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Bounded Arithmetic and Lower Bounds in Boolean Complexity
 Feasible Mathematics II
, 1993
"... We study the question of provability of lower bounds on the complexity of explicitly given Boolean functions in weak fragments of Peano Arithmetic. To that end, we analyze what is the right fragment capturing the kind of techniques existing in Boolean complexity at present. We give both formal and i ..."
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Cited by 46 (5 self)
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We study the question of provability of lower bounds on the complexity of explicitly given Boolean functions in weak fragments of Peano Arithmetic. To that end, we analyze what is the right fragment capturing the kind of techniques existing in Boolean complexity at present. We give both formal and informal arguments supporting the claim that a conceivable answer is V 1 (which, in view of RSUV isomorphism, is equivalent to S 2 ), although some major results about the complexity of Boolean functions can be proved in (presumably) weaker subsystems like U 1 . As a byproduct of this analysis, we give a more constructive version of the proof of Hastad Switching Lemma which probably is interesting in its own right.
An Equivalence between Second Order Bounded Domain Bounded Arithmetic and First Order Bounded Arithmetic
, 1993
"... We introduce a bounded domain version V 2 (BD) of Buss's second order theory V 2 of bounded arithmetic and show that this version is equivalent to the rst order theory S 3 : More precisely, we construct two natural interpretations V 3 and S 2 (BD) which are inverse to each other and pr ..."
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Cited by 28 (4 self)
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We introduce a bounded domain version V 2 (BD) of Buss's second order theory V 2 of bounded arithmetic and show that this version is equivalent to the rst order theory S 3 : More precisely, we construct two natural interpretations V 3 and S 2 (BD) which are inverse to each other and preserve the syntactic structure of bounded formulae. As a corollary, for the bounded domain case we obtain Buss's result concerning 1 expressibility in V 2 as a direct consequence of his main result for rst order theories. Using only plain corollaries of the cut elimination theorem, we show that V 2 (BD) prove the same formulae where 8 stand for rst order quanti ers. Combined with the above mentioned result this gives an alternative proof of Buss's characterization of 2 functions. All this readily extends to the case V k (BD) vs. S k+1 (i; k 1).
Structure and Definability in General Bounded Arithmetic Theories
, 1999
"... This paper is motivated by the questions: what are the \Sigma ..."
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Cited by 18 (6 self)
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This paper is motivated by the questions: what are the \Sigma
A Bounded Arithmetic Theory for Constant Depth Threshold Circuits
, 1996
"... . We define an extension R 0 2 of the bounded arithmetic theory R 0 2 and show that the class of functions \Sigma b 1 definable in R 0 2 coincides with the computational complexity class TC 0 of functions computable by polynomial size, constant depth threshold circuits. 1 Introduction Th ..."
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Cited by 8 (4 self)
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. We define an extension R 0 2 of the bounded arithmetic theory R 0 2 and show that the class of functions \Sigma b 1 definable in R 0 2 coincides with the computational complexity class TC 0 of functions computable by polynomial size, constant depth threshold circuits. 1 Introduction The theories S i 2 , for i 2 N, of Bounded Arithmetic were introduced by Buss [3]. The language of these theories is the language of Peano Arithmetic extended by symbols for the functions b 1 2 xc, jxj := dlog 2 (x + 1)e and x#y := 2 jxj\Deltajyj . A quantifier of the form 8xt , 9x t with x not occurring in t is called a bounded quantifier. Furthermore, a quantifier of the form 8x jtj , 9x jtj is called sharply bounded. A formula is called (sharply) bounded if all quantifiers in it are (sharply) bounded. The class of bounded formulae is divided into an hierarchy analogous to the arithmetical hierarchy: The class of sharply bounded formulae is denoted \Sigma b 0 or \Pi b 0 . For i...
A ModelTheoretic Property of Sharply Bounded Formulae, with some Applications
, 1998
"... We define a property of substructures of models of arithmetic, that of being lengthinitial , and show that sharply bounded formulae are absolute between a model and its lengthinitial submodels. We use this to prove independence results for some weak fragments of bounded arithmetic by construct ..."
