Proof-theoretic analysis of termination proofs

Buchholz, Wilfried

doi:10.1016/0168-0072(94)00056-9

Cited by 38 publications

(46 citation statements)

References 2 publications

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“…Here is a proof of of the wellfoundedness property (4), taken from on Buchholz [2]. (We generalized it slightly from the one-sorted to the many-sorted setting and from the assumption that < is a finite linear order on Σ to the assumption that < is any wellfounded order.…”

Section: < Is Monotonic (It Has Thementioning

confidence: 99%

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The Logic of Public Announcements, Common Knowledge, and Private Suspicions

Baltag

Moss

Solecki

2016

Readings in Formal Epistemology

236

433

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This paper presents a logical system in which various group-level epistemic actions are incorporated into the object language. That is, we consider the standard modeling of knowledge among a set of agents by multimodal Kripke structures. One might want to consider actions that take place, such as announcements to groups privately, announcements with suspicious outsiders, etc. In our system, such actions correspond to additional modalities in the object language. That is, we do not add machinery on top of models (as in Fagin et al [4]), but we reify aspects of the machinery in the logical language.Special cases of our logic have been considered in Plaza [13], Gerbrandy [5,6], and Gerbrandy and Groeneveld [7]. The latter group of papers introduce a language in which one can faithfully represent all of the reasoning in examples such as the Muddy Children scenario. In that paper we find operators for updating worlds via announcements to groups of agents who are isolated from all others. We advance this by considering many more actions, and by using a more general semantics.Our logic contains the infinitary operators used in the standard modeling of common knowledge. We present a sound and complete logical system for the logic, and we study its expressive power.

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Section: < Is Monotonic (It Has Thementioning

confidence: 99%

“…For part (1), note that if [α]P * C ψ is a normal form, then each sentence P A [β]P * C ψ and all subsentences of this sentence are normal forms. For part (2), note that when ϕ is a normal form, ϕ ∈ f (ϕ).…”

Section: (δ) and Nf ([δ]χ)mentioning

confidence: 99%

The Logic of Public Announcements, Common Knowledge, and Private Suspicions

Baltag

Moss

Solecki

2016

Readings in Formal Epistemology

236

433

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“…However, because these results rely on proving termination of term rewriting systems, they cannot be scaled to arithmetic (by Gödel's second incompleteness theorem and [Buc95]). …”

Section: Resultsmentioning

confidence: 99%

“…The subterm, monotonicity and transitivity properties are shown in [KL80], well-foundedness is shown in [Buc95]. The big head property can be shown by a straightforward induction on the structure of t.…”

Section: The Lexicographic Path Orderingmentioning

confidence: 99%

“…Buchholz [Buc95] has shown that if one proves the termination of a TRS using the LPO, then the proof can be modified such that it is valid in a fragment of arithmetic. By Gödel's second incompleteness theorem, one cannot prove the consistency of arithmetic from within a fragment of arithmetic, therefore a cut-elimination procedure for arithmetic cannot be shown terminating formulated as a TRS via the LPO.…”

Section: Introductionmentioning

confidence: 99%

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Lexicographic Path Induction

Sarnat

Schürmann

2009

Lecture Notes in Computer Science

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Abstract. Programming languages theory is full of problems that reduce to proving the consistency of a logic, such as the normalization of typed lambda-calculi, the decidability of equality in type theory, equivalence testing of traces in security, etc. Although the principle of transfinite induction is routinely employed by logicians in proving such theorems, it is rarely used by programming languages researchers who often prefer alternatives such as proofs by logical relations and model theoretic constructions. In this paper we harness the well-foundedness of the lexicographic path ordering to derive an induction principle that combines the comfort of structural induction with the expressive strength of transfinite induction. Using lexicographic path induction, we give a consistency proof of Martin-Löf's intuitionistic theory of inductive definitions. The consistency of Heyting arithmetic follows directly, and weak normalization for Gödel's T follows indirectly; both have been formalized in a prototypical extension of Twelf.

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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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Proof-theoretic analysis of termination proofs

Cited by 38 publications

References 2 publications

The Logic of Public Announcements, Common Knowledge, and Private Suspicions

The Logic of Public Announcements, Common Knowledge, and Private Suspicions

Lexicographic Path Induction

A lexicographic path order with slow growing derivation bounds

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