Towards logical foundations for probabilistic computation

Abstract

The overall purpose of the present work is to lay the foundations for a new approach to bridge logic and probabilistic computation. To this aim we introduce extensions of classical and intuitionistic propositional logic with counting quantifiers, that is, quantifiers that measure to which extent a formula is true. The resulting systems, called $\mathrm{cCPL}$ and $\mathrm{iCPL}$, respectively, admit a natural semantics, based on the Borel σ-algebra of the Cantor space, together with a sound and complete proof system. Our main results consist in relating $\mathrm{cCPL}$ and $\mathrm{iCPL}$ with some central concepts in the study of probabilistic computation. On the one hand, the validity of $\mathrm{cCPL}$-formulae in prenex form characterizes the corresponding level of Wagner's hierarchy of counting complexity classes, closely related to probabilistic complexity. On the other hand, proofs in $\mathrm{iCPL}$ correspond, in the sense of Curry and Howard, to typing derivations for a randomized extension of the λ-calculus, so that counting quantifiers reveal the probability of termination of the underlying probabilistic programs.