11,397 research outputs found
Polynomial Path Orders
This paper is concerned with the complexity analysis of constructor term
rewrite systems and its ramification in implicit computational complexity. We
introduce a path order with multiset status, the polynomial path order POP*,
that is applicable in two related, but distinct contexts. On the one hand POP*
induces polynomial innermost runtime complexity and hence may serve as a
syntactic, and fully automatable, method to analyse the innermost runtime
complexity of term rewrite systems. On the other hand POP* provides an
order-theoretic characterisation of the polytime computable functions: the
polytime computable functions are exactly the functions computable by an
orthogonal constructor TRS compatible with POP*.Comment: LMCS version. This article supersedes arXiv:1209.379
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
Coherent presentation for the hypoplactic monoid of rank n
In this thesis, we construct a coherent presentation for the hypoplactic monoid of rank
n and characterize the confluence diagrams associated with it, then we use the theory
of quasi-Kashiwara operators and quasi-crystal graphs to prove that all confluence diagrams
can be obtained from those diagrams whose vertices are highest-weight words. To
do so, we first give a complete rewriting system for the hypoplactic monoid of rank n,
then, using an extension of the Knuth–Bendix completion procedure called the homotopical
completion procedure, we compute the previously mentioned coherent presentation,
which, from a viewpoint of Monoidal Category Theory, gives us a family of generators of
the relations amongst the relations. These coherent presentations are used for representations
of monoids and are particularly useful to describe actions of monoids on categories.
The theoretical background is given without proof, since the main purpose of this thesis
is to present new results
Star Games and Hydras
The recursive path ordering is an established and crucial tool in term
rewriting to prove termination. We revisit its presentation by means of some
simple rules on trees (or corresponding terms) equipped with a 'star' as
control symbol, signifying a command to make that tree (or term) smaller in the
order being defined. This leads to star games that are very convenient for
proving termination of many rewriting tasks. For instance, using already the
simplest star game on finite unlabeled trees, we obtain a very direct proof of
termination of the famous Hydra battle, direct in the sense that there is not
the usual mention of ordinals. We also include an alternative road to setting
up the star games, using a proof method of Buchholz, adapted by van Oostrom,
resulting in a quantitative version of the star as control symbol. We conclude
with a number of questions and future research directions
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