First steps in synthetic guarded domain theory: step-indexing in the topos of trees

We present the topos S of trees as a model of guarded recursion. We study the internal dependently-typed higher-order logic of S and show that S models two modal operators, on predicates and types, which serve as guards in recursive definitions of terms, predicates, and types. In particular, we show how to solve recursive type equations involving dependent types. We propose that the internal logic of S provides the right setting for the synthetic construction of abstract versions of step-indexed models of programming languages and program logics. As an example, we show how to construct a model of a programming language with higher-order store and recursive types entirely inside the internal logic of S. Moreover, we give an axiomatic categorical treatment of models of synthetic guarded domain theory and prove that, for any complete Heyting algebra A with a well-founded basis, the topos of sheaves over A forms a model of synthetic guarded domain theory, generalizing the results for S

Step-Indexed Normalization for a Language with General Recursion

The Trellys project has produced several designs for practical dependently typed languages. These languages are broken into two fragments-a_logical_fragment where every term normalizes and which is consistent when interpreted as a logic, and a_programmatic_fragment with general recursion and other convenient but unsound features. In this paper, we present a small example language in this style. Our design allows the programmer to explicitly mention and pass information between the two fragments. We show that this feature substantially complicates the metatheory and present a new technique, combining the traditional Girard-Tait method with step-indexed logical relations, which we use to show normalization for the logical fragment.Comment: In Proceedings MSFP 2012, arXiv:1202.240

Categorical Semantics for Functional Reactive Programming with Temporal Recursion and Corecursion

Functional reactive programming (FRP) makes it possible to express temporal aspects of computations in a declarative way. Recently we developed two kinds of categorical models of FRP: abstract process categories (APCs) and concrete process categories (CPCs). Furthermore we showed that APCs generalize CPCs. In this paper, we extend APCs with additional structure. This structure models recursion and corecursion operators that are related to time. We show that the resulting categorical models generalize those CPCs that impose an additional constraint on time scales. This constraint boils down to ruling out $\omega$-supertasks, which are closely related to Zeno's paradox of Achilles and the tortoise.Comment: In Proceedings MSFP 2014, arXiv:1406.153

Guarded Cubical Type Theory: Path Equality for Guarded Recursion

This paper improves the treatment of equality in guarded dependent type theory (GDTT), by combining it with cubical type theory (CTT). GDTT is an extensional type theory with guarded recursive types, which are useful for building models of program logics, and for programming and reasoning with coinductive types. We wish to implement GDTT with decidable type-checking, while still supporting non-trivial equality proofs that reason about the extensions of guarded recursive constructions. CTT is a variation of Martin-L\"of type theory in which the identity type is replaced by abstract paths between terms. CTT provides a computational interpretation of functional extensionality, is conjectured to have decidable type checking, and has an implemented type-checker. Our new type theory, called guarded cubical type theory, provides a computational interpretation of extensionality for guarded recursive types. This further expands the foundations of CTT as a basis for formalisation in mathematics and computer science. We present examples to demonstrate the expressivity of our type theory, all of which have been checked using a prototype type-checker implementation, and present semantics in a presheaf category.Comment: 17 pages, to be published in proceedings of CSL 201

Sequent Calculus in the Topos of Trees

Nakano's "later" modality, inspired by G\"{o}del-L\"{o}b provability logic, has been applied in type systems and program logics to capture guarded recursion. Birkedal et al modelled this modality via the internal logic of the topos of trees. We show that the semantics of the propositional fragment of this logic can be given by linear converse-well-founded intuitionistic Kripke frames, so this logic is a marriage of the intuitionistic modal logic KM and the intermediate logic LC. We therefore call this logic $\mathrm{KM}_{\mathrm{lin}}$. We give a sound and cut-free complete sequent calculus for $\mathrm{KM}_{\mathrm{lin}}$ via a strategy that decomposes implication into its static and irreflexive components. Our calculus provides deterministic and terminating backward proof-search, yields decidability of the logic and the coNP-completeness of its validity problem. Our calculus and decision procedure can be restricted to drop linearity and hence capture KM.Comment: Extended version, with full proof details, of a paper accepted to FoSSaCS 2015 (this version edited to fix some minor typos

A Ghost at ω1\omega_1

In the final chain of the countable powerset functor, we show that the set at index $\omega_1$, regarded as a transition system, is not strongly extensional because it contains a "ghost" element that has no successor even though its component at each successor index is inhabited. The method, adapted from a construction of Forti and Honsell, also gives ghosts at larger ordinals in the final chain of other subfunctors of the powerset functor. This leads to a precise description of which sets in these final chains are strongly extensional