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NP search problems in low fragments of bounded arithmetic. (English) Zbl 1118.03051

The paper studies \(\forall\Sigma^b_1\)-consequences (i.e., definable NP search problems) of some fragments of bounded arithmetic. The known characterizations of \(\Sigma^b_1(T^i_2)\) for \(i\geq2\) in terms of reflection principles, or Herbrandization of induction axioms, are rather artificial and lack a direct combinatorial appeal useful for separation of the theories and other applications. The authors provide several combinatorial descriptions of the NP search problems definable in \(T^2_2\) and \(T^3_2\). Specifically, \(\Sigma^b_1(T^2_2)\) is characterized by a generalization of polynomial local search problems (PLS) called coloured PLS, a reflection principle for exponentially long resolution proofs in a succinct representation, or a statement of totality of so-called verifiable recursion (VR) programs. \(\Sigma^b_1(T^3_2)\) is similarly characterized by a reflection principle for the \(\roman{PK}_1=R(n)\) proof system, or a variant of the VR-totality problem.

MSC:

03F30 First-order arithmetic and fragments
03F20 Complexity of proofs
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