that can be independently verified. In this paper, we report on
our progress in developing a standalone tool for checking the correctness of
certificates for the termination of term rewrite systems, and formally proving
its correctness in the proof assistant Coq. To this end, we use the extraction
mechanism of Coq and the library on rewriting theory and termination called
CoLoR
Generic Encodings of Constructor Rewriting Systems
Rewriting is a formalism widely used in computer science and mathematical
logic. The classical formalism has been extended, in the context of functional
languages, with an order over the rules and, in the context of rewrite based
languages, with the negation over patterns. We propose in this paper a concise
and clear algorithm computing the difference over patterns which can be used to
define generic encodings of constructor term rewriting systems with negation
and order into classical term rewriting systems. As a direct consequence,
established methods used for term rewriting systems can be applied to analyze
properties of the extended systems. The approach can also be seen as a generic
compiler which targets any language providing basic pattern matching
primitives. The formalism provides also a new method for deciding if a set of
patterns subsumes a given pattern and thus, for checking the presence of
useless patterns or the completeness of a set of patterns.Comment: Added appendix with proofs and extended example
12th International Workshop on Termination (WST 2012) : WST 2012, February 19â23, 2012, Obergurgl, Austria / ed. by Georg Moser
This volume contains the proceedings of the 12th International Workshop on Termination (WST 2012), to be held February 19â23, 2012 in Obergurgl, Austria. The goal of the Workshop on Termination is to be a venue for presentation and discussion of all topics in and around termination. In this way, the workshop tries to bridge the gaps between different communities interested and active in research in and around termination. The 12th International Workshop on Termination in Obergurgl continues the successful workshops held in St. Andrews (1993), La Bresse (1995), Ede (1997), Dagstuhl (1999), Utrecht (2001), Valencia (2003), Aachen (2004), Seattle (2006), Paris (2007), Leipzig (2009), and Edinburgh (2010). The 12th International Workshop on Termination did welcome contributions on all aspects of termination and complexity analysis. Contributions from the imperative, constraint, functional, and logic programming communities, and papers investigating applications of complexity or termination (for example in program transformation or theorem proving) were particularly welcome. We did receive 18 submissions which all were accepted. Each paper was assigned two reviewers. In addition to these 18 contributed talks, WST 2012, hosts three invited talks by Alexander Krauss, Martin Hofmann, and Fausto Spoto
Quantifier-Free Interpolation of a Theory of Arrays
The use of interpolants in model checking is becoming an enabling technology
to allow fast and robust verification of hardware and software. The application
of encodings based on the theory of arrays, however, is limited by the
impossibility of deriving quantifier- free interpolants in general. In this
paper, we show that it is possible to obtain quantifier-free interpolants for a
Skolemized version of the extensional theory of arrays. We prove this in two
ways: (1) non-constructively, by using the model theoretic notion of
amalgamation, which is known to be equivalent to admit quantifier-free
interpolation for universal theories; and (2) constructively, by designing an
interpolating procedure, based on solving equations between array updates.
(Interestingly, rewriting techniques are used in the key steps of the solver
and its proof of correctness.) To the best of our knowledge, this is the first
successful attempt of computing quantifier- free interpolants for a variant of
the theory of arrays with extensionality
The data-exchange chase under the microscope
In this paper we take closer look at recent developments for the chase
procedure, and provide additional results. Our analysis allows us create a
taxonomy of the chase variations and the properties they satisfy. Two of the
most central problems regarding the chase is termination, and discovery of
restricted classes of sets of dependencies that guarantee termination of the
chase. The search for the restricted classes has been motivated by a fairly
recent result that shows that it is undecidable to determine whether the chase
with a given dependency set will terminate on a given instance. There is a
small dissonance here, since the quest has been for classes of sets of
dependencies guaranteeing termination of the chase on all instances, even
though the latter problem was not known to be undecidable. We resolve the
dissonance in this paper by showing that determining whether the chase with a
given set of dependencies terminates on all instances is coRE-complete. For the
hardness proof we use a reduction from word rewriting systems, thereby also
showing the close connection between the chase and word rewriting. The same
reduction also gives us the aforementioned instance-dependent RE-completeness
result as a byproduct. For one of the restricted classes guaranteeing
termination on all instances, the stratified sets dependencies, we provide new
complexity results for the problem of testing whether a given set of
dependencies belongs to it. These results rectify some previous claims that
have occurred in the literature.Comment: arXiv admin note: substantial text overlap with arXiv:1303.668
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