## 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

### Keywords:

bounded arithmetic; search problems; resolution
Full Text:

### References:

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