Temporal Type Theory (original) (raw)

Overview

Authors:

• David I. Spivak 1

1. Patrick Schultz
   1. Massachusetts Institute of Technology, Cambridge, USA
2. David I. Spivak
   1. Massachusetts Institute of Technology, Cambridge, USA

• Describes a new approach, developed by the authors, for proving properties about the behaviors of systems using temporal logic

• Review of the prerequisite material included where needed in order to make a difficult topic more accessible

• A case study involving aircraft separation illustrates a real-world application of temporal type theory

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About this book

This innovative monograph explores a new mathematical formalism in higher-order temporal logic for proving properties about the behavior of systems. Developed by the authors, the goal of this novel approach is to explain what occurs when multiple, distinct system components interact by using a category-theoretic description of behavior types based on sheaves. The authors demonstrate how to analyze the behaviors of elements in continuous and discrete dynamical systems so that each can be translated and compared to one another. Their temporal logic is also flexible enough that it can serve as a framework for other logics that work with similar models.

The book begins with a discussion of behavior types, interval domains, and translation invariance, which serves as the groundwork for temporal type theory. From there, the authors lay out the logical preliminaries they need for their temporal modalities and explain the soundness of those logical semantics. These results are then applied to hybrid dynamical systems, differential equations, and labeled transition systems. A case study involving aircraft separation within the National Airspace System is provided to illustrate temporal type theory in action.

Researchers in computer science, logic, and mathematics interested in topos-theoretic and category-theory-friendly approaches to system behavior will find this monograph to be an important resource. It can also serve as a supplemental text for a specialized graduate topics course.

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Table of contents (8 chapters)

1. Axiomatics

   • Patrick Schultz, David I. Spivak
     Pages 87-114

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Authors and Affiliations

• Massachusetts Institute of Technology, Cambridge, USA

Patrick Schultz, David I. Spivak

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Bibliographic Information

• Book Title: Temporal Type Theory
• Book Subtitle: A Topos-Theoretic Approach to Systems and Behavior
• Authors: Patrick Schultz, David I. Spivak
• Series Title: Progress in Computer Science and Applied Logic
• DOI: https://doi.org/10.1007/978-3-030-00704-1
• Publisher: Birkhäuser Cham
• eBook Packages: Mathematics and Statistics, Mathematics and Statistics (R0)
• Copyright Information: The Author(s) 2019
• Hardcover ISBN: 978-3-030-00703-4Published: 11 February 2019
• eBook ISBN: 978-3-030-00704-1Published: 29 January 2019
• Series ISSN: 2297-0576
• Series E-ISSN: 2297-0584
• Edition Number: 1
• Number of Pages: VIII, 235
• Number of Illustrations: 60 b/w illustrations, 7 illustrations in colour
• Topics: Category Theory, Homological Algebra, Mathematical Logic and Foundations, Systems Theory, Control, Mathematical Logic and Formal Languages, Aerospace Technology and Astronautics

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