Logics for probabilistic programming (Extended Abstract)

This paper introduces a logic for probabilistic programming<supscrpt>+</supscrpt> PROB-DL (for <underline>prob</underline>abilistic <underline>d</underline>ynamic <underline>l</underline>ogic; see Section 2 for a formal definition). This logic has “dynamic” modal operators in which programs appear, as in Pratt's [1976] dynamic logic DL. However the programs of PROB-DL contain constructs for probabilistic branching and looping whereas DL is restricted to nondeterministic programs. The formula {a}<subscrpt>&sgr;</subscrpt>p of PROB-DL denotes “with measure ≥&sgr;, formula p holds after executing program a.” In Section 3, we show that PROB-DL has a complete and consistent axiomatization, (using techniques derived from Parikh's [1978] completeness proof for the propositional dynamic logic). Section 4 presents a <underline>prob</underline>abilistic <underline>q</underline>uantified <underline>b</underline>oolean <underline>l</underline>ogic (PROB-QBF) which also has applications to probabilistic programming.

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