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Describing systems in terms of choices and their resulting costs and rewardsoffers the promise of freeing algorithm designers and programmers fromspecifying how those choices should be made; in implementations, the choicescan be realized by optimization techniques and, increasingly, bymachine-learning methods. We study this approach from a programming-languageperspective. We define two small languages that support decision-makingabstractions: one with choices and rewards, and the other additionally withprobabilities. We give both operational and denotational semantics.  In the case of the second language we consider three denotational semantics,with varying degrees of correlation between possible program values andexpected rewards. The operational semantics combine the usual semantics ofstandard constructs with optimization over spaces of possible executionstrategies. The denotational semantics, which are compositional, rely on theselection monad, to handle choice, augmented with an auxiliary monad to handleother effects, such as rewards or probability.  We establish adequacy theorems that the two semantics coincide in all cases.We also prove full abstraction at base types, with varying notions ofobservation in the probabilistic case corresponding to the various degrees ofcorrelation. We present axioms for choice combined with rewards andprobability, establishing completeness at base types for the case of rewardswithout probability.

Smart Choices and the Selection Monad