Improved witnessing and local improvement principles for second-order bounded arithmetic

This article concerns the second-order systems U12 and V12 of bounded arithmetic, which have proof-theoretic strengths corresponding to polynomial-space and exponential-time computation. We formulate improved witnessing theorems for these two theories by using S12 as a base theory for proving the correctness of the polynomial-space or exponential-time witnessing functions. We develop the theory of nondeterministic polynomial-space computation, including Savitch's theorem, in U12. Kołodziejczyk et al. [2011] have introduced local improvement properties to characterize the provably total NP functions of these second-order theories. We show that the strengths of their local improvement principles over U12 and V12 depend primarily on the topology of the underlying graph, not the number of rounds in the local improvement games. The theory U12 proves the local improvement principle for linear graphs even without restricting to logarithmically many rounds. The local improvement principle for grid graphs with only logarithmically-many rounds is complete for the provably total NP search problems of V12. Related results are obtained for local improvement principles with one improvement round and for local improvement over rectangular grids.

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