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The provably total NP search problems of weak second order bounded arithmetic

2011
*
Annals of Pure and Applied Logic
*

Throughout our

doi:10.1016/j.apal.2010.12.002
fatcat:wjxvw6tqrvhnll5c7l62344ae4
*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. ...##
###
Approximate counting and NP search problems
[article]

2021
*
arXiv
*
pre-print

Our model for this kind

arXiv:1812.10771v2
fatcat:h34i55ge45c3rgkvs645f7et4q
*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 ...##
###
A proof of P!=NP
[article]

2021
*
arXiv
*
pre-print

We show that it is

arXiv:2005.10080v12
fatcat:xv65opyhwree5aqt6q4nyifstu
*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. ...##
###
Incompleteness in the finite domain
[article]

2017
*
arXiv
*
pre-print

Motivated by

arXiv:1601.01487v2
fatcat:k7u6y6rrjndzjobf6tnxvlgvra
*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. ...##
###
The NP Search Problems of Frege and Extended Frege Proofs

2017
*
ACM Transactions on Computational Logic
*

*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. ...

##
###
Improved witnessing and local improvement principles for second-order bounded arithmetic

2014
*
ACM Transactions on Computational Logic
*

*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). ...

##
###
Weak Arithmetics

2001
*
Theoretical Computer Science
*

*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. ...

##
###
NP search problems in low fragments of bounded arithmetic

2007
*
Journal of Symbolic Logic (JSL)
*

We give combinatorial and computational characterizations

doi:10.2178/jsl/1185803628
fatcat:dea47unxbvenbhgymbihsqovtq
*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] . ...##
###
Computational Flows in Arithmetic
[article]

2017
*
arXiv
*
pre-print

their

arXiv:1711.01735v1
fatcat:aigaig7vw5fvjdfm7iyc2ybni4
*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. ...##
###
CONSISTENCY OF CIRCUIT EVALUATION, EXTENDED RESOLUTION AND TOTAL NP SEARCH PROBLEMS

2016
*
Forum of Mathematics, Sigma
*

${\it\Gamma}(0,s,k)$when interpreted as a relativized

doi:10.1017/fms.2016.13
fatcat:2vzpx4dgtnhpdgnghtrofaxr24
*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. ...##
###
Alternating minima and maxima, Nash equilibria and Bounded Arithmetic

2012
*
Annals of Pure and Applied Logic
*

We show simple prenex forms

doi:10.1016/j.apal.2011.06.014
fatcat:b3uhj7amjnbqlmtfobkla3yvdq
*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. ...##
###
New formally undecidable propositions: non-trivial lower bounds on proof complexity and related theorems

1991
*
Theoretical Computer Science
*

Finally we see that Cook's thesis

doi:10.1016/0304-3975(91)90272-4
fatcat:sxrrh4cxefhknccipacsn4r3nu
*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. ...##
###
Bounded Arithmetic, Cryptography and Complexity

2008
*
Theoria
*

This survey discusses theories

doi:10.1111/j.1755-2567.1997.tb00745.x
fatcat:kof42mxifrdqfdqaqbcyapg7yq
*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. ...##
###
FRAGMENTS OF APPROXIMATE COUNTING

2014
*
Journal of Symbolic Logic (JSL)
*

We study

doi:10.1017/jsl.2013.37
fatcat:wajuhk5earfm3bqo6kzkznafle
*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. ...##
###
An axiomatic approach to algebrization

2009
*
Proceedings of the 41st annual ACM symposium on Symposium on theory of computing - STOC '09
*

So

doi:10.1145/1536414.1536509
dblp:conf/stoc/ImpagliazzoKK09
fatcat:o6tthbtumnde3criihmkxnzxbm
*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. ...
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