8 research outputs found
Admissibility in Finitely Generated Quasivarieties
Checking the admissibility of quasiequations in a finitely generated (i.e.,
generated by a finite set of finite algebras) quasivariety Q amounts to
checking validity in a suitable finite free algebra of the quasivariety, and is
therefore decidable. However, since free algebras may be large even for small
sets of small algebras and very few generators, this naive method for checking
admissibility in \Q is not computationally feasible. In this paper,
algorithms are introduced that generate a minimal (with respect to a multiset
well-ordering on their cardinalities) finite set of algebras such that the
validity of a quasiequation in this set corresponds to admissibility of the
quasiequation in Q. In particular, structural completeness (validity and
admissibility coincide) and almost structural completeness (validity and
admissibility coincide for quasiequations with unifiable premises) can be
checked. The algorithms are illustrated with a selection of well-known finitely
generated quasivarieties, and adapted to handle also admissibility of rules in
finite-valued logics
Checking Admissibility Using Natural Dualities
This paper presents a new method for obtaining small algebras to check the
admissibility-equivalently, validity in free algebras-of quasi-identities in a
finitely generated quasivariety. Unlike a previous algebraic approach of
Metcalfe and Rothlisberger that is feasible only when the relevant free algebra
is not too large, this method exploits natural dualities for quasivarieties to
work with structures of smaller cardinality and surjective rather than
injective morphisms. A number of case studies are described here that could not
be be solved using the algebraic approach, including (quasi)varieties of
MS-algebras, double Stone algebras, and involutive Stone algebras
Franks' dichotomy for toric manifolds, Hofer-Zehnder conjecture, and gauged linear sigma model
We prove that for any compact toric symplectic manifold, if a Hamiltonian
diffeomorphism admits more fixed points, counted homologically, than the total
Betti number, then it has infinitely many simple periodic points. This provides
a vast generalization of Franks' famous two or infinity dichotomy for periodic
orbits of area-preserving diffeomorphisms on the two-sphere, and establishes a
conjecture attributed to Hofer-Zehnder in the case of toric manifolds. The key
novelty is the application of gauged linear sigma model and its bulk
deformations to the study of Hamiltonian dynamics of symplectic quotients.Comment: v2: 94 pages, 6 figures. New title, with expository changes in the
introduction and main part. Comments are welcome
On the Existence of Characterization Logics and Fundamental Properties of Argumentation Semantics
Given the large variety of existing logical formalisms it is of utmost importance
to select the most adequate one for a specific purpose, e.g. for representing
the knowledge relevant for a particular application or for using the formalism
as a modeling tool for problem solving. Awareness of the nature of a logical
formalism, in other words, of its fundamental intrinsic properties, is indispensable
and provides the basis of an informed choice.
One such intrinsic property of logic-based knowledge representation languages
is the context-dependency of pieces of knowledge. In classical propositional
logic, for example, there is no such context-dependence: whenever two
sets of formulas are equivalent in the sense of having the same models (ordinary
equivalence), then they are mutually replaceable in arbitrary contexts (strong
equivalence). However, a large number of commonly used formalisms are not
like classical logic which leads to a series of interesting developments. It turned
out that sometimes, to characterize strong equivalence in formalism L, we can
use ordinary equivalence in formalism L0: for example, strong equivalence in
normal logic programs under stable models can be characterized by the standard
semantics of the logic of here-and-there. Such results about the existence of
characterizing logics has rightly been recognized as important for the study of
concrete knowledge representation formalisms and raise a fundamental question:
Does every formalism have one? In this thesis, we answer this question
with a qualified “yes”. More precisely, we show that the important case of
considering only finite knowledge bases guarantees the existence of a canonical
characterizing formalism. Furthermore, we argue that those characterizing
formalisms can be seen as classical, monotonic logics which are uniquely determined (up to isomorphism) regarding their model theory.
The other main part of this thesis is devoted to argumentation semantics
which play the flagship role in Dung’s abstract argumentation theory. Almost
all of them are motivated by an easily understandable intuition of what should
be acceptable in the light of conflicts. However, although these intuitions equip
us with short and comprehensible formal definitions it turned out that their
intrinsic properties such as existence and uniqueness, expressibility, replaceability
and verifiability are not that easily accessible. We review the mentioned
properties for almost all semantics available in the literature. In doing so we
include two main axes: namely first, the distinction between extension-based
and labelling-based versions and secondly, the distinction of different kind of
argumentation frameworks such as finite or unrestricted ones