Correctness by construction for probabilistic programs

07/30/2020

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by   Annabelle McIver, et al.

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The "correct by construction" paradigm is an important component of modern Formal Methods, and here we use the probabilistic Guarded-Command Language 𝑝𝐺𝐶𝐿 to illustrate its application to 𝑝𝑟𝑜𝑏𝑎𝑏𝑖𝑙𝑖𝑠𝑡𝑖𝑐 programming. 𝑝𝐺𝐶𝐿 extends Dijkstra's guarded-command language 𝐺𝐶𝐿 with probabilistic choice, and is equipped with a correctness-preserving refinement relation (⊑) that enables compact, abstract specifications of probabilistic properties to be transformed gradually to concrete, executable code by applying mathematical insights in a systematic and layered way. Characteristically for "correctness by construction", as far as possible the reasoning in each refinement-step layer does not depend on earlier layers, and does not affect later ones. We demonstrate the technique by deriving a fair-coin implementation of any given discrete probability distribution. In the special case of simulating a fair die, our correct-by-construction algorithm turns out to be "within spitting distance" of Knuth and Yao's optimal solution.