© 2017

Semantics of the Probabilistic Typed Lambda Calculus

Markov Chain Semantics, Termination Behavior, and Denotational Semantics


  • Provides an in-depth discussion of the semantics of the probabilistic typed lambda calculus and its termination behavior

  • Self-contained, offering a recapitulation of the basic mathematical tools needed

  • Includes an extensive list of further references for researchers


Table of contents

  1. Front Matter
    Pages I-VIII
  2. Dirk Draheim
    Pages 1-16
  3. Dirk Draheim
    Pages 17-64
  4. Dirk Draheim
    Pages 65-92
  5. Dirk Draheim
    Pages 93-133
  6. Dirk Draheim
    Pages 135-191
  7. Back Matter
    Pages 193-218

About this book


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 distributions as domains.

The work mostly appeals to researchers in theoretical computer science focusing on probabilistic programming, randomized algorithms, or programming language theory.


Markov chains Probabilistic computation Lambda calculus Computability Randomized algorithms Programming language theory Verification by model checking

Authors and affiliations

  1. 1.Large-Scale Systems GroupTallinn University of TechnologyTallinnEstonia

About the authors

Dirk Draheim is full professor of information society technologies and head of the large-scale systems group at Tallinn University of Technology. From to 1990 to 2006 he worked as an IT project manager, IT consultant and IT author in Berlin. In summer 2006, he was Lecturer at the University of Auckland and from 2006-2008 he was area manager for database systems at the Software Competence Center Hagenberg as well as Adjunct Lecturer in information systems at the Johannes-Kepler-University Linz. From 2008 to 2016 he was head of the data center of the University of Innsbruck and, in parallel, from 2010 to 2016, Adjunct Reader at the Faculty of Information Systems of the University of Mannheim. Dirk is co-author of the Springer book "Form-Oriented Analysis" and author of the Springer book "Business Process Technology", and a member of the ACM.

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