137 research outputs found
Polynomial Interpretations over the Natural, Rational and Real Numbers Revisited
Polynomial interpretations are a useful technique for proving termination of
term rewrite systems. They come in various flavors: polynomial interpretations
with real, rational and integer coefficients. As to their relationship with
respect to termination proving power, Lucas managed to prove in 2006 that there
are rewrite systems that can be shown polynomially terminating by polynomial
interpretations with real (algebraic) coefficients, but cannot be shown
polynomially terminating using polynomials with rational coefficients only. He
also proved the corresponding statement regarding the use of rational
coefficients versus integer coefficients. In this article we extend these
results, thereby giving the full picture of the relationship between the
aforementioned variants of polynomial interpretations. In particular, we show
that polynomial interpretations with real or rational coefficients do not
subsume polynomial interpretations with integer coefficients. Our results hold
also for incremental termination proofs with polynomial interpretations.Comment: 28 pages; special issue of RTA 201
Decreasing Diagrams and Relative Termination
In this paper we use the decreasing diagrams technique to show that a
left-linear term rewrite system R is confluent if all its critical pairs are
joinable and the critical pair steps are relatively terminating with respect to
R. We further show how to encode the rule-labeling heuristic for decreasing
diagrams as a satisfiability problem. Experimental data for both methods are
presented.Comment: v3: missing references adde
Labelings for Decreasing Diagrams
This article is concerned with automating the decreasing diagrams technique
of van Oostrom for establishing confluence of term rewrite systems. We study
abstract criteria that allow to lexicographically combine labelings to show
local diagrams decreasing. This approach has two immediate benefits. First, it
allows to use labelings for linear rewrite systems also for left-linear ones,
provided some mild conditions are satisfied. Second, it admits an incremental
method for proving confluence which subsumes recent developments in automating
decreasing diagrams. The techniques proposed in the article have been
implemented and experimental results demonstrate how, e.g., the rule labeling
benefits from our contributions
Automating the First-Order Theory of Rewriting for Left-Linear Right-Ground Rewrite Systems
The first-order theory of rewriting is decidable for finite left-linear
right-ground rewrite systems. We present a new tool that implements the
decision procedure for this theory. It is based on tree automata
techniques. The tool offers the possibility to synthesize rewrite systems
that satisfy properties that are expressible in the first-order theory of
rewriting
Hydra Battles and AC Termination
We present a new encoding of the Battle of Hercules and Hydra as a rewrite system with AC symbols. Unlike earlier term rewriting encodings, it faithfully models any strategy of Hercules to beat Hydra. To prove the termination of our encoding, we employ type introduction in connection with many-sorted semantic labeling for AC rewriting and AC-RPO
Completion for Logically Constrained Rewriting
We propose an abstract completion procedure for logically constrained term rewrite systems (LCTRSs). This procedure can be instantiated to both standard Knuth-Bendix completion and ordered completion for LCTRSs, and we present a succinct and uniform correctness proof. A prototype implementation illustrates the viability of the new completion approach
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