172,287 research outputs found
From nominal sets binding to functions and lambda-abstraction: connecting the logic of permutation models with the logic of functions
Permissive-Nominal Logic (PNL) extends first-order predicate logic with
term-formers that can bind names in their arguments. It takes a semantics in
(permissive-)nominal sets. In PNL, the forall-quantifier or lambda-binder are
just term-formers satisfying axioms, and their denotation is functions on
nominal atoms-abstraction.
Then we have higher-order logic (HOL) and its models in ordinary (i.e.
Zermelo-Fraenkel) sets; the denotation of forall or lambda is functions on full
or partial function spaces.
This raises the following question: how are these two models of binding
connected? What translation is possible between PNL and HOL, and between
nominal sets and functions?
We exhibit a translation of PNL into HOL, and from models of PNL to certain
models of HOL. It is natural, but also partial: we translate a restricted
subsystem of full PNL to HOL. The extra part which does not translate is the
symmetry properties of nominal sets with respect to permutations. To use a
little nominal jargon: we can translate names and binding, but not their
nominal equivariance properties. This seems reasonable since HOL---and ordinary
sets---are not equivariant.
Thus viewed through this translation, PNL and HOL and their models do
different things, but they enjoy non-trivial and rich subsystems which are
isomorphic
A Complete and Recursive Feature Theory
Various feature descriptions are being employed in logic programming
languages and constrained-based grammar formalisms. The common notational
primitive of these descriptions are functional attributes called features. The
descriptions considered in this paper are the possibly quantified first-order
formulae obtained from a signature of binary and unary predicates called
features and sorts, respectively. We establish a first-order theory FT by means
of three axiom schemes, show its completeness, and construct three elementarily
equivalent models. One of the models consists of so-called feature graphs, a
data structure common in computational linguistics. The other two models
consist of so-called feature trees, a record-like data structure generalizing
the trees corresponding to first-order terms. Our completeness proof exhibits a
terminating simplification system deciding validity and satisfiability of
possibly quantified feature descriptions.Comment: Short version appeared in the 1992 Annual Meeting of the Association
for Computational Linguistic
Open spin chains with generic integrable boundaries: Baxter equation and Bethe ansatz completeness from SOV
We solve the longstanding problem to define a functional characterization of
the spectrum of the transfer matrix associated to the most general spin-1/2
representations of the 6-vertex reflection algebra for general inhomogeneous
chains. The corresponding homogeneous limit reproduces the spectrum of the
Hamiltonian of the spin-1/2 open XXZ and XXX quantum chains with the most
general integrable boundaries. The spectrum is characterized by a second order
finite difference functional equation of Baxter type with an inhomogeneous term
which vanishes only for some special but yet interesting non-diagonal boundary
conditions. This functional equation is shown to be equivalent to the known
separation of variable (SOV) representation hence proving that it defines a
complete characterization of the transfer matrix spectrum. The polynomial
character of the Q-function allows us then to show that a finite system of
equations of generalized Bethe type can be similarly used to describe the
complete transfer matrix spectrum.Comment: 28 page
Some Relations for Quark Confinement and Chiral Symmetry Breaking in QCD
We analytically study the relation between quark confinement and spontaneous
chiral-symmetry breaking in QCD. In terms of the Dirac eigenmodes, we derive
some formulae for the Polyakov loop, its fluctuations, and the string tension
from the Wilson loop. We also investigate the Polyakov loop in terms of the
eigenmodes of the Wilson, the clover and the domain wall fermion kernels,
respectively. For the confinement quantities, the low-lying Dirac/fermion
eigenmodes are found to give negligible contribution, while they are essential
for chiral symmetry breaking. These relations indicate no direct one-to-one
correspondence between confinement and chiral symmetry breaking in QCD, which
seems to be natural because confinement is realized independently of the quark
mass
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