Random 2 XORSAT Phase Transition

Abstract

We consider the 2-XOR satisfiability problem, in which each instance is a formula that is a conjunction of Boolean equations of the form x ⊕ y=0 or x ⊕ y=1. Formula of size m on n Boolean variables are chosen uniformly at random from among all ${n(n-1)\choose m}$possible choices. When c<1/2 and as n tends to infinity, the probability p(n,m=cn) that a random 2-XOR formula is satisfiable, tends to the threshold function exp (c/2)⋅(1−2c)1/4. This gives the asymptotic behavior of random 2-XOR formula in the SAT/UNSAT subcritical phase transition. In this paper, we first prove that the error term in this subcritical region is O(n −1). Then, in the critical region c=1/2, we prove that p(n,n/2)=Θ(n −1/12). Our study relies on the symbolic method and analytical tools coming from generating function theory which also enable us to describe the evolution of $n^{1/12}\ p(n,\frac{n}{2}(1+\mu n^{-1/3}))$as a function of μ. Thus, we propose a complete picture of the finite size scaling associated to the subcritical and critical regions of 2-XORSAT transition.