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.

Hilbert's Tenth problem concerned the decidability of Diophantine equations over the integers. Its negative solution, the MRDP theorem, amounted to showing that the class of formula of the form (9~y)P (~x; ~y) = Q(~x; ~y) over the natural numbers where P; Q are polynomials is equivalent to the class of recursively enumerable sets. Provability of this theorem in weak fragments of arithmetic is known to imply NP=co-NP. The bounded form of Hilbert's Tenth problem is whether the NP-predicates are the class D of predicates given by formulas of the form (9~y)[( X j y j 2 j P i x i j k ) ^ P (~x; ~y) = Q(~x; ~y)] where P; Q are polynomials with coecients in N. This problem is related to the average case completeness of certain NP-problems. In this paper we give lower bounds on the provability of both these problems in weak fragments of arithmetic and we show certain closure properties of the D-predicates. We show the theory I 5 E1 can not prove D =NP. Here I m E1 has a nite ...

definable, and the following theorem shows that these are the only values for which it is. Theorem 2 (Tada and Tatsuta [3]). S xc if and only if k 2 N is a power of 2. Using a model-theoretic method, the author proved in [2] that L 2 is properly weaker than S 2 and that 8x 9y y = b 3 xc is not provable in R 2 . Moreover, a model of L 2 that does not satisfy 8x (x = 0 9y y = Sx) was constructed in that paper. Here we show that a minor modification of the latter model construction gives a model of S 2 that does not satisfy 8x (x = 0 9y y = Sx), hence a model-theoretic proof of Theorem 1. Furthermore we show that R 2 can also define division only by powers of two. Theorem 3. R xc only if k 2 N is a power of 2. In fact, the model of R 2 constructed in [2] does not satisfy 8x9yy = b xc for any k that has an odd prime factor. We review the essential notions and constructions from [2]. For a model M of BASIC, let log M := f a 2 M ; a jbj for some b 2 M g : We c

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.

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