Probabilistic Logic over Equations and Domain Restrictions

We propose and study a probabilistic logic over an algebraic basis, including equations and domain restrictions. The logic combines aspects from classical logic and equational logic with an exogenous approach to quantitative probabilistic reasoning. We present a sound and weakly complete axiomatization for the logic, parameterized by an equational specification of the algebraic basis coupled with the intended domain restrictions. We show that the satisfiability problem for the logic is decidable, under the assumption that its algebraic basis is given by means of a convergent rewriting system, and, additionally, that the axiomatization of domain restrictions enjoys a suitable subterm property. For this purpose, we provide a polynomial reduction to Satisfiability Modulo Theories. As a consequence, we get that validity in the logic is also decidable. Furthermore, under the assumption that the rewriting system that defines the equational basis underlying the logic is also subterm convergent, we show that the resulting satisfiability problem is NP-complete, and thus the validity problem is coNP-complete. We test the logic with meaningful examples in information security, namely by verifying and estimating the probability of the existence of o✏ine guessing attacks to cryptographic protocols.