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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
 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.
Provably Total Functions in
"... This paper investigates the provably total functions of fragments of first and secondorder Bounded Arithmetic. The (strongly) 3 are precisely the (strong) ] functions. The # and U are the EXPTIME [wit, poly] functions and the # definable functions of V 2 are the EXPTI ..."
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This paper investigates the provably total functions of fragments of first and secondorder Bounded Arithmetic. The (strongly) 3 are precisely the (strong) ] functions. The # and U are the EXPTIME [wit, poly] functions and the # definable functions of V 2 are the EXPTIME functions. We give witnessing theorems for these theories and prove conservation results 3 over S 3 and for U 2 over V 2 .