scholarly journals Notes on polynomially bounded arithmetic

1996 ◽  
Vol 61 (3) ◽  
pp. 942-966 ◽  
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.

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

1995 ◽  
Vol 60 (1) ◽  
pp. 103-121 ◽  
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).

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.

scholarly journals Satisfiability Allows No Nontrivial Sparsification unless the Polynomial-Time Hierarchy Collapses

2014 ◽  
Vol 61 (4) ◽  
pp. 1-27 ◽  
Author(s):  
Holger Dell ◽  
Dieter Van Melkebeek
Keyword(s):  
Polynomial Time ◽  
Polynomial Time Hierarchy

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.

The polynomial-time hierarchy and sparse oracles

1986 ◽  
Vol 33 (3) ◽  
pp. 603-617 ◽  
Author(s):  
Jose L. Balcázar ◽  
Ronald V. Book ◽  
Uwe Schöning
Keyword(s):  
Polynomial Time ◽  
Polynomial Time Hierarchy

ON HIGHER ARTHUR-MERLIN CLASSES

2004 ◽  
Vol 15 (01) ◽  
pp. 3-19
Author(s):  
JIN-YI CAI ◽  
DENIS CHARLES ◽  
A. PAVAN ◽  
SAMIK SENGUPTA
Keyword(s):  
Polynomial Time ◽  
Upper Bounds ◽  
Complexity Classes ◽  
Polynomial Time Hierarchy

We study higher Arthur-Merlin classes defined via several natural probabilistic operators BP, R and coR. We investigate the complexity classes they define, and a number of interactions between these operators and the standard polynomial time hierarchy. We prove a hierarchy theorem for these higher Arthur-Merlin classes involving interleaving operators, and a theorem giving non-trivial upper bounds to the intersection of the complementary classes in the hierarchy.

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