Dual weak pigeonhole principle, Boolean complexity, and derandomization

Abstract

We study the extension (introduced as BT in [5]) of the theory S 1 2 by instances of the dual (onto) weak pigeonhole principle for p-time functions, dWPHP (PV ) xx 2 . We propose a natural framework for formalization of randomized algorithms in bounded arithmetic, and use it to provide a strengthening of Wilkie's witnessing theorem for S 1 2 +dWPHP (PV ). We construct a propositional proof system WF (based on a reformulation of Extended Frege in terms of Boolean circuits), which captures the ∀Π b 1 -consequences of S 1 2 +dWPHP (PV ). We also show that WF p-simulates the Unstructured Extended Nullstellensatz proof system of [2].We prove that dWPHP (PV ) is (over S 1 2 ) equivalent to a statement asserting the existence of a family of Boolean functions with exponential circuit complexity. Building on this result, we formalize the Nisan-Wigderson construction (derandomization of probabilistic p-time algorithms) in a conservative extension of S 1 2 + dWPHP (PV ).

Formalizing Randomized Matching Algorithms

“…This gives us a polytime random sampling algorithm for approximating the size of Φ. Since a counting argument [13] can be formalized within VPV+ sWPHP(L FP ) to show the existence of suitable average-case hard functions for constructing Nisan-Wigderson generators, this random sampling algorithm can be derandomized to show the existence of an approximate cardinality S of Φ for any given error E = 2 n /poly(n) in the following sense. The theory VPV + sWPHP(L FP ) proves the existence of S, y and a pair of P/poly "counting functions" (F, G)…”

“…Using this method, Jeřábek developed tools for describing algorithms in ZPP and RP. He also showed in [13,14] that the theory VPV + sWPHP(L FP ) is powerful enough to formalize proofs of very sophisticated derandomization results, e.g. the Nisan-Wigderson theorem [24] and the Impagliazzo-Wigderson theorem [12].…”

“…Jeřábek's framework for probabilistic reasoning. In this section, we give a brief and simplified overview of Jeřábek's framework [13,14,15] for probabilistic reasoning within VPV + sWPHP(L FP ). For more complete definitions and results, the reader is referred to Jeřábek's work.…”

“…The method proposed by Jeřábek in [13,14,15] is based on the following simple observation: if Γ(n) and Φ(n) are definable sets and there is a function F mapping Γ(n) onto Φ(n), then the cardinality of Φ(n) is at most the cardinality of Γ(n). Thus instead of counting the sets Γ(n) and Φ(n) directly, we can compare the sizes of Γ(n) and Φ(n) by showing the existence of a surjection F , which in many cases can be easily carried out within weak theories of bounded arithmetic.…”

Approximate counting in bounded arithmetic

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AbstractWe develop approximate counting of sets definable by Boolean circuits in bounded arithmetic using the dual weak pigeonhole principle (dWPHP (P V )), as a generalization of results from [15]. We discuss applications to formalization of randomized complexity classes (such as BPP , APP , MA, AM ) in P V 1 + dWPHP (P V ).

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