1,906 research outputs found
The Unification Hierarchy is Undecidable
In unification theory, equational theories can be classified according to the existence and cardinality of minimal complete solution sets for equation systems. For unitary, finitary, and infinitary theories minimal complete solution sets always exist and are singletons, finite, or possibly infinite sets, respectively. In nullary theories, minimal complete sets do not exist for some equation systems. These classes form the unification hierarchy.
We show that it is not possible to decide where a given equational theory resides in the unification hierarchy. Moreover, it is proved that for some classes this problem is not even recursively enumerable
Nearly degenerate neutrinos, Supersymmetry and radiative corrections
If neutrinos are to play a relevant cosmological role, they must be
essentially degenerate with a mass matrix of the bimaximal mixing type. We
study this scenario in the MSSM framework, finding that if neutrino masses are
produced by a see-saw mechanism, the radiative corrections give rise to mass
splittings and mixing angles that can accommodate the atmospheric and the
(large angle MSW) solar neutrino oscillations. This provides a natural origin
for the hierarchy. On the other hand,
the vacuum oscillation solution to the solar neutrino problem is always
excluded. We discuss also in the SUSY scenario other possible effects of
radiative corrections involving the new neutrino Yukawa couplings, including
implications for triviality limits on the Majorana mass, the infrared fixed
point value of the top Yukawa coupling, and gauge coupling and bottom-tau
unification.Comment: 32 pages, 12 Postscript figures, uses psfig.st
Type classes for efficient exact real arithmetic in Coq
Floating point operations are fast, but require continuous effort on the part
of the user in order to ensure that the results are correct. This burden can be
shifted away from the user by providing a library of exact analysis in which
the computer handles the error estimates. Previously, we [Krebbers/Spitters
2011] provided a fast implementation of the exact real numbers in the Coq proof
assistant. Our implementation improved on an earlier implementation by O'Connor
by using type classes to describe an abstract specification of the underlying
dense set from which the real numbers are built. In particular, we used dyadic
rationals built from Coq's machine integers to obtain a 100 times speed up of
the basic operations already. This article is a substantially expanded version
of [Krebbers/Spitters 2011] in which the implementation is extended in the
various ways. First, we implement and verify the sine and cosine function.
Secondly, we create an additional implementation of the dense set based on
Coq's fast rational numbers. Thirdly, we extend the hierarchy to capture order
on undecidable structures, while it was limited to decidable structures before.
This hierarchy, based on type classes, allows us to share theory on the
naturals, integers, rationals, dyadics, and reals in a convenient way. Finally,
we obtain another dramatic speed-up by avoiding evaluation of termination
proofs at runtime.Comment: arXiv admin note: text overlap with arXiv:1105.275
Language, logic and ontology: uncovering the structure of commonsense knowledge
The purpose of this paper is twofold: (i) we argue that the structure of commonsense knowledge must be discovered, rather than invented; and (ii) we argue that natural
language, which is the best known theory of our (shared) commonsense knowledge, should itself be used as a guide to discovering the structure of commonsense knowledge. In addition to suggesting a systematic method to the discovery of the structure of commonsense knowledge, the method we propose seems to also provide an explanation for a number of phenomena in natural language, such as metaphor, intensionality, and the semantics of nominal compounds. Admittedly, our ultimate goal is quite ambitious, and it is no less than the systematic ‘discovery’ of a well-typed
ontology of commonsense knowledge, and the subsequent formulation of the longawaited goal of a meaning algebra
A Bi-Directional Refinement Algorithm for the Calculus of (Co)Inductive Constructions
The paper describes the refinement algorithm for the Calculus of
(Co)Inductive Constructions (CIC) implemented in the interactive theorem prover
Matita. The refinement algorithm is in charge of giving a meaning to the terms,
types and proof terms directly written by the user or generated by using
tactics, decision procedures or general automation. The terms are written in an
"external syntax" meant to be user friendly that allows omission of
information, untyped binders and a certain liberal use of user defined
sub-typing. The refiner modifies the terms to obtain related well typed terms
in the internal syntax understood by the kernel of the ITP. In particular, it
acts as a type inference algorithm when all the binders are untyped. The
proposed algorithm is bi-directional: given a term in external syntax and a
type expected for the term, it propagates as much typing information as
possible towards the leaves of the term. Traditional mono-directional
algorithms, instead, proceed in a bottom-up way by inferring the type of a
sub-term and comparing (unifying) it with the type expected by its context only
at the end. We propose some novel bi-directional rules for CIC that are
particularly effective. Among the benefits of bi-directionality we have better
error message reporting and better inference of dependent types. Moreover,
thanks to bi-directionality, the coercion system for sub-typing is more
effective and type inference generates simpler unification problems that are
more likely to be solved by the inherently incomplete higher order unification
algorithms implemented. Finally we introduce in the external syntax the notion
of vector of placeholders that enables to omit at once an arbitrary number of
arguments. Vectors of placeholders allow a trivial implementation of implicit
arguments and greatly simplify the implementation of primitive and simple
tactics
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