Cost analysis of games, using program logic

Summary form only given. Recent work in probabilistic programming semantics has provided a relatively simple probabilistic extension to predicate transformers, making it possible to treat small imperative probabilistic programs containing both demonic and angelic nondeterminism. That work in turn has extended to provide a probabilistic basis for the modal μ-calculus of Kozen (1983), and leads to a quantitative μ-calculus. Standard (non-probabilistic) μ-calculus can be interpreted either 'normally', over its semantic domain, or as a two-player game between an 'angel' and a 'demon' representing the two forms of choice. Stirling (1995) has argued that the two interpretations correspond. Quantitative p-calculus too can be interpreted both ways, with the novel interpretation being the second: a probabilistic game involving an angel and a demon. Each player seeks a strategy to maximise (resp. minimise) the game's 'outcome', with the steps in the game now being stochastic. That suggests a connection with Markov decision processes, in which players compete for high (resp. low) 'rewards' over a Markov transition system. In this paper we explore that connection, showing how for example discounted Markov decision processes (MDP's) and terminating MDP's can be written as quantitative p-formulae. The 'normal' interpretation of those formulae (i.e. over the semantic domain) then seems to give a much more direct access to existence theorems than the presentation usually associated with MDP's. Our technical contribution is to explain the coding of MDP's as quantitative p-formulae, to discuss the extension of the latte in incorporate 'rewards', and to illustrate the resulting reformulation of several existence theorems. In an appendix we give an algebraic characterisation of the new quantitative-with-reward form of the calculus.