7,424 research outputs found
Complete independence of an axiom system for central translations
A recently proposed axiom system for Andr\'e's central translation structures
is improved upon. First, one of its axioms turns out to be dependent (derivable
from the other axioms). Without this axiom, the axiom system is indeed
independent. Second, whereas most of the original independence models were
infinite, finite independence models are available. Moreover, the independence
proof for one of the axioms employed proof-theoretic techniques rather than
independence models; for this axiom, too, a finite independence model exists.
For every axiom, then, there is a finite independence model. Finally, the axiom
system (without its single dependent axiom) is not only independent, but
completely independent.Comment: 10 pages. Submitted to Note di Matematic
Constructing Hamiltonian quantum theories from path integrals in a diffeomorphism invariant context
Osterwalder and Schrader introduced a procedure to obtain a (Lorentzian)
Hamiltonian quantum theory starting from a measure on the space of (Euclidean)
histories of a scalar quantum field. In this paper, we extend that construction
to more general theories which do not refer to any background, space-time
metric (and in which the space of histories does not admit a natural linear
structure). Examples include certain gauge theories, topological field theories
and relativistic gravitational theories. The treatment is self-contained in the
sense that an a priori knowledge of the Osterwalder-Schrader theorem is not
assumed.Comment: Plain Latex, 25 p., references added, abstract and title changed
(originally :``Osterwalder Schrader Reconstruction and Diffeomorphism
Invariance''), introduction extended, one appendix with illustrative model
added, accepted by Class. Quantum Gra
Alternative axiomatics and complexity of deliberative STIT theories
We propose two alternatives to Xu's axiomatization of the Chellas STIT. The
first one also provides an alternative axiomatization of the deliberative STIT.
The second one starts from the idea that the historic necessity operator can be
defined as an abbreviation of operators of agency, and can thus be eliminated
from the logic of the Chellas STIT. The second axiomatization also allows us to
establish that the problem of deciding the satisfiability of a STIT formula
without temporal operators is NP-complete in the single-agent case, and is
NEXPTIME-complete in the multiagent case, both for the deliberative and the
Chellas' STIT.Comment: Submitted to the Journal of Philosophical Logic; 13 pages excluding
anne
Towards MKM in the Large: Modular Representation and Scalable Software Architecture
MKM has been defined as the quest for technologies to manage mathematical
knowledge. MKM "in the small" is well-studied, so the real problem is to scale
up to large, highly interconnected corpora: "MKM in the large". We contend that
advances in two areas are needed to reach this goal. We need representation
languages that support incremental processing of all primitive MKM operations,
and we need software architectures and implementations that implement these
operations scalably on large knowledge bases.
We present instances of both in this paper: the MMT framework for modular
theory-graphs that integrates meta-logical foundations, which forms the base of
the next OMDoc version; and TNTBase, a versioned storage system for XML-based
document formats. TNTBase becomes an MMT database by instantiating it with
special MKM operations for MMT.Comment: To appear in The 9th International Conference on Mathematical
Knowledge Management: MKM 201
Cut-free Calculi and Relational Semantics for Temporal STIT Logics
We present cut-free labelled sequent calculi for a central formalism in logics of agency: STIT logics with temporal operators. These include sequent systems for Ldm , Tstit and Xstit. All calculi presented possess essential structural properties such as contraction- and cut-admissibility. The labelled calculi G3Ldm and G3Tstit are shown sound and complete relative to irreflexive temporal frames. Additionally, we extend current results by showing that also Xstit can be characterized through relational frames, omitting the use of BT+AC frames
Laver's results and low-dimensional topology
In connection with his interest in selfdistributive algebra, Richard Laver
established two deep results with potential applications in low-dimensional
topology, namely the existence of what is now known as the Laver tables and the
well-foundedness of the standard ordering of positive braids. Here we present
these results and discuss the way they could be used in topological
applications
Weighted Dirac combs with pure point diffraction
A class of translation bounded complex measures, which have the form of
weighted Dirac combs, on locally compact Abelian groups is investigated. Given
such a Dirac comb, we are interested in its diffraction spectrum which emerges
as the Fourier transform of the autocorrelation measure. We present a
sufficient set of conditions to ensure that the diffraction measure is a pure
point measure. Simultaneously, we establish a natural link to the theory of the
cut and project formalism and to the theory of almost periodic measures. Our
conditions are general enough to cover the known theory of model sets, but also
to include examples such as the visible lattice points.Comment: 44 pages; several corrections and improvement
Axioms: Mathematical and Spiritual: What Says the Parable?
Relational structure A is compact provided for any structure Jffi of the same signature, if every finite substructure of Jffi has a homomorphism to A then so does Jffi. The Constraint Satisfaction Problem (CSP) for A is the computational problem of determining whether finite structures have homomorphisms into A. We explore a connection between the hierarchy of logical axioms and the complexity hierarchy of CSPs: It appears that the complexity of CSP for A corresponds to the strength of the axiom A is compact . At the top, the statement K3 is compacts is logically equivalent to the compactness theorem. Thus the compactness of K3 implies the compactness of all finite relational structures. Moreover, the CSP for K3 is NP-complete. At the bottom are width-one structures; these are provably complete from ZF and their corresponding CPSs are polynomial-time solvable
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