PMAF: an algebraic framework for static analysis of probabilistic programs

Automatically establishing that a probabilistic program satisfies some property ϕ is a challenging problem. While a sampling-based approach—which involves running the program repeatedly—can suggest that ϕ holds, to establish that the program satisfies ϕ, analysis techniques must be used. Despite recent successes, probabilistic static analyses are still more difficult to design and implement than their deterministic counterparts. This paper presents a framework, called PMAF, for designing, implementing, and proving the correctness of static analyses of probabilistic programs with challenging features such as recursion, unstructured control-flow, divergence, nondeterminism, and continuous distributions. PMAF introduces pre-Markov algebras to factor out common parts of different analyses. To perform interprocedural analysis and to create procedure summaries, PMAF extends ideas from non-probabilistic interprocedural dataflow analysis to the probabilistic setting. One novelty is that PMAF is based on a semantics formulated in terms of a control-flow hyper-graph for each procedure, rather than a standard control-flow graph. To evaluate its effectiveness, PMAF has been used to reformulate and implement existing intraprocedural analyses for Bayesian-inference and the Markov decision problem, by creating corresponding interprocedural analyses. Additionally, PMAF has been used to implement a new interprocedural linear expectation-invariant analysis. Experiments with benchmark programs for the three analyses demonstrate that the approach is practical.