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

Leszek Aleksander Kołodziejczyk, Phuong Nguyen, Neil Thapen
2011 Annals of Pure and Applied Logic
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.  ...

### Approximate counting and NP search problems [article]

Leszek Aleksander Kołodziejczyk, Neil Thapen
2021 arXiv   pre-print
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  ...

### A proof of P!=NP [article]

Rupert McCallum
2021 arXiv   pre-print
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.  ...

### Incompleteness in the finite domain [article]

Pavel Pudlak
2017 arXiv   pre-print
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.  ...

### The NP Search Problems of Frege and Extended Frege Proofs

Arnold Beckmann, Sam Buss
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

Arnold Beckmann, Samuel R. Buss
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

J.-P. Ressayre
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

Jan Krajíček, Alan Skelley, Neil Thapen
2007 Journal of Symbolic Logic (JSL)
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  .  ...

### Computational Flows in Arithmetic [article]

Amirhossein Akbar Tabatabai
2017 arXiv   pre-print
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.  ...

### CONSISTENCY OF CIRCUIT EVALUATION, EXTENDED RESOLUTION AND TOTAL NP SEARCH PROBLEMS

JAN KRAJÍČEK
2016 Forum of Mathematics, Sigma
${\it\Gamma}(0,s,k)$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.  ...

### Alternating minima and maxima, Nash equilibria and Bounded Arithmetic

Pavel Pudlák, Neil Thapen
2012 Annals of Pure and Applied Logic
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.  ...

### New formally undecidable propositions: non-trivial lower bounds on proof complexity and related theorems

H. Luckhardt
1991 Theoretical Computer Science
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.  ...

### Bounded Arithmetic, Cryptography and Complexity

SAMUEL R. BUSS
2008 Theoria
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.  ...

### FRAGMENTS OF APPROXIMATE COUNTING

SAMUEL R. BUSS, LESZEK ALEKSANDER KOŁODZIEJCZYK, NEIL THAPEN
2014 Journal of Symbolic Logic (JSL)
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.  ...

### An axiomatic approach to algebrization

Russell Impagliazzo, Valentine Kabanets, Antonina Kolokolova
2009 Proceedings of the 41st annual ACM symposium on Symposium on theory of computing - STOC '09
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 , 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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