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Cited by 7 (0 self)
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We define a property of substructures of models of arithmetic, that of being lengthinitial , and show that sharply bounded formulae are absolute between a model and its lengthinitial submodels. We use this to prove independence results for some weak fragments of bounded arithmetic by constructing appropriate models as lengthinitial submodels of some given model. Mathematics Subject Classification: 03F30, 03H15 Introduction First we review the definitions of the theories S i 2 and T i 2 of Bounded Arithmetic introduced by S. Buss [2]: The language of these theories is the language of Peano Arithmetic extended by symbols for the functions b 1 2 xc, jxj := dlog 2 (x + 1)e and x#y := 2 jxj\Deltajyj . A quantifier of the form 8x t , 9x t with x not occurring in t is called a bounded quantifier. Furthermore, a quantifier of the form 8x jtj, 9x jtj is called sharply bounded. A formula is called sharply bounded if all quantifiers in it are sharply bounded. The class of sh...
On the b 1 bitcomprehension rule
 Logic Colloquium 98
, 2000
"... Summary. The theory � b 1CR of Bounded Arithmetic axiomatized by the � b 1bitcomprehension rule is defined and shown to be strongly related to the complexity class TC 0. The � b 1definable functions of � b 1CR are those in uniform TC 0, and the � b 2definable functions are computable by counte ..."
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Summary. The theory � b 1CR of Bounded Arithmetic axiomatized by the � b 1bitcomprehension rule is defined and shown to be strongly related to the complexity class TC 0. The � b 1definable functions of � b 1CR are those in uniform TC 0, and the � b 2definable functions are computable by counterexample computations using TC 0functions. The latter is used to show that a collapse of stronger theories to � b 1CR implies that NP is contained in nonuniform TC 0. 1
The Witness Function Method and Provably Recursive Functions of Peano
 Logic, Methodology and Philosophy of Science IX
, 1994
"... This paper presents a new proof of the characterization of the provably recursive functions of the fragments I# n of Peano arithmetic. The proof method also characterizes the # k definable functions of I# n and of theories axiomatized by transfinite induction on ordinals. The proofs are complete ..."
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This paper presents a new proof of the characterization of the provably recursive functions of the fragments I# n of Peano arithmetic. The proof method also characterizes the # k definable functions of I# n and of theories axiomatized by transfinite induction on ordinals. The proofs are completely prooftheoretic and use the method of witness functions and witness oracles.
Parallel computable higher type functionals (Extended Abstract)
 In Proceedings of IEEE 34th Annual Symposium on Foundations of Computer Science, Nov 35
, 1993
"... ) Peter Clote A. Ignjatovic y B. Kapron z 1 Introduction to higher type functionals The primary aim of this paper is to introduce higher type analogues of some familiar parallel complexity classes, and to show that these higher type classes can be characterized in significantly different way ..."
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Cited by 4 (4 self)
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) Peter Clote A. Ignjatovic y B. Kapron z 1 Introduction to higher type functionals The primary aim of this paper is to introduce higher type analogues of some familiar parallel complexity classes, and to show that these higher type classes can be characterized in significantly different ways. Recursiontheoretic, prooftheoretic and machinetheoretic characterizations are given for various classes, providing evidence of their naturalness. In this section, we motivate the approach of our work. In proof theory, primitive recursive functionals of higher type were introduced in Godel's Dialectica [13] paper, where they were used to "witness" the truth of arithmetic formulas. For instance, a function f witnesses the formula 8x9y\Phi(x; y), where \Phi is quantifierfree, provided that 8x\Phi(x; f(x)); while a type 2 functional F witnesses the formula 8x9y8u9v\Phi(x; y; u; v), provided that 8x8u\Phi(x; f(x); u; F (x; f(x); u)): Godel's formal system T is a variant of the finit...
A Note on the Relation between Polynomial Time Functionals and Constable's Class K
 IN KLEINEBUNING, EDITOR, COMPUTER SCIENCE LOGIC. SPRINGER LECTURE NOTES IN COMPUTER SCIENCE
, 1996
"... . A result claimed without proof by R. Constable in a STOC73 paper is here corrected: a strictly increasing function f is presented for which Constable's class K(f) is properly contained in FP (f ), the collection of functions polynomial time computable in f . ..."
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Cited by 3 (1 self)
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. A result claimed without proof by R. Constable in a STOC73 paper is here corrected: a strictly increasing function f is presented for which Constable's class K(f) is properly contained in FP (f ), the collection of functions polynomial time computable in f .