281,687 research outputs found
Neighbourhood Abstraction in GROOVE - Tool Paper
In this paper we discuss the implementation of neighbourhood graph abstraction in the GROOVE tool set. Important classes of graph grammars may have unbounded state spaces and therefore cannot be verified with traditional model checking techniques. One way to address this problem is to perform graph abstraction, which allows us to generate a finite abstract state space that over-approximates the original one. In previous work we presented the theory of neighbourhood abstraction. In this paper, we present the implementation of this theory in GROOVE and illustrate its applicability with a case study that models a single-linked list
Abstracting Asynchronous Multi-Valued Networks: An Initial Investigation
Multi-valued networks provide a simple yet expressive qualitative state based
modelling approach for biological systems. In this paper we develop an
abstraction theory for asynchronous multi-valued network models that allows the
state space of a model to be reduced while preserving key properties of the
model. The abstraction theory therefore provides a mechanism for coping with
the state space explosion problem and supports the analysis and comparison of
multi-valued networks. We take as our starting point the abstraction theory for
synchronous multi-valued networks which is based on the finite set of traces
that represent the behaviour of such a model. The problem with extending this
approach to the asynchronous case is that we can now have an infinite set of
traces associated with a model making a simple trace inclusion test infeasible.
To address this we develop a decision procedure for checking asynchronous
abstractions based on using the finite state graph of an asynchronous
multi-valued network to reason about its trace semantics. We illustrate the
abstraction techniques developed by considering a detailed case study based on
a multi-valued network model of the regulation of tryptophan biosynthesis in
Escherichia coli.Comment: Presented at MeCBIC 201
Fragments of Frege's Grundgesetze and G\"odel's Constructible Universe
Frege's Grundgesetze was one of the 19th century forerunners to contemporary
set theory which was plagued by the Russell paradox. In recent years, it has
been shown that subsystems of the Grundgesetze formed by restricting the
comprehension schema are consistent. One aim of this paper is to ascertain how
much set theory can be developed within these consistent fragments of the
Grundgesetze, and our main theorem shows that there is a model of a fragment of
the Grundgesetze which defines a model of all the axioms of Zermelo-Fraenkel
set theory with the exception of the power set axiom. The proof of this result
appeals to G\"odel's constructible universe of sets, which G\"odel famously
used to show the relative consistency of the continuum hypothesis. More
specifically, our proofs appeal to Kripke and Platek's idea of the projectum
within the constructible universe as well as to a weak version of
uniformization (which does not involve knowledge of Jensen's fine structure
theory). The axioms of the Grundgesetze are examples of abstraction principles,
and the other primary aim of this paper is to articulate a sufficient condition
for the consistency of abstraction principles with limited amounts of
comprehension. As an application, we resolve an analogue of the joint
consistency problem in the predicative setting.Comment: Forthcoming in The Journal of Symbolic Logi
Fregean abstraction in Zermelo-Fraenkel set theory: a deflationary account
The standard treatment of sets and definable classes in first-order
Zermelo-Fraenkel set theory accords in many respects with the Fregean
foundational framework, such as the distinction between objects and concepts.
Nevertheless, in set theory we may define an explicit association of definable
classes with set objects in such a way, I shall prove,
to realize Frege's Basic Law V as a ZF theorem scheme, Russell notwithstanding.
A similar analysis applies to the Cantor-Hume principle and to Fregean
abstraction generally. Because these extension and abstraction operators are
definable, they provide a deflationary account of Fregean abstraction, one
expressible in and reducible to set theory -- every assertion in the language
of set theory allowing the extension and abstraction operators ,
, is equivalent to an assertion not using them. The analysis
thus sidesteps Russell's argument, which is revealed not as a refutation of
Basic Law~V as such, but rather as a version of Tarski's theorem on the
nondefinability of truth, showing that the proto-truth-predicate " falls
under the concept of which is the extension" is not expressible.Comment: 22 pages. Commentary can be made on the author's blog at
http://jdh.hamkins.org/fregean-abstraction-deflationary-accoun
High-Order Metaphysics as High-Order Abstractions and Choice in Set Theory
The link between the high-order metaphysics and abstractions, on the one hand, and choice in the foundation of set theory, on the other hand, can distinguish unambiguously the āgoodā principles of abstraction from the ābadā ones and thus resolve the ābad company problemā as to set theory. Thus it implies correspondingly a more precise definition of the relation between the axiom of choice and āall companyā of axioms in set theory concerning directly or indirectly abstraction: the principle of abstraction, axiom of comprehension, axiom scheme of specification, axiom scheme of separation, subset axiom scheme, axiom scheme of replacement, axiom of unrestricted comprehension, axiom of extensionality, etc
Guidelines For Pursuing and Revealing Data Abstractions
Many data abstraction types, such as networks or set relationships, remain
unfamiliar to data workers beyond the visualization research community. We
conduct a survey and series of interviews about how people describe their data,
either directly or indirectly. We refer to the latter as latent data
abstractions. We conduct a Grounded Theory analysis that (1) interprets the
extent to which latent data abstractions exist, (2) reveals the far-reaching
effects that the interventionist pursuit of such abstractions can have on data
workers, (3) describes why and when data workers may resist such explorations,
and (4) suggests how to take advantage of opportunities and mitigate risks
through transparency about visualization research perspectives and agendas. We
then use the themes and codes discovered in the Grounded Theory analysis to
develop guidelines for data abstraction in visualization projects. To continue
the discussion, we make our dataset open along with a visual interface for
further exploration
Abstraction in parameterised Boolean equation systems
We present a general theory of abstraction for a variety of verification problems. Our theory is set in the framework of parameterized Boolean equation systems. The power of our abstraction theory is compared to that of generalised Kripke modal transition systems (GTSs). We show that for model checking the modal Āµ-calculus, our abstractions can be exponentially more succinct than GTSs and our theory is as complete as the GTS framework for abstraction. Furthermore, we investigate the completeness of our theory for verification problems other than the modal Āµ-calculus. We illustrate the potential of our theory through case studies using the first-order modal Āµ-calculus and a real-time extension thereof, conducted using a prototype implementation of a new syntactic transformation for equation systems
The gauge action, DG Lie algebra and identities for Bernoulli numbers
In this paper we prove a family of identities for Bernoulli numbers
parameterized by triples of integers with , .
These identities are deduced while translating into homotopical terms the gauge
action on the Maurer Cartan Set which can be seen an abstraction of the
behaviour of gauge infinitesimal transformations in classical gauge theory. We
show that Euler and Miki's identities, well known and apparently non related
formulas, are linear combinations of our family and they satisfy a particular
symmetry relation.Comment: Small modifications. To appear in Forum Mathematicu
- ā¦