On Measure Quantifiers in First-Order Arithmetic

Abstract

We study the logic obtained by endowing the language of first-order arithmetic with second-order measure quantifiers. This new kind of quantification allows us to express that the argument formula is true in a certain portion of all possible interpretations of the quantified variable. We show that first-order arithmetic with measure quantifiers is capable of formalizing simple results from probability theory and, most importantly, of representing every recursive random function. Moreover, we introduce a realizability interpretation of this logic in which programs have access to an oracle from the Cantor space.

An Arithmetic Theory to Characterize Poly-Time Random Functions

“…The global structure of our proof follows a similar path, with an algebra of oracle recursive function, called POR, playing the role of our Cobham-style function algebra. In our case, functions are poly-time computable by PTMs and the theory is randomized RS 1 2 . After introducing these classes, we show that the random functions which are Σ b 1 -representable in RS 1 2 are precisely those in POR, and that POR is equivalent (in a very specific sense) to the class of functions computed by PTMs running in polynomial time.…”

“…Indeed, functions in POR access randomness in a rather different way with respect to PTMs, and relating these models requires some effort, that involves long chains of intermediate simulations. Concretely, we start by defining the class of oracle functions over strings, the new theory RS 1 2 , strongly inspired by [9], but over a "probabilistic word language", and considering a slightly modified notion of Σ b i -representability, fitting the domain of our peculiar oracle functions. Then, we prove that the class of random functions computable in polynomial time, called RFP, is precisely the class of functions which are Σ b 1 -representable in RS 1 2 in three steps:…”

Curry and Howard Meet Borel

“…i.e. balanced 3 and {[], hn}-free. The rule (@ ∩ ) is a standard extension of rule (@) of C → to finite intersections.…”

“…Recently, the authors have proposed to use counting quantifiers [2,3] as a means to express probabilities within a logical language. These quantifiers, unlike standard ones, determine not only the existence of an assignment of values to variables with certain characteristics, but rather count how many of those assignments exist.…”

“…These quantifiers, unlike standard ones, determine not only the existence of an assignment of values to variables with certain characteristics, but rather count how many of those assignments exist. Recent works show that classical propositional logic enriched with counting quantifiers corresponds to Wagner's hierarchy of counting complexity classes [55] (itself intimately linked to probabilistic complexity), and that Peano Arithmetics enriched with similar quantitative quantifiers yields a system capable of speaking of randomized computation in the same sense in which standard PA models deterministic computation [3]. One may now wonder whether all this scale to something like a proper CHC.…”