955 research outputs found
The Biequivalence of Locally Cartesian Closed Categories and Martin-L\"of Type Theories
Seely's paper "Locally cartesian closed categories and type theory" contains
a well-known result in categorical type theory: that the category of locally
cartesian closed categories is equivalent to the category of Martin-L\"of type
theories with Pi-types, Sigma-types and extensional identity types. However,
Seely's proof relies on the problematic assumption that substitution in types
can be interpreted by pullbacks. Here we prove a corrected version of Seely's
theorem: that the B\'enabou-Hofmann interpretation of Martin-L\"of type theory
in locally cartesian closed categories yields a biequivalence of 2-categories.
To facilitate the technical development we employ categories with families as a
substitute for syntactic Martin-L\"of type theories. As a second result we
prove that if we remove Pi-types the resulting categories with families are
biequivalent to left exact categories.Comment: TLCA 2011 - 10th Typed Lambda Calculi and Applications, Novi Sad :
Serbia (2011
Dual-Context Calculi for Modal Logic
We present natural deduction systems and associated modal lambda calculi for
the necessity fragments of the normal modal logics K, T, K4, GL and S4. These
systems are in the dual-context style: they feature two distinct zones of
assumptions, one of which can be thought as modal, and the other as
intuitionistic. We show that these calculi have their roots in in sequent
calculi. We then investigate their metatheory, equip them with a confluent and
strongly normalizing notion of reduction, and show that they coincide with the
usual Hilbert systems up to provability. Finally, we investigate a categorical
semantics which interprets the modality as a product-preserving functor.Comment: Full version of article previously presented at LICS 2017 (see
arXiv:1602.04860v4 or doi: 10.1109/LICS.2017.8005089
Extensional Collapse Situations I: non-termination and unrecoverable errors
We consider a simple model of higher order, functional computation over the
booleans. Then, we enrich the model in order to encompass non-termination and
unrecoverable errors, taken separately or jointly. We show that the models so
defined form a lattice when ordered by the extensional collapse situation
relation, introduced in order to compare models with respect to the amount of
"intensional information" that they provide on computation. The proofs are
carried out by exhibiting suitable applied {\lambda}-calculi, and by exploiting
the fundamental lemma of logical relations
Semantics of a Typed Algebraic Lambda-Calculus
Algebraic lambda-calculi have been studied in various ways, but their
semantics remain mostly untouched. In this paper we propose a semantic analysis
of a general simply-typed lambda-calculus endowed with a structure of vector
space. We sketch the relation with two established vectorial lambda-calculi.
Then we study the problems arising from the addition of a fixed point
combinator and how to modify the equational theory to solve them. We sketch an
algebraic vectorial PCF and its possible denotational interpretations
- …