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

Abstract

This article concerns the second-order systems U 1 2 and V 1 2 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 S 1 2 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 U 1 2 . 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 U 1 2 and V 1 2 depend primarily on the topology of the underlying graph, not the number of rounds in the local improvement games. The theory U 1 2 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 V 1 2 . Related results are obtained for local improvement principles with one improvement round and for local improvement over rectangular grids.

Formalizing Termination Proofs under Polynomial Quasi-interpretations

“…For an easy example, assume that two sets U and V encode binary strings respectively of length m and n in such a way that j ∈ U ⇔ "the j th bit of the string U is 1" and j ∈ U ⇔ "the j th bit of the string U is 0" for each j < m. Then the concatenation W = U V , the string U followed by V , is defined by (Σ b,1 0 -CA) as follows. To readers who are not familiar with second order bounded arithmetic, it might be of interest to outline the proof that every polynomial-space computable function can be defined in U 1 2 . The argument is commonly known as the divide-and-conquer method, which was originally used to show the classical inclusion NPSPACE ⊆ PSPACE [18].…”

Propositional Proofs in Frege and Extended Frege Systems (Abstract)

“…The results of [19] include that the LI principle is many-one complete for the NP search problems of V 1 2 ; it follows that LI is equivalent to partial consistency statements for extended Frege systems. Beckmann and Buss [6] subsequently proved that LI log is provably equivalent (in S 1 2 ) to LI and that the linear local improvement principle LLI is provable in U 1 2 . The LLI principle thus has quasipolynomial size Frege proofs.…”

Total Search Problems in Bounded Arithmetic and Improved Witnessing

“…The aim of this paper is to provide characterizations of ∀Σ B 1 (VC), for C below P, and ∀Σ B 1 (V 1 ) in terms of subclasses of TFNP, using new-style witnessing theorems in which the correctness of witnessing functions is provable in V 0these new-style witnessing theorems are similar to the ones in [2][3][4]15,24], except for the correctness of witnessing functions that is now proved over a weaker theory. To achieve our aim, we define the class of total N-AC 0 search problem as those total NP search problems for which solutions are verifiable in AC 0 rather than in P. We denote this class as ∀∃AC 0 .…”