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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 / ..."
Abstract
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Cited by 27 (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 .
Comparing Constructive Arithmetical Theories Based On NP-PIND and coNP-PIND
"... In this note we show that the intuitionistic theory of polynomial induction on Π b+ 1-formulas does not imply the intuitionistic theory IS1 2 of polynomial induction on Σ b+ 1-formulas. We also show the converse assuming the Polynomial Hierarchy does not collapse. Similar results hold also for lengt ..."
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Cited by 4 (4 self)
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In this note we show that the intuitionistic theory of polynomial induction on Π b+ 1-formulas does not imply the intuitionistic theory IS1 2 of polynomial induction on Σ b+ 1-formulas. We also show the converse assuming the Polynomial Hierarchy does not collapse. Similar results hold also for length induction in place of polynomial induction. We also investigate the relation between various other intuitionistic first-order theories of bounded arithmetic. Our method is mostly semantical, we use Kripke models of the theories.
