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The provably total NP search problems of weak second order bounded arithmetic
2011
Annals of Pure and Applied Logic
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Throughout our search problems are "type-2" NP search problems, which take second-order objects as parameters. ...
We define a new NP search problem, the "local improvement" principle, about labellings of an acyclic, bounded-degree graph. ...
In this way we think of the set of NP search problems provably total in a theory, which we identify with the set of such sentences, as characterizing the algorithmic strength of the theory. ...
doi:10.1016/j.apal.2010.12.002
fatcat:wjxvw6tqrvhnll5c7l62344ae4
Approximate counting and NP search problems
[article]
2021
arXiv
pre-print
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2018 pre-print arXiv:1812.10771v1 fatcat:zbvejvh4ubgrbl43zjre4crt4m
Our model for this kind of reasoning is the bounded arithmetic theory APC_2 of [Jeřábek 2009]. In particular, the Ramsey and weak pigeonhole search problems lie in the new class. ...
The introduction to the paper is intended to make the statements and context of the results accessible to someone unfamiliar with NP search problems or with bounded arithmetic. ...
Intuitively, APC 2 is a combination of T 1 2 and the weak pigeonhole principle, and what we show is that the NP search problems provably total in APC 2 arise as a combination of PLS (which is known to ...
arXiv:1812.10771v2
fatcat:h34i55ge45c3rgkvs645f7et4q
A proof of P!=NP
[article]
2021
arXiv
pre-print
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We show that it is provable in PA that there is an arithmetically definable sequence {ϕ_n:n ∈ω} of Π^0_2-sentences, such that - PRA+{ϕ_n:n ∈ω} is Π^0_2-sound and Π^0_1-complete - the length of ϕ_n is bounded ...
The argument establishes the result that it is provable in PA that P ≠ NP. We indicate how to pull the argument all the way down into EFA. ...
Our conclusion is that the search does provably terminate in Σ 0 2 -induction arithmetic. Hence the function g is provably total in Σ 0 2 -induction arithmetic as claimed. Proof of Proposition 3.9. ...
arXiv:2005.10080v12
fatcat:xv65opyhwree5aqt6q4nyifstu
Incompleteness in the finite domain
[article]
2017
arXiv
pre-print
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Motivated by the problem of finding finite versions of classical incompleteness theorems, we present some conjectures that go beyond NP≠ co NP. ...
that the sentence is not provable in a weak theory, or requires a long proof. ...
Acknowledgment I would like to thank Pavel Hrubeš, Emil Jeřábek, Jan Krajíček and Neil Thapen for their useful comments on the draft of this paper. ...
arXiv:1601.01487v2
fatcat:k7u6y6rrjndzjobf6tnxvlgvra
The NP Search Problems of Frege and Extended Frege Proofs
2017
ACM Transactions on Computational Logic
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The consistency search problems for Frege and extended Frege systems are shown to be many-one complete for the provably total NP search problems of the second order bounded arithmetic theories U 1 2 and ...
We study consistency search problems for Frege and extended Frege proofs, namely the NP search problems of finding syntactic errors in Frege and extended Frege proofs of contradictions. ...
ACKNOWLEDGMENTS We thank Jan Krajíček, Neil Thapen, and the two referees for helpful substantive comments on earlier drafts of this paper. ...
doi:10.1145/3060145
fatcat:va6osjcwifcmvb3fs7azlgwp5i
Improved witnessing and local improvement principles for second-order bounded arithmetic
2014
ACM Transactions on Computational Logic
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The local improvement principle for grid graphs with only logarithmically rounds is complete for the provably total NP search problems of V 1 2 . ...
Ko lodziejczyk, Nguyen, and Thapen have introduced local improvement properties to characterize the provably total NP functions of these second order theories. ...
The provably total NP search problems of a theory T of bounded arithmetic are the multifunctions which have polynomial time graph G f (x, y) such that T proves (∀x)(∃y)G f (x, y). ...
doi:10.1145/2559950
fatcat:4v3m6somhzcw7fwlsgzgoofh4a
Weak Arithmetics
2001
Theoretical Computer Science
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Second Order Arithmetic; of Real Closed Fields axioms; and of other powerful yet pathology-free theories. ...
This is a survey of the systematic study of weak axiom systems for Arithmetic (in short: "Weak Arithmetics"), which plays a growing and unifying role in various kinds of investigations: 1. exploring the ...
Every model R of R(E)CF has an Integral Part Z; in other words, (R; Z) is model of the weak 2d order Arithmetic R(E)CFI. ...
doi:10.1016/s0304-3975(00)00106-7
fatcat:jj6a6dgl7jg2paxrgfdbxz547a
NP search problems in low fragments of bounded arithmetic
2007
Journal of Symbolic Logic (JSL)
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We give combinatorial and computational characterizations of the NP search problems definable in the bounded arithmetic theories and . ...
It turns out that if we can define an NP search problem in a theory of bounded arithmetic, we learn something about the complexity of solving the search problem. ...
