Abstract.
βThe paper establishes lower bounds on the provability of π=NP and the MRDP theorem in weak fragments of arithmetic. The theory I 5 E 1 is shown to be unable to prove π=NP. This non-provability result is used to show that I 5 E 1 cannot prove the MRDP theorem. On the other hand it is shown that I 1 E 1 proves π contains all predicates of the form (βiβ€|b|)P(i,x)^Q(i,x) where ^ is =, <, or β€, and I 0 E 1 proves π contains all predicates of the form (βiβ€b)P(i,x)=Q(i,x). Here P and Q are polynomials. A conjecture is made that π contains NLOGTIME. However, it is shown that this conjecture would not be sufficient to imply π=N P. Weak reductions to equality are then considered as a way of showing π=NP. It is shown that the bit-wise less than predicate, β€2, and equality are both co-NLOGTIME complete under FDLOGTIME reductions. This is used to show that if the FDLOGTIME functions are definable in π then π=N P.
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Received: 13 July 2001 / Revised version: 9 April 2002 / Published online: 19 December 2002
Key words or phrases:βBounded Arithmetic β Bounded Diophantine Complexity
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Pollett, C. On the bounded version of Hilbert's tenth problem. Arch. Math. Logic 42, 469β488 (2003). https://doi.org/10.1007/s00153-002-0162-y
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DOI: https://doi.org/10.1007/s00153-002-0162-y