27,473 research outputs found
A Category Theoretic View of Contextual Types: from Simple Types to Dependent Types
We describe the categorical semantics for a simply typed variant and a
simplified dependently typed variant of Cocon, a contextual modal type theory
where the box modality mediates between the weak function space that is used to
represent higher-order abstract syntax (HOAS) trees and the strong function
space that describes (recursive) computations about them. What makes Cocon
different from standard type theories is the presence of first-class contexts
and contextual objects to describe syntax trees that are closed with respect to
a given context of assumptions. Following M. Hofmann's work, we use a presheaf
model to characterise HOAS trees. Surprisingly, this model already provides the
necessary structure to also model Cocon. In particular, we can capture the
contextual objects of Cocon using a comonad that restricts presheaves
to their closed elements. This gives a simple semantic characterisation of the
invariants of contextual types (e.g. substitution invariance) and identifies
Cocon as a type-theoretic syntax of presheaf models. We further extend this
characterisation to dependent types using categories with families and show
that we can model a fragment of Cocon without recursor in the Fitch-style
dependent modal type theory presented by Birkedal et. al.
A constructive modal semantics for contextual verification
This paper introduces a non-standard semantics for a modal version of constructive KT for contextual (assumptions-based) verification. The modal fragment expresses verifiability under extensions of contexts, enjoying adapted validity and (weak) monotonicity properties depending on satisfaction of the contextual data
Multi-level Contextual Type Theory
Contextual type theory distinguishes between bound variables and
meta-variables to write potentially incomplete terms in the presence of
binders. It has found good use as a framework for concise explanations of
higher-order unification, characterize holes in proofs, and in developing a
foundation for programming with higher-order abstract syntax, as embodied by
the programming and reasoning environment Beluga. However, to reason about
these applications, we need to introduce meta^2-variables to characterize the
dependency on meta-variables and bound variables. In other words, we must go
beyond a two-level system granting only bound variables and meta-variables.
In this paper we generalize contextual type theory to n levels for arbitrary
n, so as to obtain a formal system offering bound variables, meta-variables and
so on all the way to meta^n-variables. We obtain a uniform account by
collapsing all these different kinds of variables into a single notion of
variabe indexed by some level k. We give a decidable bi-directional type system
which characterizes beta-eta-normal forms together with a generalized
substitution operation.Comment: In Proceedings LFMTP 2011, arXiv:1110.668
Modalities in homotopy type theory
Univalent homotopy type theory (HoTT) may be seen as a language for the
category of -groupoids. It is being developed as a new foundation for
mathematics and as an internal language for (elementary) higher toposes. We
develop the theory of factorization systems, reflective subuniverses, and
modalities in homotopy type theory, including their construction using a
"localization" higher inductive type. This produces in particular the
(-connected, -truncated) factorization system as well as internal
presentations of subtoposes, through lex modalities. We also develop the
semantics of these constructions
Hilbert Space Quantum Mechanics is Contextual (Reply to R. B. Griffiths)
In a recent paper Griffiths [38] has argued, based on the consistent
histories interpretation, that Hilbert space quantum mechanics (QM) is
noncontextual. According to Griffiths the problem of contextuality disappears
if the apparatus is "designed and operated by a competent experimentalist" and
we accept the Single Framework Rule (SFR). We will argue from a
representational realist stance that the conclusion is incorrect due to the
misleading understanding provided by Griffiths to the meaning of quantum
contextuality and its relation to physical reality and measurements. We will
discuss how the quite general incomprehension of contextuality has its origin
in the "objective-subjective omelette" created by Heisenberg and Bohr. We will
argue that in order to unscramble the omelette we need to disentangle, firstly,
representational realism from naive realism, secondly, ontology from
epistemology, and thirdly, the different interpretational problems of QM. In
this respect, we will analyze what should be considered as Meaningful Physical
Statements (MPS) within a theory and will argue that Counterfactual Reasoning
(CR) -considered by Griffiths as "tricky"- must be accepted as a necessary
condition for any representational realist interpretation of QM. Finally we
discuss what should be considered as a problem (and what not) in QM from a
representational realist perspective.Comment: arXiv admin note: substantial text overlap with arXiv:1502.0531
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