Results 1 
3 of
3
Bounded Arithmetic and Propositional Proof Complexity
 in Logic of Computation
, 1995
"... This is a survey of basic facts about bounded arithmetic and about the relationships between bounded arithmetic and propositional proof complexity. We introduce the theories S 2 of bounded arithmetic and characterize their proof theoretic strength and their provably total functions in terms of t ..."
Abstract

Cited by 13 (0 self)
 Add to MetaCart
(Show Context)
This is a survey of basic facts about bounded arithmetic and about the relationships between bounded arithmetic and propositional proof complexity. We introduce the theories S 2 of bounded arithmetic and characterize their proof theoretic strength and their provably total functions in terms of the polynomial time hierarchy. We discuss other axiomatizations of bounded arithmetic, such as minimization axioms. It is shown that the bounded arithmetic hierarchy collapses if and only if bounded arithmetic proves that the polynomial hierarchy collapses. We discuss Frege and extended Frege proof length, and the two translations from bounded arithmetic proofs into propositional proofs. We present some theorems on bounding the lengths of propositional interpolants in terms of cutfree proof length and in terms of the lengths of resolution refutations. We then define the RazborovRudich notion of natural proofs of P NP and discuss Razborov's theorem that certain fragments of bounded arithmetic cannot prove superpolynomial lower bounds on circuit size, assuming a strong cryptographic conjecture. Finally, a complete presentation of a proof of the theorem of Razborov is given. 1 Review of Computational Complexity 1.1 Feasibility This article will be concerned with various "feasible" forms of computability and of provability. For something to be feasibly computable, it must be computable in practice in the real world, not merely e#ectively computable in the sense of being recursively computable.
On the Bounded Version of Hilbert's Tenth Problem
, 2002
"... Hilbert's Tenth problem concerned the decidability of Diophantine equations over the integers. Its negative solution, the MRDP theorem, amounted to showing that the class of formula of the form (9~y)P (~x; ~y) = Q(~x; ~y) over the natural numbers where P; Q are polynomials is equivalent to the ..."
Abstract

Cited by 3 (1 self)
 Add to MetaCart
Hilbert's Tenth problem concerned the decidability of Diophantine equations over the integers. Its negative solution, the MRDP theorem, amounted to showing that the class of formula of the form (9~y)P (~x; ~y) = Q(~x; ~y) over the natural numbers where P; Q are polynomials is equivalent to the class of recursively enumerable sets. Provability of this theorem in weak fragments of arithmetic is known to imply NP=coNP. The bounded form of Hilbert's Tenth problem is whether the NPpredicates are the class D of predicates given by formulas of the form (9~y)[( X j y j 2 j P i x i j k ) ^ P (~x; ~y) = Q(~x; ~y)] where P; Q are polynomials with coecients in N. This problem is related to the average case completeness of certain NPproblems. In this paper we give lower bounds on the provability of both these problems in weak fragments of arithmetic and we show certain closure properties of the Dpredicates. We show the theory I 5 E1 can not prove D =NP. Here I m E1 has a nite ...
A National Users Facility
"... This report was prepared as an account of work sponsored by an agency of the United States Government. Neither the United States Government nor any agency thereof, nor any of their employees, makes any warranty, express or implied, or assumes any legal liability or responsibility for the accuracy, ..."
Abstract
 Add to MetaCart
This report was prepared as an account of work sponsored by an agency of the United States Government. Neither the United States Government nor any agency thereof, nor any of their employees, makes any warranty, express or implied, or assumes any legal liability or responsibility for the accuracy, completeness, or usefulness of any information, apparatus, product, or process disclosed, or represents that its use would not infringe privately owned rights. Reference herein to any specific commercial product, process, or service by trade name, trademark, manufacturer, or otherwise does not necessarily constitute or imply its endorsement, recommendation, or favoring by the United States Government or any agency thereof. The views and opinions of authors expressed herein do not necessarily state or reflect those of the United States Government or any agency thereof.