8,891 research outputs found
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
Proof Orders for Decreasing Diagrams
We present and compare some well-founded proof orders for decreasing diagrams. These proof orders order a conversion above another conversion if the latter is obtained by filling any peak in the former by a (locally) decreasing diagram. Therefore each such proof order entails the decreasing diagrams technique for proving confluence. The proof orders differ with respect to monotonicity and complexity. Our results are developed in the setting of involutive monoids. We extend these results to obtain a decreasing diagrams technique for confluence modulo
Certified Rule Labeling
The rule labeling heuristic aims to establish confluence of (left-)linear term rewrite systems via decreasing diagrams. We present a formalization of a confluence criterion based on the interplay of relative termination and the rule labeling in the theorem prover Isabelle. Moreover, we report on the integration of this result into the certifier CeTA, facilitating the checking of confluence certificates based on decreasing diagrams for the first time. The power of the method is illustrated by an experimental evaluation on a (standard) collection of confluence problems
CERTIFYING CONFLUENCE PROOFS VIA RELATIVE TERMINATION AND RULE LABELING
The rule labeling heuristic aims to establish confluence of (left-)linear
term rewrite systems via decreasing diagrams. We present a formalization of a
confluence criterion based on the interplay of relative termination and the
rule labeling in the theorem prover Isabelle. Moreover, we report on the
integration of this result into the certifier CeTA, facilitating the checking
of confluence certificates based on decreasing diagrams. The power of the
method is illustrated by an experimental evaluation on a (standard) collection
of confluence problems
Certified rule labeling
© Julian Nagele and Harald Zankl. The rule labeling heuristic aims to establish confluence of (left-)linear term rewrite systems via decreasing diagrams. We present a formalization of a confluence criterion based on the interplay of relative termination and the rule labeling in the theorem prover Isabelle. Moreover, we report on the integration of this result into the certifier CeTA, facilitating the checking of confluence certificates based on decreasing diagrams for the first time. The power of the method is illustrated by an experimental evaluation on a (standard) collection of confluence problems
Automated Confluence Proof by Decreasing Diagrams based on Rule-Labelling
Decreasing diagrams technique (van Oostrom, 1994) is a technique that
can be widely applied to prove confluence of rewrite systems. To
directly apply the decreasing diagrams technique to prove confluence
of rewrite systems, rule-labelling heuristic has been proposed by van
Oostrom (2008). We show how constraints for ensuring confluence of
term rewriting systems constructed based on the rule-labelling
heuristic are encoded as linear arithmetic constraints suitable for
solving the satisfiability of them by external SMT solvers. We point
out an additional constraint omitted in (van Oostrom, 2008) that is
needed to guarantee the soundness of confluence proofs based on the
rule-labelling heuristic extended to deal with non-right-linear rules.
We also present several extensions of the rule-labelling heuristic by
which the applicability of the technique is enlarged
Type Preservation as a Confluence Problem
This paper begins with recent work by Kuan, MacQueen, and Findler,
which shows how standard type systems, such as the simply typed lambda
calculus, can be viewed as abstract reduction systems operating on
terms. The central idea is to think of the process of typing a term
as the computation of an abstract value for that term. The standard
metatheoretic property of type preservation can then be seen as a
confluence problem involving the concrete and abstract operational
semantics, viewed as abstract reduction systems (ARSs).
In this paper, we build on the work of Kuan et al. by showing show how
modern ARS theory, in particular the theory of decreasing diagrams,
can be used to establish type preservation via confluence. We
illustrate this idea through several examples of solving such problems
using decreasing diagrams. We also consider how automated tools for
analysis of term-rewriting systems can be applied in testing typ
Decreasing Diagrams for Confluence and Commutation
Like termination, confluence is a central property of rewrite systems. Unlike
for termination, however, there exists no known complexity hierarchy for
confluence. In this paper we investigate whether the decreasing diagrams
technique can be used to obtain such a hierarchy. The decreasing diagrams
technique is one of the strongest and most versatile methods for proving
confluence of abstract rewrite systems. It is complete for countable systems,
and it has many well-known confluence criteria as corollaries.
So what makes decreasing diagrams so powerful? In contrast to other
confluence techniques, decreasing diagrams employ a labelling of the steps with
labels from a well-founded order in order to conclude confluence of the
underlying unlabelled relation. Hence it is natural to ask how the size of the
label set influences the strength of the technique. In particular, what class
of abstract rewrite systems can be proven confluent using decreasing diagrams
restricted to 1 label, 2 labels, 3 labels, and so on? Surprisingly, we find
that two labels suffice for proving confluence for every abstract rewrite
system having the cofinality property, thus in particular for every confluent,
countable system.
Secondly, we show that this result stands in sharp contrast to the situation
for commutation of rewrite relations, where the hierarchy does not collapse.
Thirdly, investigating the possibility of a confluence hierarchy, we
determine the first-order (non-)definability of the notion of confluence and
related properties, using techniques from finite model theory. We find that in
particular Hanf's theorem is fruitful for elegant proofs of undefinability of
properties of abstract rewrite systems
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
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