Any NP search problem can be defined by an open formula φ(x, y) in the language of the bounded arithmetic theory PV of Cook [6] . ...
doi:10.2178/jsl/1185803628
fatcat:dea47unxbvenbhgymbihsqovtq
Computational Flows in Arithmetic
[article]
2017
arXiv
pre-print
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their total NP search problems as a consequence. ...
This property helps us to decompose a first order arithmetical proof to a sequence of computational reductions by which we can extract the computational content of low complexity statements in some bounded ...
We wish to thank Pavel Pudlak for his support, his suggestions and the invaluable discussions that we have had since the beginning of this project. ...
arXiv:1711.01735v1
fatcat:aigaig7vw5fvjdfm7iyc2ybni4
CONSISTENCY OF CIRCUIT EVALUATION, EXTENDED RESOLUTION AND TOTAL NP SEARCH PROBLEMS
2016
Forum of Mathematics, Sigma
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${\it\Gamma}(0,s,k)$when interpreted as a relativized NP search problem is complete among all such problems provably total in bounded arithmetic theory$V_{1}^{1}$. ...
Symbolic Logic,70(2) (2005), 619–630] to define from${\it\Gamma}(0,s,k)$a nonrelativized NP search problem$i{\it\Gamma}$and we show that it is complete among all such problems provably total in bounded ...
Buss for comments on the draft of this paper and to N. Thapen for discussions about the topic. ...
doi:10.1017/fms.2016.13
fatcat:2vzpx4dgtnhpdgnghtrofaxr24
Alternating minima and maxima, Nash equilibria and Bounded Arithmetic
2012
Annals of Pure and Applied Logic
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We show simple prenex forms of these formulas whose herbrandizations (by polynomial time functions) are ∀Σ b 1 formulas that characterize ∀Σ b 1 theorems of the levels T k 2 of the Bounded Arithmetic Hierarchy ...
, and we derive from this another characterization, in terms of a search problem about finding pure Nash equilibria in k-turn games. ...
This work was partially supported by Institutional Research Plan AV0Z10190503 and grant IAA100190902 of GA AV ČR and by a grant from the John Templeton Foundation. ...
doi:10.1016/j.apal.2011.06.014
fatcat:b3uhj7amjnbqlmtfobkla3yvdq
New formally undecidable propositions: non-trivial lower bounds on proof complexity and related theorems
1991
Theoretical Computer Science
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Finally we see that Cook's thesis NP# P implies the existence of { F,} c TAUT having a lower bound LB,({F"}) of an order of growth comparable to that in our examples. ...
A proof of this conjecture has to overcome the proof-theoretic difficulty that if validity + is replaced by provability t, in T, then this NP # P-variant cannot be proved in T. * The main results of this ...
(F2) Let (NP # P)T denote the formula SAT& P (arithmetically
expressed in T)
where validity + is replaced by provability
kr in T. ...
doi:10.1016/0304-3975(91)90272-4
fatcat:sxrrh4cxefhknccipacsn4r3nu
Bounded Arithmetic, Cryptography and Complexity
2008
Theoria
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This survey discusses theories of bounded arithmetic, growth rates of definable functions, natural proofs, interpolation theorems, connections to cryptography, and the difficulty of obtaining independence ...
These two questions are not entirely independent, since it is known that the theories of bounded arithmetic collapse to some finite level if and only if S 2 is finitely axiomatized and if and only if S ...
Impagliazzo for pointing out the construction of section 4.4 and J. Krajíček, P. Pudlák and C. Pollett for comments on a preliminary version of this paper. ...
doi:10.1111/j.1755-2567.1997.tb00745.x
fatcat:kof42mxifrdqfdqaqbcyapg7yq
FRAGMENTS OF APPROXIMATE COUNTING
2014
Journal of Symbolic Logic (JSL)
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We study the long-standing open problem of giving $\forall {\rm{\Sigma }}_1^b$ separations for fragments of bounded arithmetic in the relativized setting. ...
We show that the $\forall {\rm{\Sigma }}_1^b$ Herbrandized ordering principle is unprovable in a fragment of bounded arithmetic that includes the injective weak pigeonhole principle for polynomial time ...
However, in Section 4 of this paper we study sWPHP for PV 2 functions over the theory PV 1 , which is too weak to prove the totality of FP NP functions. ...
doi:10.1017/jsl.2013.37
fatcat:wajuhk5earfm3bqo6kzkznafle
An axiomatic approach to algebrization
2009
Proceedings of the 41st annual ACM symposium on Symposium on theory of computing - STOC '09
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So provability in the proposed theories can serve as a surrogate for provability using the arithmetization technique. ...
(ii) Most of the algebrizing collapses and separations from [1], such as IP = PSPACE, NP ⊂ ZKIP if one-way functions exist, MA-EXP ⊂ P/poly, etc., are provable from Arithmetic Checkability. ...
We want to thank Scott Aaronson, Lance Fortnow, and the anonymous referees for their comments. ...
doi:10.1145/1536414.1536509
dblp:conf/stoc/ImpagliazzoKK09
fatcat:o6tthbtumnde3criihmkxnzxbm
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