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Witnessing functions in bounded arithmetic and search problems
Published online by Cambridge University Press: 12 March 2014
Abstract
We investigate the possibility to characterize (multi)functions that are -definable with small i (i = 1, 2, 3) in fragments of bounded arithmetic T2 in terms of natural search problems defined over polynomial-time structures. We obtain the following results:
(1) A reformulation of known characterizations of (multi)functions that are and
-definable in the theories
and
.
(2) New characterizations of (multi)functions that are and
-definable in the theory
.
(3) A new non-conservation result: the theory is not
-conservative over the theory
.
To prove that the theory is not
-conservative over the theory
, we present two examples of a
-principle separating the two theories:
(a) the weak pigeonhole principle WPHP(a2, f, g) formalizing that no function f is a bijection between a2 and a with the inverse g,
(b) the iteration principle Iter(a, R, f) formalizing that no function f defined on a strict partial order ({0,…, a}, R) can have increasing iterates.
- Type
- Research Article
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- Copyright
- Copyright © Association for Symbolic Logic 1998
References
REFERENCES
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