Batz, Kevin Stefan; (2024) Automated Deductive Verification of Probabilistic Programs. Doctoral thesis (Ph.D), RWTH Aachen University.

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Abstract

We study both foundational and practical aspects of the automated deductive verification of discrete probabilistic programs. Our special emphasis is on the verification of possibly unbounded loops by quantitative loop invariants. We build upon Kozen's, McIver & Morgan's, and Kaminski's weakest preexpectation calculus for reasoning about expected outcomes such as the probability of terminating in some postcondition or the expected final value of a program variable. The weakest preexpectation calculus replaces predicates from classical program verification by more general expectations --- functions mapping program states to numbers instead of truth values. We study a syntax for specifying expectations, providing a foundation for automated deductive probabilistic program verifiers. We prove that our syntax is expressive and hence obtain a relatively complete verification system for reasoning about expected outcomes. We then present three different approaches for automating the verification of bounds on expected outcomes of linear loops, all of which combine weakest preexpectation reasoning with Satisfiability Modulo Theories (SMT) techniques: 1. We revisit two well-established verification techniques for transition systems, k-induction and bounded model checking (BMC), in the more general setting of bounding least fixpoints of functions over complete lattices, yielding latticed k-induction and latticed BMC. Instantiating our latticed techniques with the weakest preexpectation calculus enables the fully automatic verification of linear loops taken from the literature. 2. We present a counterexample-guided inductive synthesis approach for the synthesis of quantitative loop invariants. This enables the fully automatic verification of bounds on both expected outcomes and expected runtimes. Our implementation synthesizes quantitative invariants of various linear loops taken from the literature, can beat state-of-the-art probabilistic model checkers on finite-state programs, and is competitive with modern tools dedicated to invariant synthesis or expected runtime reasoning. 3. We present PrIC3 --- the first truly quantitative extension of the state-of-the-art qualitative symbolic model checking algorithm IC3 to symbolic model checking of Markov decision processes. By marrying PrIC3 with the weakest preexpectation calculus, we obtain a symbolic model checking algorithm for probabilistic programs. Alongside, we present an implementation of PrIC3 featuring IC3-specific techniques such as generalization.

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