19 research outputs found
Homotopy theory with bornological coarse spaces
We propose an axiomatic characterization of coarse homology theories defined
on the category of bornological coarse spaces. We construct a category of
motivic coarse spectra. Our focus is the classification of coarse homology
theories and the construction of examples. We show that if a transformation
between coarse homology theories induces an equivalence on all discrete
bornological coarse spaces, then it is an equivalence on bornological coarse
spaces of finite asymptotic dimension. The example of coarse K-homology will be
discussed in detail.Comment: 220 pages (complete revision
The infinite random simplicial complex
We study the Fraisse limit of the class of all finite simplicial complexes.
Whilst the natural model-theoretic setting for this class uses an infinite
language, a range of results associated with Fraisse limits of structures for
finite languages carry across to this important example. We introduce the
notion of a local class, with the class of finite simplicial complexes as an
archetypal example, and in this general context prove the existence of a 0-1
law and other basic model-theoretic results. Constraining to the case where all
relations are symmetric, we show that every direct limit of finite groups, and
every metrizable profinite group, appears as a subgroup of the automorphism
group of the Fraisse limit. Finally, for the specific case of simplicial
complexes, we show that the geometric realisation is topologically surprisingly
simple: despite the combinatorial complexity of the Fraisse limit, its
geometric realisation is homeomorphic to the infinite simplex.Comment: 33 page
Towards a Relativistic KMS Condition
It is shown that, under quite general conditions, thermal correlation
functions in relativistic quantum field theory have stronger analyticity
properties in configuration space than those imposed by the KMS-condition.
These analyticity properties may be understood as a remnant of the relativistic
spectrum condition in the vacuum sector and lead to a Lorentz-covariant
formulation of the KMS-condition involving all space-time variables.Comment: TEX, 23 pages, figures omitted (e-print version of published paper
Interdefinability of defeasible logic and logic programming under the well-founded semantics
We provide a method of translating theories of Nute's defeasible logic into
logic programs, and a corresponding translation in the opposite direction.
Under certain natural restrictions, the conclusions of defeasible theories
under the ambiguity propagating defeasible logic ADL correspond to those of the
well-founded semantics for normal logic programs, and so it turns out that the
two formalisms are closely related. Using the same translation of logic
programs into defeasible theories, the semantics for the ambiguity blocking
defeasible logic NDL can be seen as indirectly providing an ambiguity blocking
semantics for logic programs. We also provide antimonotone operators for both
ADL and NDL, each based on the Gelfond-Lifschitz (GL) operator for logic
programs. For defeasible theories without defeaters or priorities on rules, the
operator for ADL corresponds to the GL operator and so can be seen as partially
capturing the consequences according to ADL. Similarly, the operator for NDL
captures the consequences according to NDL, though in this case no restrictions
on theories apply. Both operators can be used to define stable model semantics
for defeasible theories.Comment: 36 pages; To appear in Theory and Practice of Logic Programming
(TPLP
Linear-Time Verification of Data-Aware Processes Modulo Theories via Covers and Automata (Extended Version)
The need to model and analyse dynamic systems operating over complex data is
ubiquitous in AI and neighboring areas, in particular business process
management. Analysing such data-aware systems is a notoriously difficult
problem, as they are intrinsically infinite-state. Existing approaches work for
specific datatypes, and/or limit themselves to the verification of safety
properties. In this paper, we lift both such limitations, studying for the
first time linear-time verification for so-called data-aware processes modulo
theories (DMTs), from the foundational and practical point of view. The DMT
model is very general, as it supports processes operating over variables that
can store arbitrary types of data, ranging over infinite domains and equipped
with domain-specific predicates. Specifically, we provide four contributions.
First, we devise a semi-decision procedure for linear-time verification of
DMTs, which works for a very large class of datatypes obeying to mild
model-theoretic assumptions. The procedure relies on a unique combination of
automata-theoretic and cover computation techniques to respectively deal with
linear-time properties and datatypes. Second, we identify an abstract, semantic
property that guarantees the existence of a faithful finite-state abstraction
of the original system, and show that our method becomes a decision procedure
in this case. Third, we identify concrete, checkable classes of systems that
satisfy this property, generalising several results in the literature. Finally,
we present an implementation and a first experimental evaluation
Admissibility of Î <sub>2</sub>-inference rules: Interpolation, model completion, and contact algebras
We devise three strategies for recognizing admissibility of non-standard inference rules via interpolation, uniform interpolation, and model completions. We apply our machinery to the case of symmetric implication calculus S2IC, where we also supply a finite axiomatization of the model completion of its algebraic counterpart, via the equivalent theory of contact algebras. Using this result we obtain a finite basis for admissible Î 2-rules
Some attempts at a direct reduction of the infinite to the (large) finite
I survey some endeavors which have been made to attain a sort of direct reduction of the usual notion of countable infinity to some reasonable notion of finiteness, in terms of nonstandard arithmetic, feasibility, pseudo-models of derivations, Ehrenfeucht star-models, etc. I maintain that although many interesting results have been obtained in these attempts, they ultimately show that (at least by the means considered here) no satisfactory reduction is possible
Linearly ordered sets with only one operator have the amalgamation property
The class of linearly ordered sets with one order preserving unary operation
has the Strong Amalgamation Property (SAP). The class of linearly ordered sets
with one strict order preserving unary operation has AP but not SAP. The class
of linearly ordered sets with two order preserving unary operations has not AP.
For every set , the class of linearly ordered sets with an -indexed
family of automorphisms has SAP. Corresponding results are proved in the case
of order reversing operations. Various subclasses of the above classes are
considered and some model-theoretical consequences are presented.Comment: v2 minor fixe