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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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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 .
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.