Termination proofs by multiset path orderings imply primitive recursive derivation lengths

Hofbauer, Dieter

doi:10.1007/3-540-53162-9_50

Cited by 19 publications

(34 citation statements)

References 5 publications

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“…Weiermann has shown in [Wei93] Cichon's claim which states that each finite rewrite system whose termination is proved by a lexicographic path ordering induced by an order over a finite signature yields a subrecursive (multiple recursive) function on the depth of terms which bound the lengths of derivations of terms. A similar and earlier result was achieved by Hofbauer in [Hof92]. These bounding functions are defined recursively in the extended Grzegorczyk-hierarchy; we refer the reader to [Ros84] for a study on subrecursive hierarchies.…”

Section: Strong Normalization Proofs By Natural Interpretationssupporting

confidence: 56%

“…Lemma 4.14 ( [Hof92]). Given a finite rewrite system R on a finite signature F and an interpretation I of elements of F such that the interpretation I prove termination of R. Given Φ : N −→ N a strictly monotonic function such that for each f ∈ F and for every k ∈ N the inequality I(f )(k, .…”

Section: Strong Normalization Proofs By Natural Interpretationsmentioning

confidence: 99%

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Strong normalization proofs for cut elimination in Gentzen's sequent calculi

Bittar¹

1999

Banach Center Publ.

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We define an equivalent variant LK sp of the Gentzen sequent calculus LK. In LK sp weakenings or contractions can be performed in parallel. This modification allows us to interpret a symmetrical system of mix elimination rules E LKsp by a finite rewriting system; the termination of this rewriting system can be machine checked. We give also a self-contained strong normalization proof by structural induction. We give another strong normalization proof by a strictly monotone subrecursive interpretation; this interpretation gives subrecursive bounds for the length of derivations. We give a strong normalization proof by applying orthogonal term rewriting results for a confluent restriction of the mix elimination system E LKsp .

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Section: Strong Normalization Proofs By Natural Interpretationssupporting

confidence: 56%

Section: Strong Normalization Proofs By Natural Interpretationsmentioning

confidence: 99%

Strong normalization proofs for cut elimination in Gentzen's sequent calculi

Bittar¹

1999

Banach Center Publ.

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“…Note that in our notion of !-termination there is no restriction on the kind of monotone functions allowed. Hofbauer (1992) proved that any TRS which can be proven terminating by a recursive path order, is !-terminating. By giving restrictions on the kind of functions allowed, a more detailed hierarchy between polynomial termination and !termination can be given.…”

Section: Hencementioning

confidence: 99%

“…x + (s(s(y)) + z) s(x) + (y + (z + w)) ! x + (z + (y + w)) from Hofbauer and Lautemann (1989). The termination proof can be given via the typing s : s 1 !…”

Section: Applicationsmentioning

confidence: 99%

Termination of term rewriting: interpretation and type elimination

Zantema¹

1994

Journal of Symbolic Computation

126

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We investigate proving termination of term rewriting systems by interpretation of terms in a well-founded monotone algebra. The well-known polynomial interpretations can be considered as a particular case in this framework. A classi cation of types of termination, including simple termination, is proposed based on properties in the semantic level. A transformation on term rewriting systems eliminating distributive rules is introduced. Using this distribution elimination a new termination proof of the system SUBST of Hardin and Laville (1986) is given. This system describes explicit substitution incalculus. Another tool for proving termination is based on introduction and removal of type restrictions. A property of many-sorted term rewriting systems is called persistent if it is not a ected by removing the corresponding typing restriction. Persistence turns out to be a generalization of direct sum modularity, but is more powerful for both proving con uence and termination. Termination is proved to be persistent for the class of term rewriting systems for which not both duplicating rules and collapsing rules occur, generalizing a similar result of Rusinowitch for modularity. This result has nice applications, in particular in undecidability proofs.

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“…Both (i) and (ii) were stated in Cichon [7] together with Girard's Hierarchy Comparison Theorem. Afterwards (i) has been verified by Hofbauer [10] and (ii) by Weiermann [11]. Furthermore, MPO and LPO are complete, i. e., a finite rewrite system for each primitive recursive function is reducing under MPO, and a finite rewrite system for each multiply recursive function is reducing under LPO.…”

Section: Introductionmentioning

confidence: 97%

A lexicographic path order with slow growing derivation bounds

Eguchi¹

2009

Mathematical Logic Qtrly

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This paper is concerned with implicit computational complexity of the exptime computable functions. Modifying the lexicographic path order, we introduce a path order EPO. It is shown that a termination proof for a term rewriting system via EPO implies an exponential bound on the lengths of derivations. The path order EPO is designed so that every exptime function is representable as a term rewrite system compatible with EPO.

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Termination proofs by multiset path orderings imply primitive recursive derivation lengths

Cited by 19 publications

References 5 publications

Strong normalization proofs for cut elimination in Gentzen's sequent calculi

Strong normalization proofs for cut elimination in Gentzen's sequent calculi

Termination of term rewriting: interpretation and type elimination

A lexicographic path order with slow growing derivation bounds

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