20 research outputs found
Cubical Syntax for Reflection-Free Extensional Equality
We contribute XTT, a cubical reconstruction of Observational Type Theory
which extends Martin-L\"of's intensional type theory with a dependent equality
type that enjoys function extensionality and a judgmental version of the
unicity of identity types principle (UIP): any two elements of the same
equality type are judgmentally equal. Moreover, we conjecture that the typing
relation can be decided in a practical way. In this paper, we establish an
algebraic canonicity theorem using a novel cubical extension (independently
proposed by Awodey) of the logical families or categorical gluing argument
inspired by Coquand and Shulman: every closed element of boolean type is
derivably equal to either 'true' or 'false'.Comment: Extended version; International Conference on Formal Structures for
Computation and Deduction (FSCD), 201
Multimodal Dependent Type Theory
We introduce MTT, a dependent type theory which supports multiple modalities.
MTT is parametrized by a mode theory which specifies a collection of modes,
modalities, and transformations between them. We show that different choices of
mode theory allow us to use the same type theory to compute and reason in many
modal situations, including guarded recursion, axiomatic cohesion, and
parametric quantification. We reproduce examples from prior work in guarded
recursion and axiomatic cohesion, thereby demonstrating that MTT constitutes a
simple and usable syntax whose instantiations intuitively correspond to
previous handcrafted modal type theories. In some cases, instantiating MTT to a
particular situation unearths a previously unknown type theory that improves
upon prior systems. Finally, we investigate the metatheory of MTT. We prove the
consistency of MTT and establish canonicity through an extension of recent
type-theoretic gluing techniques. These results hold irrespective of the choice
of mode theory, and thus apply to a wide variety of modal situations
Cartesian Cubical Computational Type Theory: Constructive Reasoning with Paths and Equalities
We present a dependent type theory organized around a Cartesian notion of cubes (with faces, degeneracies, and diagonals), supporting both fibrant and non-fibrant types. The fibrant fragment validates Voevodsky\u27s univalence axiom and includes a circle type, while the non-fibrant fragment includes exact (strict) equality types satisfying equality reflection. Our type theory is defined by a semantics in cubical partial equivalence relations, and is the first two-level type theory to satisfy the canonicity property: all closed terms of boolean type evaluate to either true or false