9,985 research outputs found
Nets, relations and linking diagrams
In recent work, the author and others have studied compositional algebras of
Petri nets. Here we consider mathematical aspects of the pure linking algebras
that underly them. We characterise composition of nets without places as the
composition of spans over appropriate categories of relations, and study the
underlying algebraic structures.Comment: 15 pages, Proceedings of 5th Conference on Algebra and Coalgebra in
Computer Science (CALCO), Warsaw, Poland, 3-6 September 201
Embedded graph invariants in Chern-Simons theory
Chern-Simons gauge theory, since its inception as a topological quantum field
theory, has proved to be a rich source of understanding for knot invariants. In
this work the theory is used to explore the definition of the expectation value
of a network of Wilson lines - an embedded graph invariant. Using a slight
generalization of the variational method, lowest-order results for invariants
for arbitrary valence graphs are derived; gauge invariant operators are
introduced; and some higher order results are found. The method used here
provides a Vassiliev-type definition of graph invariants which depend on both
the embedding of the graph and the group structure of the gauge theory. It is
found that one need not frame individual vertices. Though, without a global
projection of the graph, there is an ambiguity in the relation of the
decomposition of distinct vertices. It is suggested that framing may be seen as
arising from this ambiguity - as a way of relating frames at distinct vertices.Comment: 20 pages; RevTex; with approx 50 ps figures; References added,
introduction rewritten, version to be published in Nuc. Phys.
Canonical quantum gravity in the Vassiliev invariants arena: I. Kinematical structure
We generalize the idea of Vassiliev invariants to the spin network context,
with the aim of using these invariants as a kinematical arena for a canonical
quantization of gravity. This paper presents a detailed construction of these
invariants (both ambient and regular isotopic) requiring a significant
elaboration based on the use of Chern-Simons perturbation theory which extends
the work of Kauffman, Martin and Witten to four-valent networks. We show that
this space of knot invariants has the crucial property -from the point of view
of the quantization of gravity- of being loop differentiable in the sense of
distributions. This allows the definition of diffeomorphism and Hamiltonian
constraints. We show that the invariants are annihilated by the diffeomorphism
constraint. In a companion paper we elaborate on the definition of a
Hamiltonian constraint, discuss the constraint algebra, and show that the
construction leads to a consistent theory of canonical quantum gravity.Comment: 21 Pages, RevTex, many figures included with psfi
Quantum deformation of quantum gravity
We describe a deformation of the observable algebra of quantum gravity in
which the loop algebra is extended to framed loops. This allows an alternative
nonperturbative quantization which is suitable for describing a phase of
quantum gravity characterized by states which are normalizable in the measure
of Chern-Simons theory. The deformation parameter, q, depends on the
cosmological constant. The Mandelstam identities are extended to a set of
relations which are governed by the Kauffman bracket so that the spin network
basis is deformed to a basis of SU(2)q spin networks. Corrections to the
actions of operators in non-perturbative quantum gravity may be readily
computed using recoupling theory; the example of the area observable is treated
here. Finally, eigenstates of the q-deformed Wilson loops are constructed,
which may make possible the construction of a q-deformed connection
representation through an inverse transform.Comment: 12 pages, many figure
MALL proof equivalence is Logspace-complete, via binary decision diagrams
Proof equivalence in a logic is the problem of deciding whether two proofs
are equivalent modulo a set of permutation of rules that reflects the
commutative conversions of its cut-elimination procedure. As such, it is
related to the question of proofnets: finding canonical representatives of
equivalence classes of proofs that have good computational properties. It can
also be seen as the word problem for the notion of free category corresponding
to the logic.
It has been recently shown that proof equivalence in MLL (the multiplicative
with units fragment of linear logic) is PSPACE-complete, which rules out any
low-complexity notion of proofnet for this particular logic.
Since it is another fragment of linear logic for which attempts to define a
fully satisfactory low-complexity notion of proofnet have not been successful
so far, we study proof equivalence in MALL- (multiplicative-additive without
units fragment of linear logic) and discover a situation that is totally
different from the MLL case. Indeed, we show that proof equivalence in MALL-
corresponds (under AC0 reductions) to equivalence of binary decision diagrams,
a data structure widely used to represent and analyze Boolean functions
efficiently.
We show these two equivalent problems to be LOGSPACE-complete. If this
technically leaves open the possibility for a complete solution to the question
of proofnets for MALL-, the established relation with binary decision diagrams
actually suggests a negative solution to this problem.Comment: in TLCA 201
A review of information flow diagrammatic models for product-service systems
A product-service system (PSS) is a combination of products and services to
create value for both customers and manufacturers. Modelling a PSS based on
function orientation offers a useful way to distinguish system inputs and
outputs with regards to how data are consumed and information is used, i.e.
information flow. This article presents a review of diagrammatic information
flow tools, which are designed to describe a system through its functions. The
origin, concept and applications of these tools are investigated, followed by an
analysis of information flow modelling with regards to key PSS properties. A
case study of selection laser melting technology implemented as PSS will then be
used to show the application of information flow modelling for PSS design. A
discussion based on the usefulness of the tools in modelling the key elements of
PSS and possible future research directions are also presented
Understanding Science Through Knowledge Organizers: An Introduction
We propose, in this paper, a teaching program based on a grammar of scientific language borrowed mostly from the area of knowledge representation in computer science and logic. The paper introduces an operationizable framework for understanding knowledge using knowledge representation (KR) methodology. We start with organizing concepts based on their cognitive function, followed by assigning valid and authentic semantic relations to the concepts. We propose that in science education, students can understand better if they organize their knowledge using the KR principles. The process, we claim, can help them to align their conceptual framework with that of experts which we assume is the goal of science education
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