On the bounded version of Hilbert's tenth problem

ABSTRACT Hilbert's Tenth problem on erned the de idability of Diophantine equations over the integers. Its negative solution, the MRDP theorem, amounted to showing that the lass 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 lass of re ursively enumerable sets. Provability of this theorem in weak fragments of arithmeti is known to imply NP= o-NP. The bounded form of Hilbert's Tenth problem is whether the NP-predi ates are the lass D of predi ates given by formulas of the form (9~y)[(Xj yj 2jPi xijk) ^ P (~x; ~y) = Q(~x; ~y)℄ where P;Q are polynomials with oeÆ ients in N. This problem is related to the average ase ompleteness of ertain NP-problems. In this paper we give lower bounds on the provability of both these problems in weak fragments of arithmeti and we show ertain losure properties of the D-predi ates. We show the theory I5E1 an not prove D =NP. Here ImE1 has a nite set of axioms for the language L2 := f , 0, S, +, x y, jxj, 2jxjjyj, bx=2i , and x : yg together with indu tion on bounded existential formulas up to m lengths of a number. We use the non-provability of D = NP to show that I5E1 annot prove the MRDP theorem. As eviden e that I5E1 is not that weak a theory we show that I1E1 proves D ontains all predi ates of the form (8~i jbj)P (~i; ~x) Æ Q(~i; ~x) where Æ is =, <, or , and I0E1 proves D ontains all predi ates of the form (8~i b)P (~i; ~x) Æ Q(~i; ~x) where Æ is = or . Here P and Q are polynomials. We onje ture that D ontains NLOGTIME. We show, however, that ENTIME[tO(1)℄ =NTIME[tO(1)℄ for t(jxj) in our language su h that t(jxj) log jxj, t(jxjk) 2 O([t(jxj)℄k) o(jxj) for all k. Here EC means the losure of the lass C under quanti ers of the form (9y < t) for t an L2-term. This implies ENPOLYLOGTIME(NP and hen e our onje ture would not be suÆ ient to imply D = NP . We then onsider weak redu tions to equality as a way of showing D =NP. We show the predi ates 2 of Jones and Matiyasevi h [8℄ and equality are both o-NLOGTIME omplete under FDLOGTIME redu tions. We show that if the FDLOGTIME fun tions are de nable in D then D = NP .