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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 a ..."
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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).
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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Cited by 5 (0 self)
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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.
The Witness Function Method and Provably Recursive Functions of Peano Arithmetic
"... Abstract This paper presents a new proof of the characterization of the provably recursive functions of the fragments I\Sigma n of Peano arithmetic. The proof method also characterizes the \Sigma kdefinable functions of I\Sigma n and of theories axiomatized by transfinite induction on ordinals. The ..."
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Abstract This paper presents a new proof of the characterization of the provably recursive functions of the fragments I\Sigma n of Peano arithmetic. The proof method also characterizes the \Sigma kdefinable functions of I\Sigma 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. Similar methods also yield a new proof of Parson's theorem on the conservativity of the \Sigma n+1induction rule over the \Sigma ninduction axioms. A new proof of the conservativity of B\Sigma n+1 over I\Sigma n is given. The proof methods provide new analogies between Peano arithmetic and bounded arithmetic.