1,279 research outputs found
An algorithmic approach to the existence of ideal objects in commutative algebra
The existence of ideal objects, such as maximal ideals in nonzero rings,
plays a crucial role in commutative algebra. These are typically justified
using Zorn's lemma, and thus pose a challenge from a computational point of
view. Giving a constructive meaning to ideal objects is a problem which dates
back to Hilbert's program, and today is still a central theme in the area of
dynamical algebra, which focuses on the elimination of ideal objects via
syntactic methods. In this paper, we take an alternative approach based on
Kreisel's no counterexample interpretation and sequential algorithms. We first
give a computational interpretation to an abstract maximality principle in the
countable setting via an intuitive, state based algorithm. We then carry out a
concrete case study, in which we give an algorithmic account of the result that
in any commutative ring, the intersection of all prime ideals is contained in
its nilradical
Helmut Brandt and Manfred Beyer, eds.: Ansichten der deutschen Klassik
Berlin and Weimar: Aufbau, 1981. 456 p., 12,- M
Existential witness extraction in classical realizability and via a negative translation
We show how to extract existential witnesses from classical proofs using
Krivine's classical realizability---where classical proofs are interpreted as
lambda-terms with the call/cc control operator. We first recall the basic
framework of classical realizability (in classical second-order arithmetic) and
show how to extend it with primitive numerals for faster computations. Then we
show how to perform witness extraction in this framework, by discussing several
techniques depending on the shape of the existential formula. In particular, we
show that in the Sigma01-case, Krivine's witness extraction method reduces to
Friedman's through a well-suited negative translation to intuitionistic
second-order arithmetic. Finally we discuss the advantages of using call/cc
rather than a negative translation, especially from the point of view of an
implementation.Comment: 52 pages. Accepted in Logical Methods for Computer Science (LMCS),
201
Magnetic ground state and magnon-phonon interaction in multiferroic h-YMnO
Inelastic neutron scattering has been used to study the magneto-elastic
excitations in the multiferroic manganite hexagonal YMnO. An avoided
crossing is found between magnon and phonon modes close to the Brillouin zone
boundary in the -plane. Neutron polarization analysis reveals that this
mode has mixed magnon-phonon character. An external magnetic field along the
-axis is observed to cause a linear field-induced splitting of one of the
spin wave branches. A theoretical description is performed, using a Heisenberg
model of localized spins, acoustic phonon modes and a magneto-elastic coupling
via the single-ion magnetostriction. The model quantitatively reproduces the
dispersion and intensities of all modes in the full Brillouin zone, describes
the observed magnon-phonon hybridized modes, and quantifies the magneto-elastic
coupling. The combined information, including the field-induced magnon
splitting, allows us to exclude several of the earlier proposed models and
point to the correct magnetic ground state symmetry, and provides an effective
dynamic model relevant for the multiferroic hexagonal manganites.Comment: 12 pages, 10 figure
On a class of invariant coframe operators with application to gravity
Let a differential 4D-manifold with a smooth coframe field be given. Consider
the operators on it that are linear in the second order derivatives or
quadratic in the first order derivatives of the coframe, both with coefficients
that depend on the coframe variables. The paper exhibits the class of operators
that are invariant under a general change of coordinates, and, also, invariant
under the global SO(1,3)-transformation of the coframe. A general class of
field equations is constructed. We display two subclasses in it. The subclass
of field equations that are derivable from action principles by free variations
and the subclass of field equations for which spherical-symmetric solutions,
Minkowskian at infinity exist. Then, for the spherical-symmetric solutions, the
resulting metric is computed. Invoking the Geodesic Postulate, we find all the
equations that are experimentally (by the 3 classical tests) indistinguishable
from Einstein field equations. This family includes, of course, also Einstein
equations. Moreover, it is shown, explicitly, how to exhibit it. The basic tool
employed in the paper is an invariant formulation reminiscent of Cartan's
structural equations. The article sheds light on the possibilities and
limitations of the coframe gravity. It may also serve as a general procedure to
derive covariant field equations
Rejection in Łukasiewicz's and Słupecki's Sense
The idea of rejection originated by Aristotle. The notion of rejection
was introduced into formal logic by Łukasiewicz [20]. He applied it to
complete syntactic characterization of deductive systems using an axiomatic
method of rejection of propositions [22, 23]. The paper gives not only genesis,
but also development and generalization of the notion of rejection. It also
emphasizes the methodological approach to biaspectual axiomatic method of
characterization of deductive systems as acceptance (asserted) systems and
rejection (refutation) systems, introduced by Łukasiewicz and developed by
his student Słupecki, the pioneers of the method, which becomes relevant in
modern approaches to logic
Forcing-based cut-elimination for Gentzen-style intuitionistic sequent calculus
International audienceWe give a simple intuitionistic completeness proof of Kripke semantics for intuitionistic logic with implication and universal quantification with respect to cut-free intuitionistic sequent calculus. The Kripke semantics is ``simplified'' in the way that the domain remains constant. The proof has been formalised in the Coq proof assistant and by combining soundness with completeness, we obtain an executable cut-elimination procedure. The proof easily extends to the case of the absurdity connective using Kripke models with exploding nodes à la Veldman
On the computational content of Zorn's lemma
We give a computational interpretation to an abstract instance of Zorn's
lemma formulated as a wellfoundedness principle in the language of arithmetic
in all finite types. This is achieved through G\"odel's functional
interpretation, and requires the introduction of a novel form of recursion over
non-wellfounded partial orders whose existence in the model of total continuous
functionals is proven using domain theoretic techniques. We show that a
realizer for the functional interpretation of open induction over the
lexicographic ordering on sequences follows as a simple application of our main
results
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