Witnessing functions in bounded arithmetic and search problems

1998 ◽  
Vol 63 (3) ◽  
pp. 1095-1115 ◽  
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.

scholarly journals FRAGMENTS OF APPROXIMATE COUNTING

2014 ◽  
Vol 79 (2) ◽  
pp. 496-525 ◽  
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.

scholarly journals Approximate counting by hashing in bounded arithmetic

2009 ◽  
Vol 74 (3) ◽  
pp. 829-860 ◽  
Author(s):  
Emil Jeřábek
Keyword(s):  
Polynomial Time ◽  
Hash Functions ◽  
Bounded Arithmetic ◽  
Pigeonhole Principle ◽  
Approximate Counting ◽  
Arithmetic Hierarchy ◽  
Relationship Of ◽  
Polynomial Time Hierarchy

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.

Time and Duration in Noninterleaving Concurrency

1993 ◽  
Vol 19 (3-4) ◽  
pp. 403-416
Author(s):  
David Murphy
Keyword(s):  
Partial Order ◽  
Independent Interest ◽  
Underlying Event ◽  
Strict Partial Order ◽  
Event Structures ◽  
Concurrency Theory ◽  
Finite Structures

The purpose of this paper is to present a real-timed concurrency theory in the noninterleaving tradition. The theory is based on the occurrences of actions; each occurrence or event has a start and a finish. Causality is modelled by assigning a strict partial order to these starts and finishes, while timing is modelled by giving them reals. The theory is presented in some detail. All of the traditional notions found in concurrency theories (such as conflict, confusion, liveness, and so on) are found to be expressible. Four notions of causality arise naturally from the model, leading to notions of securing. Three of the notions give rise to underlying event structures, demonstrating that our model generalises Winskel’s. Infinite structures are then analysed: a poset of finite structures is defined and suitably completed to give one containing infinite structures. These infinite structures are characterised as just those arising as limits of finite ones. Our technique here, which relies on the structure of time, is of independent interest.

A strict partial order on payoff configurations and its some properties

2011 ◽  
Vol 218 (5) ◽  
pp. 2108-2112 ◽  
Author(s):  
Kentaro Kojima ◽  
Takehiro Inohara
Keyword(s):  
Partial Order ◽  
Strict Partial Order

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

2011 ◽  
Vol 162 (6) ◽  
pp. 419-446 ◽  
Author(s):  
Leszek Aleksander Kołodziejczyk ◽  
Phuong Nguyen ◽  
Neil Thapen
Keyword(s):  
Second Order ◽  
Bounded Arithmetic ◽  
Search Problems

scholarly journals Extensions of strict partial orders in N-Groups

1978 ◽  
Vol 25 (2) ◽  
pp. 241-249 ◽  
Author(s):  
K. B. Prabhakara Rao
Keyword(s):  
Partial Order ◽  
Sufficient Conditions ◽  
Partial Orders ◽  
Necessary And Sufficient Conditions ◽  
Full Extension ◽  
Partially Ordered ◽  
Strict Partial Order ◽  
Positive Elements ◽  
Necessary And Sufficient

AbstractAn attempt is made to extend the theory of extensions of partial orders in groups to strict partially ordered N-groups. Necessary and sufficient conditions, for a strict partial order of an N-group to have a strict full extension, and for a strict partial order of an N-group to be an intersection of strict full orders, are obtained when the partially ordered near-ring N and the N-group G satisfy the condition (− x) n = − xn for all elements x in G and positive elements n in N.

scholarly journals NP search problems in low fragments of bounded arithmetic

2007 ◽  
Vol 72 (2) ◽  
pp. 649-672 ◽  
Author(s):  
Jan Krajíček ◽  
Alan Skelley ◽  
Neil Thapen
Keyword(s):  
Bounded Arithmetic ◽  
Search Problems

AbstractWe give combinatorial and computational characterizations of the NP search problems definable in the bounded arithmetic theories and .

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