44,238 research outputs found
Computability and analysis: the legacy of Alan Turing
We discuss the legacy of Alan Turing and his impact on computability and
analysis.Comment: 49 page
Laver and set theory
In this commemorative article, the work of Richard Laver is surveyed in its full range and extent.Accepted manuscrip
On what I do not understand (and have something to say): Part I
This is a non-standard paper, containing some problems in set theory I have
in various degrees been interested in. Sometimes with a discussion on what I
have to say; sometimes, of what makes them interesting to me, sometimes the
problems are presented with a discussion of how I have tried to solve them, and
sometimes with failed tries, anecdote and opinion. So the discussion is quite
personal, in other words, egocentric and somewhat accidental. As we discuss
many problems, history and side references are erratic, usually kept at a
minimum (``see ... '' means: see the references there and possibly the paper
itself).
The base were lectures in Rutgers Fall'97 and reflect my knowledge then. The
other half, concentrating on model theory, will subsequently appear
Generalized probabilities in statistical theories
In this review article we present different formal frameworks for the
description of generalized probabilities in statistical theories. We discuss
the particular cases of probabilities appearing in classical and quantum
mechanics, possible generalizations of the approaches of A. N. Kolmogorov and
R. T. Cox to non-commutative models, and the approach to generalized
probabilities based on convex sets
Canonical extensions and ultraproducts of polarities
J{\'o}nsson and Tarski's notion of the perfect extension of a Boolean algebra
with operators has evolved into an extensive theory of canonical extensions of
lattice-based algebras. After reviewing this evolution we make two
contributions. First it is shown that the failure of a variety of algebras to
be closed under canonical extensions is witnessed by a particular one of its
free algebras. The size of the set of generators of this algebra can be made a
function of a collection of varieties and is a kind of Hanf number for
canonical closure. Secondly we study the complete lattice of stable subsets of
a polarity structure, and show that if a class of polarities is closed under
ultraproducts, then its stable set lattices generate a variety that is closed
under canonical extensions. This generalises an earlier result of the author
about generation of canonically closed varieties of Boolean algebras with
operators, which was in turn an abstraction of the result that a first-order
definable class of Kripke frames determines a modal logic that is valid in its
so-called canonical frames
Efficient First-Order Temporal Logic for Infinite-State Systems
In this paper we consider the specification and verification of
infinite-state systems using temporal logic. In particular, we describe
parameterised systems using a new variety of first-order temporal logic that is
both powerful enough for this form of specification and tractable enough for
practical deductive verification. Importantly, the power of the temporal
language allows us to describe (and verify) asynchronous systems, communication
delays and more complex properties such as liveness and fairness properties.
These aspects appear difficult for many other approaches to infinite-state
verification.Comment: 16 pages, 2 figure
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