A new proof of Ajtai’s completeness theorem for nonstandard finite structures

Garlík, Michal

doi:10.1007/s00153-014-0416-5

Cited by 29 publications

(3 citation statements)

References 8 publications

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“…The existence of such a model of T is characterized by the non-existence of a proof of contradiction in T that is -in a specific, rather technical, sensedefinable over A W . We refer the reader to Garlík [9] who found a simpler and more conceptual proof of Ajtai's theorem. The construction needs M satisfying (a)-(c) above and also (d) L is finite.…”

Section: Discussionmentioning

confidence: 99%

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Expansions of pseudofinite structures and circuit and proof complexity

Krajicek

2015

Preprint

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I shall describe a general model-theoretic task to construct expansions of pseudofinite structures and discuss several examples of particular relevance to computational complexity. Then I will present one specific situation where finding a suitable expansion would imply that, assuming a one-way permutation exists, the computational class N P is not closed under complementation.1 We shall discuss another example with parity later.

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Section: Discussionmentioning

confidence: 99%

“…In Claims 1 -3 we stipulated that L is finite in order to avoid the discussion how it is coded. In these claims L can be, in fact, infinite as long as A W is coded in M. But in [3,4,9] the hypothesis (d) is needed.…”

Section: Discussionmentioning

confidence: 99%

Expansions of pseudofinite structures and circuit and proof complexity

Krajicek

2015

Preprint

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“…It is well known that some problems in complexity theory can be cast as problems of constructions of expanded extensions of models of bounded arithmetic [1,2,8,12,13]. These models are usually required to satisfy some form of bounded induction but at the same time not introduce any new lengths of strings.…”

Section: Introductionmentioning

confidence: 99%

Construction of models of bounded arithmetic by restricted reduced powers

Garlík

2016

Arch. Math. Logic

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We present two constructions of models of bounded arithmetic, both in the form of a generalization of the ultrapower construction, that yield nonelementary extensions but do not introduce new lengths. As an application we show, assuming the existence of a one-way permutation g hard against polynomial-size circuits, that strictR 1 2 (g) is weaker than R 1 2 (g). In particular, if such a permutation can be defined by a term in the language of R 1 2 , then strictR 1 2 is weaker than R 1 2 .

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Proof Complexity

Krajíček¹

2019

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A new proof of Ajtai’s completeness theorem for nonstandard finite structures

Cited by 29 publications

References 8 publications

Expansions of pseudofinite structures and circuit and proof complexity

Expansions of pseudofinite structures and circuit and proof complexity

Construction of models of bounded arithmetic by restricted reduced powers

Proof Complexity

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