Approximate counting in bounded arithmetic

Abstract

We 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 ).

Formalizing Randomized Matching Algorithms

“…Intuitively the pair (F, G) witnesses that S − E ≤ |Φ| ≤ S + E. This allows him to show many properties, expected from cardinality comparison, that are satisfied by his method within VPV + sWPHP(L FP ) (see Lemmas 2.10 and 2.11 in [15]). It is worth noting that proving the uniqueness of cardinality within some error seems to be the best we can do within bounded arithmetic, where exact counting is not available.…”

“…In [15], Jeřábek developed an even more systematic approach by showing that for any bounded P/poly definable set, there exists a suitable pair of surjective "counting functions" which can approximate the cardinality of the set up to a polynomially small error. From this and other results he argued convincingly that VPV+sWPHP(L FP ) is the "right" theory for reasoning about probabilistic polynomial time algorithms.…”

“…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.…”

“…As a result, Jeřábek proposed in [15] a systematic and sophisticated framework to justify his definition of size comparison. He observed that estimating the size of a P/poly definable set Φ ⊆ [0, 2 n ) within an error 2 n /poly(n) is the same as estimating Pr X∈[0,2 n ) [X ∈ Φ] within an error 1/poly(n), which can be solved by drawing poly(n) independent random samples X ∈ [0, 2 n ) and check if X ∈ Φ.…”