Approximate counting by hashing in bounded arithmetic

AbstractWe show how to formalize approximate counting via hash functions in subsystems of bounded arithmetic, using variants of the weak pigeonhole principle. We discuss several applications, including a proof of the tournament principle, and an improvement on the known relationship of the collapse of the bounded arithmetic hierarchy to the collapse of the polynomial-time hierarchy.

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FRAGMENTS OF APPROXIMATE COUNTING

Journal of Symbolic Logic ◽

10.1017/jsl.2013.37 ◽

2014 ◽

Vol 79 (2) ◽

pp. 496-525 ◽

Cited By ~ 8

Author(s):

SAMUEL R. BUSS ◽

LESZEK ALEKSANDER KOŁODZIEJCZYK ◽

NEIL THAPEN

Keyword(s):

Polynomial Time ◽

Open Problem ◽

Bounded Arithmetic ◽

Pigeonhole Principle ◽

Approximate Counting ◽

Image Position

AbstractWe study the long-standing open problem of giving $\forall {\rm{\Sigma }}_1^b$ separations for fragments of bounded arithmetic in the relativized setting. Rather than considering the usual fragments defined by the amount of induction they allow, we study Jeřábek’s theories for approximate counting and their subtheories. 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 functions, and also in a fragment that includes the surjective weak pigeonhole principle for FPNP functions. We further give new propositional translations, in terms of random resolution refutations, for the consequences of $T_2^1$ augmented with the surjective weak pigeonhole principle for polynomial time functions.

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Witnessing functions in bounded arithmetic and search problems

Journal of Symbolic Logic ◽

10.2307/2586729 ◽

1998 ◽

Vol 63 (3) ◽

pp. 1095-1115 ◽

Cited By ~ 12

Author(s):

Mario Chiari ◽

Jan Krajíček

Keyword(s):

Partial Order ◽

Polynomial Time ◽

Bounded Arithmetic ◽

Pigeonhole Principle ◽

Search Problems ◽

Strict Partial Order ◽

Conservation Result ◽

Time Structures

AbstractWe investigate the possibility to characterize (multi)functions that are-definable with smalli(i= 1, 2, 3) in fragments of bounded arithmeticT2in terms of natural search problems defined over polynomial-time structures. We obtain the following results:(1) A reformulation of known characterizations of (multi)functions that areand-definable in the theoriesand.(2) New characterizations of (multi)functions that areand-definable in the theory.(3) A new non-conservation result: the theoryis not-conservative over the theory.To prove that the theoryis not-conservative over the theory, we present two examples of a-principle separating the two theories:(a) the weak pigeonhole principle WPHP(a2,f, g) formalizing that no functionfis a bijection betweena2andawith the inverseg,(b) the iteration principle Iter(a, R, f) formalizing that no functionfdefined on a strict partial order ({0,…, a},R) can have increasing iterates.

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Delineating classes of computational complexity via second order theories with weak set existence principles. I

Journal of Symbolic Logic ◽

10.2307/2275511 ◽

1995 ◽

Vol 60 (1) ◽

pp. 103-121 ◽

Cited By ~ 1

Author(s):

Aleksandar Ignjatović

Keyword(s):

Computational Complexity ◽

Polynomial Time ◽

Second Order ◽

Complexity Classes ◽

Bounded Arithmetic ◽

Recursive Functions ◽

Speed Up ◽

Bounded Complexity ◽

Polynomial Time Hierarchy ◽

Comprehension Axiom

AbstractIn this paper we characterize the well-known computational complexity classes of the polynomial time hierarchy as classes of provably recursive functions (with graphs of suitable bounded complexity) of some second order theories with weak comprehension axiom schemas but without any induction schemas (Theorem 6). We also find a natural relationship between our theories and the theories of bounded arithmetic (Lemmas 4 and 5). Our proofs use a technique which enables us to “speed up” induction without increasing the bounded complexity of the induction formulas. This technique is also used to obtain an interpretability result for the theories of bounded arithmetic (Theorem 4).

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Notes on polynomially bounded arithmetic

Journal of Symbolic Logic ◽

10.2307/2275794 ◽

1996 ◽

Vol 61 (3) ◽

pp. 942-966 ◽

Cited By ~ 58

Author(s):

Domenico Zambella

Keyword(s):

Polynomial Time ◽

General Model ◽

Bounded Arithmetic ◽

Theoretical Investigations ◽

Polynomial Time Hierarchy

AbstractWe characterize the collapse of Buss' bounded arithmetic in terms of the provable collapse of the polynomial time hierarchy. We include also some general model-theoretical investigations on fragments of bounded arithmetic.

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Approximate counting in bounded arithmetic

Journal of Symbolic Logic ◽

10.2178/jsl/1191333850 ◽

2007 ◽

Vol 72 (3) ◽

pp. 959-993 ◽

Cited By ~ 11

Author(s):

Emil Jeřábek

Keyword(s):

Complexity Classes ◽

Bounded Arithmetic ◽

Boolean Circuits ◽

Pigeonhole Principle ◽

Approximate Counting

AbstractWe develop approximate counting of sets definable by Boolean circuits in bounded arithmetic using the dual weak pigeonhole principle (dWPHP(PV)), as a generalization of results from [15]. We discuss applications to formalization of randomized complexity classes (such as BPP, APP, MA, AM) in PV1 + dWPHP(PV).

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