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Relating the Bounded Arithmetic and Polynomial Time Hierarchies
 Annals of Pure and Applied Logic
, 1994
"... The bounded arithmetic theory S 2 is finitely axiomatized if and only if the polynomial hierarchy provably collapses. If T 2 equals S then T 2 is equal to S 2 and proves that the polynomial time hierarchy collapses to # , and, in fact, to the Boolean hierarchy over # and to # i+1 / ..."
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Cited by 30 (1 self)
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The bounded arithmetic theory S 2 is finitely axiomatized if and only if the polynomial hierarchy provably collapses. If T 2 equals S then T 2 is equal to S 2 and proves that the polynomial time hierarchy collapses to # , and, in fact, to the Boolean hierarchy over # and to # i+1 /poly .
Provably Total Functions in Bounded Arithmetic Theories . . .
, 2002
"... This paper investigates the provably total functions of fragments of first and secondorder Bounded Arithmetic. The (strongly) \Sigma bidefinable functions of Si13 and Ri3 are precisely the (strong) FP\Sigma p i13 [wit, logO(1)] functions. The \Sigma 1,bidefinable functions of V i12 and U i2 ..."
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Cited by 18 (4 self)
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This paper investigates the provably total functions of fragments of first and secondorder Bounded Arithmetic. The (strongly) \Sigma bidefinable functions of Si13 and Ri3 are precisely the (strong) FP\Sigma p i13 [wit, logO(1)] functions. The \Sigma 1,bidefinable functions of V i12 and U i2 are the EXPTIME\Sigma 1,p i1[wit, poly] functions and the \Sigma 1,bi definable functions of V i2 are the EXPTIME\Sigma 1,p ifunctions. We give witnessing theorems for these theories and prove conservation results for Ri3 over Si13 and for U i2 over V i12.
The Witness Function Method and Provably Recursive Functions of Peano Arithmetic
"... 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 ..."
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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.