Probabilistic Recursion Theory and Implicit Computational Complexity (Long Version)

TL;DR

This paper extends classical recursion theory to probabilistic functions, providing a framework that characterizes polynomial-time sampleable distributions relevant to complexity and cryptography.

Contribution

It introduces a probabilistic recursion algebra that generalizes classical recursive functions and captures polytime sampleable distributions.

Findings

Probabilistic computable functions characterized by a generalized recursion algebra.

Framework captures polytime sampleable distributions relevant to cryptography.

Extension of classical recursion theory to probabilistic and complexity contexts.

Abstract

We show that probabilistic computable functions, i.e., those functions outputting distributions and computed by probabilistic Turing machines, can be characterized by a natural generalization of Church and Kleene's partial recursive functions. The obtained algebra, following Leivant, can be restricted so as to capture the notion of polytime sampleable distributions, a key concept in average-case complexity and cryptography.