19 research outputs found
A Sheaf Model of the Algebraic Closure
In constructive algebra one cannot in general decide the irreducibility of a
polynomial over a field K. This poses some problems to showing the existence of
the algebraic closure of K. We give a possible constructive interpretation of
the existence of the algebraic closure of a field in characteristic 0 by
building, in a constructive metatheory, a suitable site model where there is
such an algebraic closure. One can then extract computational content from this
model. We give examples of computation based on this model.Comment: In Proceedings CL&C 2014, arXiv:1409.259
The Independence of Markov's Principle in Type Theory
In this paper, we show that Markov's principle is not derivable in dependent
type theory with natural numbers and one universe. One way to prove this would
be to remark that Markov's principle does not hold in a sheaf model of type
theory over Cantor space, since Markov's principle does not hold for the
generic point of this model. Instead we design an extension of type theory,
which intuitively extends type theory by the addition of a generic point of
Cantor space. We then show the consistency of this extension by a normalization
argument. Markov's principle does not hold in this extension, and it follows
that it cannot be proved in type theory
Dynamic Newton-Puiseux Theorem
A constructive version of Newton-Puiseux theorem for computing the Puiseux
expansions of algebraic curves is presented. The proof is based on a classical
proof by Abhyankar. Algebraic numbers are evaluated dynamically; hence the base
field need not be algebraically closed and a factorization algorithm of
polynomials over the base field is not needed. The extensions obtained are a
type of regular algebras over the base field and the expansions are given as
formal power series over these algebras.Comment: 22 pag
The Clocks They Are Adjunctions: Denotational Semantics for Clocked Type Theory
Clocked Type Theory (CloTT) is a type theory for guarded recursion useful for
programming with coinductive types, allowing productivity to be encoded in
types, and for reasoning about advanced programming language features using an
abstract form of step-indexing. CloTT has previously been shown to enjoy a
number of syntactic properties including strong normalisation, canonicity and
decidability of type checking. In this paper we present a denotational
semantics for CloTT useful, e.g., for studying future extensions of CloTT with
constructions such as path types.
The main challenge for constructing this model is to model the notion of
ticks used in CloTT for coinductive reasoning about coinductive types. We build
on a category previously used to model guarded recursion, but in this category
there is no object of ticks, so tick-assumptions in a context can not be
modelled using standard tools. Instead we show how ticks can be modelled using
adjoint functors, and how to model the tick constant using a semantic
substitution
Stack semantics of type theory
We give a model of dependent type theory with one univalent universe and
propositional truncation interpreting a type as a stack, generalising the
groupoid model of type theory. As an application, we show that countable choice
cannot be proved in dependent type theory with one univalent universe and
propositional truncation
Ticking clocks as dependent right adjoints: Denotational semantics for clocked type theory
Clocked Type Theory (CloTT) is a type theory for guarded recursion useful for
programming with coinductive types, allowing productivity to be encoded in
types, and for reasoning about advanced programming language features using an
abstract form of step-indexing. CloTT has previously been shown to enjoy a
number of syntactic properties including strong normalisation, canonicity and
decidability of the equational theory. In this paper we present a denotational
semantics for CloTT useful, e.g., for studying future extensions of CloTT with
constructions such as path types.
The main challenge for constructing this model is to model the notion of
ticks on a clock used in CloTT for coinductive reasoning about coinductive
types. We build on a category previously used to model guarded recursion with
multiple clocks. In this category there is an object of clocks but no object of
ticks, and so tick-assumptions in a context can not be modelled using standard
tools. Instead we model ticks using dependent right adjoint functors, a
generalisation of the category theoretic notion of adjunction to the setting of
categories with families. Dependent right adjoints are known to model
Fitch-style modal types, but in the case of CloTT, the modal operators
constitute a family indexed internally in the type theory by clocks. We model
this family using a dependent right adjoint on the slice category over the
object of clocks. Finally we show how to model the tick constant of CloTT using
a semantic substitution.
This work improves on a previous model by the first two named authors which
not only had a flaw but was also considerably more complicated.Comment: 31 pages. Second version is a minor revision. arXiv admin note: text
overlap with arXiv:1804.0668
Ticking clocks as dependent right adjoints: Denotational semantics for clocked type theory
Clocked Type Theory (CloTT) is a type theory for guarded recursion useful for
programming with coinductive types, allowing productivity to be encoded in
types, and for reasoning about advanced programming language features using an
abstract form of step-indexing. CloTT has previously been shown to enjoy a
number of syntactic properties including strong normalisation, canonicity and
decidability of the equational theory. In this paper we present a denotational
semantics for CloTT useful, e.g., for studying future extensions of CloTT with
constructions such as path types.
The main challenge for constructing this model is to model the notion of
ticks on a clock used in CloTT for coinductive reasoning about coinductive
types. We build on a category previously used to model guarded recursion with
multiple clocks. In this category there is an object of clocks but no object of
ticks, and so tick-assumptions in a context can not be modelled using standard
tools. Instead we model ticks using dependent right adjoint functors, a
generalisation of the category theoretic notion of adjunction to the setting of
categories with families. Dependent right adjoints are known to model
Fitch-style modal types, but in the case of CloTT, the modal operators
constitute a family indexed internally in the type theory by clocks. We model
this family using a dependent right adjoint on the slice category over the
object of clocks. Finally we show how to model the tick constant of CloTT using
a semantic substitution.
This work improves on a previous model by the first two named authors which
not only had a flaw but was also considerably more complicated
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