Temporal Type Theory

@article{Schultz2019TemporalTT,
  title={Temporal Type Theory},
  author={Patrick Schultz and David I. Spivak},
  journal={Progress in Computer Science and Applied Logic},
  year={2019}
}
This book introduces a temporal type theory, the first of its kind as far as we know. It is based on a standard core, and as such it can be formalized in a proof assistant such as Coq or Lean by adding a number of axioms. Well-known temporal logics---such as Linear and Metric Temporal Logic (LTL and MTL)---embed within the logic of temporal type theory. The types in this theory represent "behavior types". The language is rich enough to allow one to define arbitrary hybrid dynamical systems… 
The Topological and Logical Structure of Concurrency and Dependency via Distributive Lattices
TLDR
The specification of branching dependency structures, which exist in fields from knowledge-representation to package management, to the specification of semantics of concurrent computation, and how such constructions may relate to important questions in complexity theory, including solutions of satisfiability problems are discussed.
Dynamical Systems and Sheaves
TLDR
A categorical framework for modeling and analyzing systems in a broad sense, which includes lax monoidal functors, which provide a language of compositionality, as well as sheaf theory, which flexibly captures the crucial notion of time.
Behavioral Mereology (Proofs and Properties).
Mereology is the study of parts and the relationships that hold between them. We introduce a behavioral approach to mereology, in which systems and their parts are known only by the types of behavior
Topos Semantics for a Higher-Order Temporal Logic of Actions
TLDR
This work uses categorical techniques to recast a real-time semantics for TLA in terms of the actions of a group of time dilations, or "stutters," and an extension by a monoid incorporating delays, or 'falters.
Behavioral Mereology: A Modal Logic for Passing Constraints
TLDR
A behavioral approach to mereology is introduced, in which systems and their parts are known only by the types of behavior they can exhibit, and an inter-modal logic is given that generalizes the usual alethic modalities in the setting of symmetric accessibility.
A cost-aware logical framework
TLDR
It is argued that the cost structure of programs motivates a phase distinction between intension and extension, which contributes a synthetic account of cost structure as a computational effect in which cost-aware programs enjoy an internal noninterference property: input/output behavior cannot depend on cost.
Chapter IV : Modal homotopy type theory
We now proceed to take the final step in our journey towards modal homotopy type theory. Analytic philosophers have put modal logic to extensive use in their exploration of so-called alethic,
Bisimulation maps in presheaf categories
Temporal Landscapes: A Graphical Temporal Logic for Reasoning
We present an elementary introduction to a new logic for reasoning about behaviors that occur over time. This logic is based on temporal type theory. The syntax of the logic is similar to the usual
Behavioral Mereology
Mereology is the study of parts and the relationships that hold between them. We introduce a behavioral approach to mereology, in which systems and their parts are known only by the types of behavior
...
...

References

SHOWING 1-10 OF 55 REFERENCES
The Logic of Topoi
A propositional modal logic of time intervals
In certain areas of artificial intelligence there is need to represent continuous change and to make statements that are interpreted with respect to time intervals rather than time points. To this
Dyck Algebras, Interval Temporal Logic, and Posets of Intervals
TLDR
A logic-theoretic interpretation of such Heyting algebras is found, showing that they are the algebraic counterpart of a certain fragment of a classical interval temporal logic (also known as Halpern-Shoham logic).
Natural models of homotopy type theory
  • S. Awodey
  • Mathematics
    Mathematical Structures in Computer Science
  • 2016
TLDR
It is shown that a category admits a natural model of type theory if it has a class of maps that behave like the abstract fibrations in axiomatic homotopy theory: They should be stable under pullback, closed under composition and relative products, and there should be weakly orthogonal factorizations into the class.
The Dedekind reals in abstract Stone duality
TLDR
The core of the paper constructs the real line using two-sided Dedekind cuts, and shows that the closed interval is compact and overt, where these concepts are defined using quantifiers.
Modular correspondence between dependent type theories and categories including pretopoi and topoi
  • M. Maietti
  • Philosophy
    Mathematical Structures in Computer Science
  • 2005
TLDR
A modular correspondence between various categorical structures and their internal languages in terms of extensional dependent type theories à la Martin-Löf is presented and formulas corresponding to subobjects can be regained as particular types that are equipped with proof-terms according to the isomorphism ‘propositions as mono types’.
Sheaves in geometry and logic: a first introduction to topos theory
This text presents topos theory as it has developed from the study of sheaves. Sheaves arose in geometry as coefficients for cohomology and as descriptions of the functions appropriate to various
Modal Intervals Revisited, Part 1: A Generalized Interval Natural Extension
TLDR
With a construction similar to the classical intervals theory, the new formulation of the modal intervals theory proposed in this paper should facilitate the understanding of the underlying mechanisms, the addition of new items to the theory and its utilization.
Homotopy Type Theory: Univalent Foundations of Mathematics
TLDR
This book is intended as a first systematic exposition of the basics of the resulting"Univalent Foundations" program, and a collection of examples of this new style of reasoning -- but without requiring the reader to know or learn any formal logic, or to use any computer proof assistant.
...
...