2-Xor revisited: satisfiability and probabilities of functions

TL;DR

This paper provides an explicit formula for the probability that a random 2-Xor expression is satisfiable and extends it to the probability of computing specific Boolean functions, using combinatorial graph models and generating functions.

Contribution

It offers a new explicit expression for satisfiability probability and refines it to compute probabilities of specific Boolean functions, employing analytic combinatorics and graph representations.

Findings

Explicit formula for 2-Xor-Sat satisfiability probability

Probability expressions for specific Boolean functions

Use of multigraphs and generating functions in analysis

Abstract

The problem 2-Xor-Sat asks for the probability that a random expression, built as a conjunction of clauses $x \oplus y$, is satisfiable. We revisit this classical problem by giving an alternative, explicit expression of this probability. We then consider a refinement of it, namely the probability that a random expression computes a specific Boolean function. The answers to both problems involve a description of 2-Xor expressions as multigraphs and use classical methods of analytic combinatorics by expressing probabilities through coefficients of generating functions.