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Semantics of the Probabilistic Typed Lambda Calculus: Markov Chain Semantics, Termination Behavior, and Denotational Semantics

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This book takes a foundational solution 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 elemental mathematical tools needed all through the book, particularly Markov chains, graph theory and domain theory, and in addition explores the subject 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. In line with 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, In line with continuous functions over probability distributions as domain names.

The work most commonly appeals to researchers in theoretical computer science that specialize in probabilistic programming, randomized algorithms, or programming language theory.

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