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Full Description
This book takes a foundational approach to the semantics of probabilistic programming. It elaborates a rigorous Markov chain semantics for the probabilistic typed lambda calculus, which is the typed lambda calculus with recursion plus probabilistic choice.
The book starts with a recapitulation of the basic mathematical tools needed throughout the book, in particular Markov chains, graph theory and domain theory, and also explores the topic of inductive definitions. It then defines the syntax and establishes the Markov chain semantics of the probabilistic lambda calculus and, furthermore, both a graph and a tree semantics. Based on that, it investigates the termination behavior of probabilistic programs. It introduces the notions of termination degree, bounded termination and path stoppability and investigates their mutual relationships. Lastly, it defines a denotational semantics of the probabilistic lambda calculus, based on continuous functions over probability distributionsas domains.
The work mostly appeals to researchers in theoretical computer science focusing on probabilistic programming, randomized algorithms, or programming language theory.
Contents
Part I: The Probabilistic Lambda-Calculus and its Semantics.- Introduction.- Syntax and Operational Semantics.- The Working Probabilistic Lambda Calculus.- Properties of the Markov Chain Semantics.- Denotational Semantics.- Semantical Correspondences.- Categorical Treatment.- Probabilism and Non-Determinism.- Part II: Natural Probabilistic Reasoning.- On Natural Two-Tier Semantics for Propositional Logics.- Natural Semantics of Propositions.- Finite Discrete Stochastics Reconsidered.- Lambda-Calculus Definitions.- Markov Chains.- Basic Logic Language and Semantics Definitions.- References.- Index.
