32,926 research outputs found
Checking Trustworthiness of Probabilistic Computations in a Typed Natural Deduction System
In this paper we present the probabilistic typed natural deduction calculus
TPTND, designed to reason about and derive trustworthiness properties of
probabilistic computational processes, like those underlying current AI
applications. Derivability in TPTND is interpreted as the process of extracting
samples of possibly complex outputs with a certain frequency from a given
categorical distribution. We formalize trust for such outputs as a form of
hypothesis testing on the distance between such frequency and the intended
probability. The main advantage of the calculus is to render such notion of
trustworthiness checkable. We present a computational semantics for the terms
over which we reason and then the semantics of TPTND, where logical operators
as well as a Trust operator are defined through introduction and elimination
rules. We illustrate structural and metatheoretical properties, with particular
focus on the ability to establish under which term evolutions and logical rules
applications the notion of trustworhtiness can be preserved
Quantum picturalism for topological cluster-state computing
Topological quantum computing is a way of allowing precise quantum
computations to run on noisy and imperfect hardware. One implementation uses
surface codes created by forming defects in a highly-entangled cluster state.
Such a method of computing is a leading candidate for large-scale quantum
computing. However, there has been a lack of sufficiently powerful high-level
languages to describe computing in this form without resorting to single-qubit
operations, which quickly become prohibitively complex as the system size
increases. In this paper we apply the category-theoretic work of Abramsky and
Coecke to the topological cluster-state model of quantum computing to give a
high-level graphical language that enables direct translation between quantum
processes and physical patterns of measurement in a computer - a "compiler
language". We give the equivalence between the graphical and topological
information flows, and show the applicable rewrite algebra for this computing
model. We show that this gives us a native graphical language for the design
and analysis of topological quantum algorithms, and finish by discussing the
possibilities for automating this process on a large scale.Comment: 18 pages, 21 figures. Published in New J. Phys. special issue on
topological quantum computin
An Arithmetization of Logical Oppositions
An arithmetic theory of oppositions is devised by comparing expressions, Boolean bitstrings, and integers. This leads to a set of correspondences between three domains of investigation, namely: logic, geometry, and arithmetic. The structural properties of each area are investigated in turn, before justifying the procedure as a whole. Io finish, I show how this helps to improve the logical calculus of oppositions, through the consideration of corresponding operations between integers
Logic in Opposition
It is claimed hereby that, against a current view of logic as a theory of consequence, opposition is a basic logical concept that can be used to define consequence itself. This requires some substantial changes in the underlying framework, including: a non-Fregean semantics of questions and answers, instead of the usual truth-conditional semantics; an extension of opposition as a relation between any structured objects; a definition of oppositions in terms of basic negation. Objections to this claim will be reviewed
Logical operators for ontological modeling
We show that logic has more to offer to ontologists than standard first order
and modal operators. We first describe some operators of linear logic which we
believe are particularly suitable for ontological modeling, and suggest how to interpret
them within an ontological framework. After showing how they can coexist
with those of classical logic, we analyze three notions of artifact from the literature
to conclude that these linear operators allow for reducing the ontological commitment
needed for their formalization, and even simplify their logical formulation
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