Probabilistic Guarded Commands Mechanized in HOL

The probabilistic guarded-command language pGCL [Carroll Morgan, Annabelle McIver. pGCL: formal reasoning for random algorithms. South African Computer Journal (1999)] contains both demonic and probabilistic nondeterminism, which makes it suitable for reasoning about distributed random algorithms [Carroll Morgan. Proof rules for probabilistic loops. In Proceedings of the BCS-FACS 7th Refinement Workshop. He Jifeng, John Cooke and Peter Wallis (eds). Springer Verlag Workshops in Computing, 1996]. Proofs are based on weakest precondition semantics, using an underlying logic of real- (rather than Boolean-) valued functions. We present a mechanization of the quantitative logic for pGCL [Carroll Morgan, Annabelle McIver, and Karen Seidel, Probabilistic predicate transformers. ACM Transactions on Programming Languages and Systems, 18(3): 325-353, May 1996] using the HOL theorem prover [M.J.C. Gordon and T.F. Melham. Introduction to HOL (A theorem-proving environment for higher order logic). Cambridge University Press, 1993], including a proof that all pGCL commands satisfy the new condition sublinearity, the quantitative generalization of conjunctivity for standard GCL [E.W. Dijkstra. A Discipline of Programming. Prentice Hall, 1976]. The mechanized theory also supports the creation of an automatic proof tool which takes as input an annotated pGCL program and its partial correctness specification, and derives from that a sufficient set of verification conditions. This is employed to verify the partial correctness of the probabilistic voting stage in Rabin's mutual-exclusion algorithm [Eyal Kushilevitz and Michael O. Rabin. Randomized mutual exclusion algorithms revisited. In Maurice Herlihy, editor, Proceedings of the 11th Annual Symposium on Principles of Distributed Computing, pages 275-283, Vancouver, BC, Canada, August 1992. ACM Press].

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