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Polynomial Induction and Length Minimization in Intuitionistic Bounded Arithmetic
"... It is shown that the feasibly constructive arithmetic theory IPV does not prove (double negation of) LMIN(NP), unless the polynomial hierarchy CPV-provably collapses. It is proved that PV plus (double negation of) LMIN(NP) intuitionistically proves PIND(coNP). It is observed that PV+ PIND(NP∪coNP) d ..."
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It is shown that the feasibly constructive arithmetic theory IPV does not prove (double negation of) LMIN(NP), unless the polynomial hierarchy CPV-provably collapses. It is proved that PV plus (double negation of) LMIN(NP) intuitionistically proves PIND(coNP). It is observed that PV+ PIND(NP∪coNP) does not intuitionistically prove NPB, a scheme which states that the extended Frege systems are not polynomially bounded.
An Independence Result For Intuitionistic Bounded Arithmetic
"... It is shown that the intuitionistic theory of polynomial induction on positive Π b 1 (coNP) formulas does not prove the sentence ¬¬∀x, y∃z ≤ y(x ≤ |y | → x = |z|). This implies the unprovability of the scheme ¬¬PIND(Σ b+ 1) in the mentioned theory. However, this theory contains the sentence ∀x, y¬¬ ..."
Abstract
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It is shown that the intuitionistic theory of polynomial induction on positive Π b 1 (coNP) formulas does not prove the sentence ¬¬∀x, y∃z ≤ y(x ≤ |y | → x = |z|). This implies the unprovability of the scheme ¬¬PIND(Σ b+ 1) in the mentioned theory. However, this theory contains the sentence ∀x, y¬¬∃z ≤ y(x ≤ |y | → x = |z|). The above independence result is proved by constructing an ω-chain of submodels of a countable model of S2 +Ω3 +¬exp such that none of the worlds in the chain satisfies the sentence, and interpreting the chain as a Kripke model.
