Reasoning About States of Probabilistic Sequential Programs

Abstract

Abstract. A complete and decidable propositional logic for reasoning about states of probabilistic sequential programs is presented. The state logic is then used to obtain a sound Hoare-style calculus for basic probabilistic sequential programs. The Hoare calculus presented herein is the first probabilistic Hoare calculus with a complete and decidable state logic that has truth-functional propositional (not arithmetical) connectives. The models of the state logic are obtained exogenously by attaching sub-probability measures to valuations over memory cells. In order to achieve complete and recursive axiomatization of the state logic, the probabilities are taken in arbitrary real closed fields.

Reasoning about probabilistic sequential programs

“…It turns out that the variables in the state language are sufficient to keep track of individual contributions of the alternate choices and it is not necessary to mark the choices explicitly to achieve completeness. Another difference between our work and the work in [8] is that is that we do not distinguish between possibility and probability. This is in accordance with standard works on probabilistic programs, and it simplifies the proof of completeness of Hoare logic.…”

“…The gap between [31] and [10] was addressed in [8], which provides a sound Hoare logic with a complete and decidable state assertion logic. The Hoare rule for alternative construct is tackled using two key ingredients.…”

“…The conditional formulas (η/bm) and (η/(¬ bm)) in the state logic can then be used to combine information of the alternative paths. Another feature of the state language in [8] is that a distinction is maintained between possibility and probability, yielding a more expressive state language. The completeness of the Hoare logic was left as an open question.…”

“…This paper addresses the gap between [31] and [10,8], providing a complete and decidable Hoare logic for iteration-free probabilistic programs using a complete and decidable truth-functional probabilistic state assertion logic. The Hoare calculus provided herein was arrived at while attempting to prove the completeness of the Hoare logic proposed in [8].…”

“…This is not a serious restriction; for instance, r is taken to be 1 2 in probabilistic Turing machines. The alternative choice construct described here is the standard if-then-else construct and not the modified marked if-then-else proposed in [8]. It turns out that the variables in the state language are sufficient to keep track of individual contributions of the alternate choices and it is not necessary to mark the choices explicitly to achieve completeness.…”