A calculus for probabilistic languages

Abstract

As probabilistic computation plays an increasing role in diverse fields in computer science, researchers have designed new languages to facilitate the development of probabilistic programs. In this paper, we develop a probabilistic calculus by extending the traditional lambda calculus. In our calculus, every expression denotes a probability distribution yet evaluates to a regular value. The most notable feature of our calculus is that it is founded upon sampling functions, which map the unit interval to probability domains. As a consequence, we achieve a unified representation scheme for all types of probability distributions. In order to support an efficient implementation of the calculus, we also develop a refinement type system which is capable of distinguishing expressions denoting regular values from expressions denoting probability distributions. We use a novel formulation of the intuitionistic modal logic S4 with an intersection connective in the refinement type system. We present preliminary evidence that a probabilistic language based upon our calculus is viable in applications involving massive probabilistic computation.