802 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
The maximum of Brownian motion with parabolic drift
We study the maximum of a Brownian motion with a parabolic drift; this is a
random variable that often occurs as a limit of the maximum of discrete
processes whose expectations have a maximum at an interior point. We give
series expansions and integral formulas for the distribution and the first two
moments, together with numerical values to high precision.Comment: 37 page
Randomness Relative to Cantor Expansions
Imagine a sequence in which the first letter comes from a binary alphabet,
the second letter can be chosen on an alphabet with 10 elements, the third
letter can be chosen on an alphabet with 3 elements and so on. When such a
sequence can be called random? In this paper we offer a solution to the above
question using the approach to randomness proposed by Algorithmic Information
Theory.Comment: several small change
Dependent Types for Pragmatics
This paper proposes the use of dependent types for pragmatic phenomena such
as pronoun binding and presupposition resolution as a type-theoretic
alternative to formalisms such as Discourse Representation Theory and Dynamic
Semantics.Comment: This version updates the paper for publication in LEU
Two kinds of procedural semantics for privative modification
In this paper we present two kinds of procedural semantics for privative modification. We do this for three reasons. The first reason is to launch a tough test case to gauge the degree of substantial agreement between a constructivist and a realist interpretation of procedural semantics; the second is to extend Martin-L Ìfâs Constructive Type Theory to privative modification, which is characteristic of natural language; the third reason is to sketch a positive characterization of privation
On the strength of dependent products in the type theory of Martin-L\"of
One may formulate the dependent product types of Martin-L\"of type theory
either in terms of abstraction and application operators like those for the
lambda-calculus; or in terms of introduction and elimination rules like those
for the other constructors of type theory. It is known that the latter rules
are at least as strong as the former: we show that they are in fact strictly
stronger. We also show, in the presence of the identity types, that the
elimination rule for dependent products--which is a "higher-order" inference
rule in the sense of Schroeder-Heister--can be reformulated in a first-order
manner. Finally, we consider the principle of function extensionality in type
theory, which asserts that two elements of a dependent product type which are
pointwise propositionally equal, are themselves propositionally equal. We
demonstrate that the usual formulation of this principle fails to verify a
number of very natural propositional equalities; and suggest an alternative
formulation which rectifies this deficiency.Comment: 18 pages; v2: final journal versio
Asymptotic normality of the size of the giant component via a random walk
In this paper we give a simple new proof of a result of Pittel and Wormald
concerning the asymptotic value and (suitably rescaled) limiting distribution
of the number of vertices in the giant component of above the scaling
window of the phase transition. Nachmias and Peres used martingale arguments to
study Karp's exploration process, obtaining a simple proof of a weak form of
this result. We use slightly different martingale arguments to obtain a much
sharper result with little extra work.Comment: 11 pages; slightly expanded, reference adde
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