320,087 research outputs found
A dependent nominal type theory
Nominal abstract syntax is an approach to representing names and binding
pioneered by Gabbay and Pitts. So far nominal techniques have mostly been
studied using classical logic or model theory, not type theory. Nominal
extensions to simple, dependent and ML-like polymorphic languages have been
studied, but decidability and normalization results have only been established
for simple nominal type theories. We present a LF-style dependent type theory
extended with name-abstraction types, prove soundness and decidability of
beta-eta-equivalence checking, discuss adequacy and canonical forms via an
example, and discuss extensions such as dependently-typed recursion and
induction principles
Modal dependent type theory and dependent right adjoints
In recent years we have seen several new models of dependent type theory
extended with some form of modal necessity operator, including nominal type
theory, guarded and clocked type theory, and spatial and cohesive type theory.
In this paper we study modal dependent type theory: dependent type theory with
an operator satisfying (a dependent version of) the K-axiom of modal logic. We
investigate both semantics and syntax. For the semantics, we introduce
categories with families with a dependent right adjoint (CwDRA) and show that
the examples above can be presented as such. Indeed, we show that any finite
limit category with an adjunction of endofunctors gives rise to a CwDRA via the
local universe construction. For the syntax, we introduce a dependently typed
extension of Fitch-style modal lambda-calculus, show that it can be interpreted
in any CwDRA, and build a term model. We extend the syntax and semantics with
universes
C-system of a module over a monad on sets
This is the second paper in a series that aims to provide mathematical
descriptions of objects and constructions related to the first few steps of the
semantical theory of dependent type systems.
We construct for any pair , where is a monad on sets and is
a left module over , a C-system (contextual category) and
describe a class of sub-quotients of in terms of objects directly
constructed from and . In the special case of the monads of expressions
associated with nominal signatures this construction gives the C-systems of
general dependent type theories when they are specified by collections of
judgements of the four standard kinds
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Modal dependent type theory and dependent right adjoints
In recent years we have seen several new models of dependent type theory
extended with some form of modal necessity operator, including nominal type
theory, guarded and clocked type theory, and spatial and cohesive type theory.
In this paper we study modal dependent type theory: dependent type theory with
an operator satisfying (a dependent version of) the K-axiom of modal logic. We
investigate both semantics and syntax. For the semantics, we introduce
categories with families with a dependent right adjoint (CwDRA) and show that
the examples above can be presented as such. Indeed, we show that any finite
limit category with an adjunction of endofunctors gives rise to a CwDRA via the
local universe construction. For the syntax, we introduce a dependently typed
extension of Fitch-style modal lambda-calculus, show that it can be interpreted
in any CwDRA, and build a term model. We extend the syntax and semantics with
universes
A Dependent Type Theory with Abstractable Names
This paper describes a version of Martin-Löf's dependent type theory extended with names and constructs for freshness and name-abstraction derived from the theory of nominal sets. We aim for a type theory for computing and proving (via a Curry-Howard correspondence) with syntactic structures which captures familiar, but informal, ‘nameful’ practices when dealing with binders.Partially supported by the UK EPSRC program grant EP/K008528/1, Rigorous Engineering for Mainstream Systems (REMS). Supported by the UK EPSRC leadership fellowship (Peter Sewell) grant EP/H005633/1, Semantic Foundations for Real-World Systems.This is the final published version of the article. It was originally published in Electronic Notes in Theoretical Computer Science (Pitts AM, Matthiesen J, Derikx J, Electronic Notes in Theoretical Computer Science 2015, 312, 19–50, doi:10.1016/j.entcs.2015.04.003) http://dx.doi.org/10.1016/j.entcs.2015.04.00
Transpension: The Right Adjoint to the Pi-type
Presheaf models of dependent type theory have been successfully applied to
model HoTT, parametricity, and directed, guarded and nominal type theory. There
has been considerable interest in internalizing aspects of these presheaf
models, either to make the resulting language more expressive, or in order to
carry out further reasoning internally, allowing greater abstraction and
sometimes automated verification. While the constructions of presheaf models
largely follow a common pattern, approaches towards internalization do not.
Throughout the literature, various internal presheaf operators (,
, , , ,
, the strictness axiom and locally fresh names) can be found and
little is known about their relative expressivenes. Moreover, some of these
require that variables whose type is a shape (representable presheaf, e.g. an
interval) be used affinely.
We propose a novel type former, the transpension type, which is right adjoint
to universal quantification over a shape. Its structure resembles a dependent
version of the suspension type in HoTT. We give general typing rules and a
presheaf semantics in terms of base category functors dubbed multipliers.
Structural rules for shape variables and certain aspects of the transpension
type depend on characteristics of the multiplier. We demonstrate how the
transpension type and the strictness axiom can be combined to implement all and
improve some of the aforementioned internalization operators (without formal
claim in the case of locally fresh names)
A Normalizing Computation Rule for Propositional Extensionality in Higher-Order Minimal Logic
The univalence axiom expresses the principle of extensionality for dependent type theory. However, if we simply add the univalence axiom to type theory, then we lose the property of canonicity - that every closed term computes to a canonical form. A computation becomes "stuck" when it reaches the point that it needs to evaluate a proof term that is an application of the univalence axiom. So we wish to find a way to compute with the univalence axiom. While this problem has been solved with the formulation of cubical type theory, where the computations are expressed using a nominal extension of lambda-calculus, it may be interesting to explore alternative solutions, which do not require such an extension.
As a first step, we present here a system of propositional higher-order minimal logic (PHOML). There are three kinds of typing judgement in PHOML. There are terms which inhabit types, which are the simple types over Omega. There are proofs which inhabit propositions, which are the terms of type Omega. The canonical propositions are those constructed from false by implication. Thirdly, there are paths which inhabit equations M =_A N, where M and N are terms of type A. There are two ways to prove an equality: reflexivity, and propositional extensionality - logically equivalent propositions are equal. This system allows for some definitional equalities that are not present in cubical type theory, namely that transport along the trivial path is identity.
We present a call-by-name reduction relation for this system, and prove that the system satisfies canonicity: every closed typable term head-reduces to a canonical form. This work has been formalised in Agda
Mechanizing the Metatheory of LF
LF is a dependent type theory in which many other formal systems can be
conveniently embedded. However, correct use of LF relies on nontrivial
metatheoretic developments such as proofs of correctness of decision procedures
for LF's judgments. Although detailed informal proofs of these properties have
been published, they have not been formally verified in a theorem prover. We
have formalized these properties within Isabelle/HOL using the Nominal Datatype
Package, closely following a recent article by Harper and Pfenning. In the
process, we identified and resolved a gap in one of the proofs and a small
number of minor lacunae in others. We also formally derive a version of the
type checking algorithm from which Isabelle/HOL can generate executable code.
Besides its intrinsic interest, our formalization provides a foundation for
studying the adequacy of LF encodings, the correctness of Twelf-style
metatheoretic reasoning, and the metatheory of extensions to LF.Comment: Accepted to ACM Transactions on Computational Logic. Preprint
A Presheaf Model of Parametric Type Theory
Abstract We extend Martin-Löf's Logical Framework with special constructions and typing rules providing internalized parametricity. Compared to previous similar proposals, this version comes with a denotational semantics which is a refinement of the standard presheaf semantics of dependent type theory. Further, this presheaf semantics is a refinement of the one used to interpret nominal sets with restrictions. The present calculus is a candidate for the core of a proof assistant with internalized parametricity
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