106 research outputs found
The non-unique Universe
The purpose of this paper is to elucidate, by means of concepts and theorems
drawn from mathematical logic, the conditions under which the existence of a
multiverse is a logical necessity in mathematical physics, and the implications
of Godel's incompleteness theorem for theories of everything.
Three conclusions are obtained in the final section: (i) the theory of the
structure of our universe might be an undecidable theory, and this constitutes
a potential epistemological limit for mathematical physics, but because such a
theory must be complete, there is no ontological barrier to the existence of a
final theory of everything; (ii) in terms of mathematical logic, there are two
different types of multiverse: classes of non-isomorphic but elementarily
equivalent models, and classes of model which are both non-isomorphic and
elementarily inequivalent; (iii) for a hypothetical theory of everything to
have only one possible model, and to thereby negate the possible existence of a
multiverse, that theory must be such that it admits only a finite model
Adding an Abstraction Barrier to ZF Set Theory
Much mathematical writing exists that is, explicitly or implicitly, based on
set theory, often Zermelo-Fraenkel set theory (ZF) or one of its variants. In
ZF, the domain of discourse contains only sets, and hence every mathematical
object must be a set. Consequently, in ZF, with the usual encoding of an
ordered pair , formulas like have truth values, and operations like have results that are sets. Such 'accidental theorems' do not match
how people think about the mathematics and also cause practical difficulties
when using set theory in machine-assisted theorem proving. In contrast, in a
number of proof assistants, mathematical objects and concepts can be built of
type-theoretic stuff so that many mathematical objects can be, in essence,
terms of an extended typed -calculus. However, dilemmas and
frustration arise when formalizing mathematics in type theory.
Motivated by problems of formalizing mathematics with (1) purely
set-theoretic and (2) type-theoretic approaches, we explore an option with much
of the flexibility of set theory and some of the useful features of type
theory. We present ZFP: a modification of ZF that has ordered pairs as
primitive, non-set objects. ZFP has a more natural and abstract axiomatic
definition of ordered pairs free of any notion of representation. This paper
presents axioms for ZFP, and a proof in ZF (machine-checked in Isabelle/ZF) of
the existence of a model for ZFP, which implies that ZFP is consistent if ZF
is. We discuss the approach used to add this abstraction barrier to ZF
Changing a semantics: opportunism or courage?
The generalized models for higher-order logics introduced by Leon Henkin, and
their multiple offspring over the years, have become a standard tool in many
areas of logic. Even so, discussion has persisted about their technical status,
and perhaps even their conceptual legitimacy. This paper gives a systematic
view of generalized model techniques, discusses what they mean in mathematical
and philosophical terms, and presents a few technical themes and results about
their role in algebraic representation, calibrating provability, lowering
complexity, understanding fixed-point logics, and achieving set-theoretic
absoluteness. We also show how thinking about Henkin's approach to semantics of
logical systems in this generality can yield new results, dispelling the
impression of adhocness. This paper is dedicated to Leon Henkin, a deep
logician who has changed the way we all work, while also being an always open,
modest, and encouraging colleague and friend.Comment: 27 pages. To appear in: The life and work of Leon Henkin: Essays on
his contributions (Studies in Universal Logic) eds: Manzano, M., Sain, I. and
Alonso, E., 201
A Multiset Rewriting Model for Specifying and Verifying Timing Aspects of Security Protocols
Catherine Meadows has played an important role in the advancement of formal methods for protocol security verification. Her insights on the use of, for example, narrowing and rewriting logic has made possible the automated discovery of new attacks and the shaping of new protocols. Meadows has also investigated other security aspects, such as, distance-bounding protocols and denial of service attacks. We have been greatly inspired by her work. This paper describes the use of Multiset Rewriting for the specification and verification of timing aspects of protocols, such as network delays, timeouts, timed intruder models and distance-bounding properties. We detail these timed features with a number of examples and describe decidable fragments of related verification problems
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Freshwater transport in the coupled ocean-atmosphere system: a passive ocean
Conservation of water demands that meridional ocean and atmosphere freshwater transports (FWT) are of equal magnitude but opposite in direction. This suggests that the atmospheric FWT and its associated latent heat (LH) transport could be thought of as a \textquotedblleft coupled ocean/atmosphere mode\textquotedblright. But what is the true nature of this coupling? Is the ocean passive or active?
Here we analyze a series of simulations with a coupled ocean-atmosphere-sea ice model employing highly idealized geometries but with markedly different coupled climates and patterns of ocean circulation. Exploiting streamfunctions in specific humidity coordinates for the atmosphere and salt coordinates for the ocean to represent FWT in their respective medium, we find that atmospheric FWT/LH transport is essentially independent of the ocean state. Ocean circulation and salinity distribution adjust to achieve a return freshwater pathway demanded of them by the atmosphere. So, although ocean and atmosphere FWTs are indeed coupled by mass conservation, the ocean is a passive component acting as a reservoir of freshwater
Revisiting Enumerative Instantiation
International audienceFormal methods applications often rely on SMT solvers to automatically discharge proof obligations. SMT solvers handle quantified formulas using incomplete heuristic techniques like E-matching, and often resort to model-based quantifier instantiation (MBQI) when these techniques fail. This paper revisits enumerative instantiation, a technique that considers instantiations based on exhaustive enumeration of ground terms. Although simple, we argue that enumer-ative instantiation can supplement other instantiation techniques and be a viable alternative to MBQI for valid proof obligations. We first present a stronger Her-brand Theorem, better suited as a basis for the instantiation loop used in SMT solvers; it furthermore requires considering less instances than classical Herbrand instantiation. Based on this result, we present different strategies for combining enumerative instantiation with other instantiation techniques in an effective way. The experimental evaluation shows that the implementation of these new techniques in the SMT solver CVC4 leads to significant improvements in several benchmark libraries, including many stemming from verification efforts
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