Bounded Arithmetic and Randomized Computation

In this work, we develop a randomized bounded arithmetic for probabilistic computation, following the approach adopted by Buss for non-randomized computation. This work relies on a notion of representability inspired by of Buss' one, but depending on a non-standard quantitative and measurable semantic. Then, we establish that the representable functions are exactly the ones in PPT. Finally, we extend the language of our arithmetic with a measure quantifier, which is true if and only if the quantified formula's semantic has measure greater than a given threshold. This allows us to define purely logical characterizations of standard probabilistic complexity classes such as BPP, RP, co-RP and ZPP.