Abstract
As mentioned in the introduction, we believe that our temporal type theory can serve as a “big tent” into which to embed many disparate formalisms for proving property of behaviors. In this chapter we discuss a few of these, including hybrid dynamical systems in Sect. 8.1 , delays in Sect. 8.2 , differential equations in Sect. 8.3 , and labeled transition systems in Sect. 8.4.4 . This last occurs in Sect. 8.4 where we briefly discuss a general framework on machines, systems, and behavior contracts. Next in Sect. 8.5 we give a toy example—an extreme simplification of the National Airspace System—and prove a safety property. The idea is to show that we really can mix continuous, discrete, and delay properties without ever leaving the temporal type theory. We conclude this chapter by discussing how some temporal logics, e.g., metric temporal logic, embeds into the temporal type theory.
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Notes
- 1.
The construction of trav described here extends to hybrid sheaves in general; it is only a bit more complicated in the general case because we cannot assume Hyb1(C, D) is π-separated.
- 2.
A term in context, such as s 1 : S 1, …, s n : S n ⊢ ϕ(s 1, …, s n) : Prop is roughly the same as a formula ϕ : (S 1 ×⋯ × S n) →Prop. The main difference is that, in the former case, the variables have been named.
- 3.
The inertiality condition for an output map p : X → S′ says that every internal trajectory produces a guaranteed output trajectory of a slightly duration. This property can be stated internally as follows:
- 4.
“Monadic” here refers to the restriction that all predicates must be unary, and has no connection to monads in the sense of category theory. The only time we use “monadic” in this sense is when discussing other temporal logics.
References
Hunter, P., Ouaknine, J., Worrell, J.: Expressive completeness for metric temporal logic. In: Proceedings of the 2013 28th Annual ACM/IEEE Symposium on Logic in Computer Science, pp. 349–357. IEEE Computer Society (2013)
Spivak, D.I., Vasilakopoulou, C., Schultz, P.: Dynamical Systems and Sheaves (2016). eprint: arXiv:1609.08086
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Schultz, P., Spivak, D.I. (2019). Applications. In: Temporal Type Theory. Progress in Computer Science and Applied Logic, vol 29. Birkhäuser, Cham. https://doi.org/10.1007/978-3-030-00704-1_8
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DOI: https://doi.org/10.1007/978-3-030-00704-1_8
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