11,024 research outputs found
Combining Decision Algorithms for Matching in the Union of Disjoint Equational Theories
AbstractThis paper addresses the problem of systematically building a matching algorithm for the union of two disjoint theoriesE1∪E2provided that matching algorithms are known in both theoriesE1andE2. In general, the blind use of combination techniques introduces unification. Two different restrictions are considered in order to reduce this unification to matching. First, we show that combining matching algorithms (with linear constant restriction) is always sufficient for solving a pure fragment of combined matching problems. Second, the investigated method is complete for the largest class of theories where unification is not needed, including regular collapse-free theories and linear theories. Syntactic conditions are given to define this class of theories in which solving the combined matching problem is performed in a modular way
Quantum gravity: unification of principles and interactions, and promises of spectral geometry
Quantum gravity was born as that branch of modern theoretical physics that
tries to unify its guiding principles, i.e., quantum mechanics and general
relativity. Nowadays it is providing new insight into the unification of all
fundamental interactions, while giving rise to new developments in modern
mathematics. It is however unclear whether it will ever become a falsifiable
physical theory, since it deals with Planck-scale physics. Reviewing a wide
range of spectral geometry from index theory to spectral triples, we hope to
dismiss the general opinion that the mere mathematical complexity of the
unification programme will obstruct that programme.Comment: This is a contribution to the Proceedings of the 2007 Midwest
Geometry Conference in honor of Thomas P. Branson, published in SIGMA
(Symmetry, Integrability and Geometry: Methods and Applications) at
http://www.emis.de/journals/SIGMA
Gravity, Geometry and the Quantum
After a brief introduction, basic ideas of the quantum Riemannian geometry
underlying loop quantum gravity are summarized. To illustrate physical
ramifications of quantum geometry, the framework is then applied to homogeneous
isotropic cosmology. Quantum geometry effects are shown to replace the big bang
by a big bounce. Thus, quantum physics does not stop at the big-bang
singularity. Rather there is a pre-big-bang branch joined to the current
post-big-bang branch by a `quantum bridge'. Furthermore, thanks to the
background independence of loop quantum gravity, evolution is deterministic
across the bridge.Comment: Minor typos corrected, including a factor of in the expression
of the critical density. 16 pages, 2 figures. To appear in the Proceedings of
the `Einstein Century' Conference, 15-22 July, Paris, edited by J-M Alimi et
al (American Institute of Physics
Opening the AC-Unification Race
This note reports about the implementation of AC-unification algorithms, based on the variable-abstraction method of Stickel and on the constant-abstraction method of Livesey, Siekmann, and Herold. We give a set of 105 benchmark examples and compare execution times for implementations of the two approaches. This documents for other researchers what we consider to be the state-of-the-art performance for elementary AC-unification problems
Wormholes and Naked Singularities in Brans-Dicke cosmology
We perform analytical and numerical study of static spherically symmetric
solutions in the context of Brans-Dicke-like cosmological model by Elizalde et
al. with an exponential potential. In this model the phantom regime arises
without the appearance of any ghost degree of freedom due to the specific form
of coupling. For the certain parameter ranges the model contains a regular
solution which we interpret as a wormhole in an otherwise dS Universe. We put
several bounds on the parameter values: . The numerical
solution could mimic the Schwarzschild one, so the original model is consistent
with astrophysical and cosmological observational data. However differences
between our solution and the Schwarzschild one can be quite large, so black
hole candidate observations could probably place further limits on the
value.Comment: 20 pages, 6 figures, typos & errors correcte
Unital Anti-Unification: Type and Algorithms
Unital equational theories are defined by axioms that assert the existence of the unit element for some function symbols. We study anti-unification (AU) in unital theories and address the problems of establishing generalization type and designing anti-unification algorithms. First, we prove that when the term signature contains at least two unital functions, anti-unification is of the nullary type by showing that there exists an AU problem, which does not have a minimal complete set of generalizations. Next, we consider two special cases: the linear variant and the fragment with only one unital symbol, and design AU algorithms for them. The algorithms are terminating, sound, complete, and return tree grammars from which the set of generalizations can be constructed. Anti-unification for both special cases is finitary. Further, the algorithm for the one-unital fragment is extended to the unrestricted case. It terminates and returns a tree grammar which produces an infinite set of generalizations. At the end, we discuss how the nullary type of unital anti-unification might affect the anti-unification problem in some combined theories, and list some open questions
Unification in the union of disjoint equational theories : combining decision procedures
Most of the work on the combination of unification algorithms for the union of disjoint equational theories has been restricted to algorithms which compute finite complete sets of unifiers. Thus the developed combination methods usually cannot be used to combine decision procedures, i.e., algorithms which just decide solvability of unification problems without computing unifiers. In this paper we describe a combination algorithm for decision procedures which works for arbitrary equational theories, provided that solvability of so-called unification problems with constant restrictions--a slight generalization of unification problems with constants--is decidable for these theories. As a consequence of this new method, we can for example show that general A-unifiability, i.e., solvability of A-unification problems with free function symbols, is decidable. Here A stands for the equational theory of one associative function symbol. Our method can also be used to combine algorithms which compute finite complete sets of unifiers. Manfred Schmidt-Schauß\u27 combination result, the until now most general result in this direction, can be obtained as a consequence of this fact. We also get the new result that unification in the union of disjoint equational theories is finitary, if general unification--i.e., unification of terms with additional free function symbols--is finitary in the single theories
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