62 research outputs found
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
Internal Universes in Models of Homotopy Type Theory
We begin by recalling the essentially global character of universes in various models of homotopy type theory, which prevents a straightforward axiomatization of their properties using the internal language of the presheaf toposes from which these model are constructed. We get around this problem by extending the internal language with a modal operator for expressing properties of global elements. In this setting we show how to construct a universe that classifies the Cohen-Coquand-Huber-Mörtberg (CCHM) notion of fibration from their cubical sets model, starting from the assumption that the interval is tiny - a property that the interval in cubical sets does indeed have. This leads to an elementary axiomatization of that and related models of homotopy type theory within what we call crisp type theory
Cubical Models of Homotopy Type Theory - An Internal Approach
This thesis presents an account of the cubical sets model of homotopy type theory using an internal type theory for elementary topoi.
Homotopy type theory is a variant of Martin-Lof type theory where we think of types as spaces, with terms as points in the space and elements of the identity type as paths. We actualise this intuition by extending type theory with Voevodsky's univalence axiom which identifies equalities between types with homotopy equivalences between spaces.
Voevodsky showed the univalence axiom to be consistent by giving a model of homotopy type theory in the category of Kan simplicial sets in a paper with Kapulkin and Lumsdaine. However, this construction makes fundamental use of classical logic in order to show certain results. Therefore this model cannot be used to explain the computational content of the univalence axiom, such as how to compute terms involving univalence.
This problem was resolved by Cohen, Coquand, Huber and Mortberg, who presented a new model of type theory in Kan cubical sets which validated the univalence axiom using a constructive metatheory. This meant that the model provided an understanding of the computational content of univalence. In fact, the authors present a new type theory, cubical type theory, where univalence is provable using a new "glueing" type former. This type former comes with appropriate definitional equalities which explain how the univalence axiom should compute. In particular, Huber proved that any term of natural number type constructed in this new type theory must reduce to a numeral.
This thesis explores models of type theory based on the cubical sets model of Cohen et al. It gives an account of this model using the internal language of toposes, where we present a series of axioms which are sufficient to construct a model of cubical type theory, and hence a model of homotopy type theory. This approach therefore generalises the original model and gives a new and useful method for analysing models of type theory.
We also discuss an alternative derivation of the univalence axiom and show how this leads to a potentially simpler proof of univalence in any model satisfying the axioms mentioned above, such as cubical sets.
Finally, we discuss some shortcomings of the internal language approach with respect to constructing univalent universes. We overcome these difficulties by extending the internal language with an appropriate modality in order to manipulate global elements of an object.UK EPSRC PhD studentship, funded by grants EP/L504920/1, EP/M506485/1
Greatest HITs: Higher Inductive Types in Coinductive Definitions via Induction under Clocks
Guarded recursion is a powerful modal approach to recursion that can be seen
as an abstract form of step-indexing. It is currently used extensively in
separation logic to model programming languages with advanced features by
solving domain equations also with negative occurrences. In its multi-clocked
version, guarded recursion can also be used to program with and reason about
coinductive types, encoding the productivity condition required for recursive
definitions in types. This paper presents the first type theory combining
multi-clocked guarded recursion with the features of Cubical Type Theory, as
well as a denotational semantics. Using the combination of Higher Inductive
Types (HITs) and guarded recursion allows for simple programming and reasoning
about coinductive types that are traditionally hard to represent in type
theory, such as the type of finitely branching labelled transition systems. For
example, our results imply that bisimilarity for these imply path equality, and
so proofs can be transported along bisimilarity proofs. Among our technical
contributions is a new principle of induction under clocks. This allows
universal quantification over clocks to commute with HITs up to equivalence of
types, and is crucial for the encoding of coinductive types. Such commutativity
requirements have been formulated for inductive types as axioms in previous
type theories with multi-clocked guarded recursion, but our present formulation
as an induction principle allows for the formulation of general computation
rules.Comment: 29 page
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