43,395 research outputs found
Z_3-graded exterior differential calculus and gauge theories of higher order
We present a possible generalization of the exterior differential calculus,
based on the operator d such that d^3=0, but d^2\not=0. The first and second
order differentials generate an associative algebra; we shall suppose that
there are no binary relations between first order differentials, while the
ternary products will satisfy the cyclic relations based on the representation
of cyclic group Z_3 by cubic roots of unity. We shall attribute grade 1 to the
first order differentials and grade 2 to the second order differentials; under
the associative multiplication law the grades add up modulo 3. We show how the
notion of covariant derivation can be generalized with a 1-form A, and we give
the expression in local coordinates of the curvature 3-form. Finally, the
introduction of notions of a scalar product and integration of the Z_3-graded
exterior forms enables us to define variational principle and to derive the
differential equations satisfied by the curvature 3-form. The Lagrangian
obtained in this way contains the invariants of the ordinary gauge field tensor
F_{ik} and its covariant derivatives D_i F_{km}.Comment: 13 pages, no figure
On P-transitive graphs and applications
We introduce a new class of graphs which we call P-transitive graphs, lying
between transitive and 3-transitive graphs. First we show that the analogue of
de Jongh-Sambin Theorem is false for wellfounded P-transitive graphs; then we
show that the mu-calculus fixpoint hierarchy is infinite for P-transitive
graphs. Both results contrast with the case of transitive graphs. We give also
an undecidability result for an enriched mu-calculus on P-transitive graphs.
Finally, we consider a polynomial time reduction from the model checking
problem on arbitrary graphs to the model checking problem on P-transitive
graphs. All these results carry over to 3-transitive graphs.Comment: In Proceedings GandALF 2011, arXiv:1106.081
An Arithmetization of Logical Oppositions
An arithmetic theory of oppositions is devised by comparing expressions, Boolean bitstrings, and integers. This leads to a set of correspondences between three domains of investigation, namely: logic, geometry, and arithmetic. The structural properties of each area are investigated in turn, before justifying the procedure as a whole. Io finish, I show how this helps to improve the logical calculus of oppositions, through the consideration of corresponding operations between integers
Algebraic Properties of Qualitative Spatio-Temporal Calculi
Qualitative spatial and temporal reasoning is based on so-called qualitative
calculi. Algebraic properties of these calculi have several implications on
reasoning algorithms. But what exactly is a qualitative calculus? And to which
extent do the qualitative calculi proposed meet these demands? The literature
provides various answers to the first question but only few facts about the
second. In this paper we identify the minimal requirements to binary
spatio-temporal calculi and we discuss the relevance of the according axioms
for representation and reasoning. We also analyze existing qualitative calculi
and provide a classification involving different notions of a relation algebra.Comment: COSIT 2013 paper including supplementary materia
To found or not to found: that is the question
Aim of this paper is to confute two views, the first about Schr\"oder's
presumptive foundationalism, according to he founded mathematics on the
calculus of relatives; the second one mantaining that Schr\"oder only in his
last years (from 1890 onwards) focused on an universal and symbolic language
(by him called pasigraphy). We will argue that, on the one hand Schr\"oder
considered the problem of founding mathematics already solved by Dedekind,
limiting himself in a mere translation of the Chain Theory in the language of
the relatives. On the other hand, we will show that Schr\"oder's pasigraphy was
connaturate to himself and that it roots in his very childhood and in his love
for foreign languages.Comment: Next to be published in Logic and Logical Philosoph
Temporal Data Modeling and Reasoning for Information Systems
Temporal knowledge representation and reasoning is a major research field in Artificial
Intelligence, in Database Systems, and in Web and Semantic Web research. The ability to
model and process time and calendar data is essential for many applications like appointment
scheduling, planning, Web services, temporal and active database systems, adaptive
Web applications, and mobile computing applications. This article aims at three complementary
goals. First, to provide with a general background in temporal data modeling
and reasoning approaches. Second, to serve as an orientation guide for further specific
reading. Third, to point to new application fields and research perspectives on temporal
knowledge representation and reasoning in the Web and Semantic Web
On Role Logic
We present role logic, a notation for describing properties of relational
structures in shape analysis, databases, and knowledge bases. We construct role
logic using the ideas of de Bruijn's notation for lambda calculus, an encoding
of first-order logic in lambda calculus, and a simple rule for implicit
arguments of unary and binary predicates. The unrestricted version of role
logic has the expressive power of first-order logic with transitive closure.
Using a syntactic restriction on role logic formulas, we identify a natural
fragment RL^2 of role logic. We show that the RL^2 fragment has the same
expressive power as two-variable logic with counting C^2 and is therefore
decidable. We present a translation of an imperative language into the
decidable fragment RL^2, which allows compositional verification of programs
that manipulate relational structures. In addition, we show how RL^2 encodes
boolean shape analysis constraints and an expressive description logic.Comment: 20 pages. Our later SAS 2004 result builds on this wor
Similarity and bisimilarity notions appropriate for characterizing indistinguishability in fragments of the calculus of relations
Motivated by applications in databases, this paper considers various
fragments of the calculus of binary relations. The fragments are obtained by
leaving out, or keeping in, some of the standard operators, along with some
derived operators such as set difference, projection, coprojection, and
residuation. For each considered fragment, a characterization is obtained for
when two given binary relational structures are indistinguishable by
expressions in that fragment. The characterizations are based on appropriately
adapted notions of simulation and bisimulation.Comment: 36 pages, Journal of Logic and Computation 201
On Tarski's axiomatic foundations of the calculus of relations
It is shown that Tarski's set of ten axioms for the calculus of relations is
independent in the sense that no axiom can be derived from the remaining
axioms. It is also shown that by modifying one of Tarski's axioms slightly, and
in fact by replacing the right-hand distributive law for relative
multiplication with its left-hand version, we arrive at an equivalent set of
axioms which is redundant in the sense that one of the axioms, namely the
second involution law, is derivable from the other axioms. The set of remaining
axioms is independent. Finally, it is shown that if both the left-hand and
right-hand distributive laws for relative multiplication are included in the
set of axioms, then two of Tarski's other axioms become redundant, namely the
second involution law and the distributive law for converse. The set of
remaining axioms is independent and equivalent to Tarski's axiom system
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