56,073 research outputs found
Randomness and semigenericity
Let L contain only the equality symbol and let L^+ be an arbitrary finite
symmetric relational language containing L . Suppose probabilities are defined
on finite L^+ structures with ''edge probability'' n^{- alpha}. By T^alpha, the
almost sure theory of random L^+-structures we mean the collection of
L^+-sentences which have limit probability 1. T_alpha denotes the theory of the
generic structures for K_alpha, (the collection of finite graphs G with
delta_{alpha}(G)=|G|- alpha. | edges of G | hereditarily nonnegative.)
THEOREM: T_alpha, the almost sure theory of random L^+-structures is the same
as the theory T_alpha of the K_alpha-generic model. This theory is complete,
stable, and nearly model complete. Moreover, it has the finite model property
and has only infinite models so is not finitely axiomatizable
Toric grammars: a new statistical approach to natural language modeling
We propose a new statistical model for computational linguistics. Rather than
trying to estimate directly the probability distribution of a random sentence
of the language, we define a Markov chain on finite sets of sentences with many
finite recurrent communicating classes and define our language model as the
invariant probability measures of the chain on each recurrent communicating
class. This Markov chain, that we call a communication model, recombines at
each step randomly the set of sentences forming its current state, using some
grammar rules. When the grammar rules are fixed and known in advance instead of
being estimated on the fly, we can prove supplementary mathematical properties.
In particular, we can prove in this case that all states are recurrent states,
so that the chain defines a partition of its state space into finite recurrent
communicating classes. We show that our approach is a decisive departure from
Markov models at the sentence level and discuss its relationships with Context
Free Grammars. Although the toric grammars we use are closely related to
Context Free Grammars, the way we generate the language from the grammar is
qualitatively different. Our communication model has two purposes. On the one
hand, it is used to define indirectly the probability distribution of a random
sentence of the language. On the other hand it can serve as a (crude) model of
language transmission from one speaker to another speaker through the
communication of a (large) set of sentences
A Generalised Quantifier Theory of Natural Language in Categorical Compositional Distributional Semantics with Bialgebras
Categorical compositional distributional semantics is a model of natural
language; it combines the statistical vector space models of words with the
compositional models of grammar. We formalise in this model the generalised
quantifier theory of natural language, due to Barwise and Cooper. The
underlying setting is a compact closed category with bialgebras. We start from
a generative grammar formalisation and develop an abstract categorical
compositional semantics for it, then instantiate the abstract setting to sets
and relations and to finite dimensional vector spaces and linear maps. We prove
the equivalence of the relational instantiation to the truth theoretic
semantics of generalised quantifiers. The vector space instantiation formalises
the statistical usages of words and enables us to, for the first time, reason
about quantified phrases and sentences compositionally in distributional
semantics
Gaussian Wavefunctional Approach in Thermofield Dynamics
The Gaussian wavefunctional approach is developed in thermofield dynamics. We
manufacture thermal vacuum wavefunctional, its creation as well as annihilation
operators,and accordingly thermo-particle excited states. For a
(D+1)-dimensional scalar field system with an arbitrary potential whose Fourier
representation exists in a sense of tempered distributions, we calculate the
finite temperature Gaussian effective potential (FTGEP), one- and
two-thermo-particle-state energies. The zero-temperature limit of each of them
is just the corresponding result in quantum field theory, and the FTGEP can
lead to the same one of each of some concrete models as calculated by the
imaginary time Green function.Comment: the revised version of hep-th/9807025, with one equation being added,
a few sentences rewritten, and some spelling mistakes corrected. 7 page,
Revtex, no figur
Computabilities of Validity and Satisfiability in Probability Logics over Finite and Countable Models
The -logic (which is called E-logic in this paper) of
Kuyper and Terwijn is a variant of first order logic with the same syntax, in
which the models are equipped with probability measures and in which the
quantifier is interpreted as "there exists a set of measure
such that for each , ...." Previously, Kuyper and
Terwijn proved that the general satisfiability and validity problems for this
logic are, i) for rational , respectively
-complete and -hard, and ii) for ,
respectively decidable and -complete. The adjective "general" here
means "uniformly over all languages."
We extend these results in the scenario of finite models. In particular, we
show that the problems of satisfiability by and validity over finite models in
E-logic are, i) for rational , respectively
- and -complete, and ii) for , respectively
decidable and -complete. Although partial results toward the countable
case are also achieved, the computability of E-logic over countable
models still remains largely unsolved. In addition, most of the results, of
this paper and of Kuyper and Terwijn, do not apply to individual languages with
a finite number of unary predicates. Reducing this requirement continues to be
a major point of research.
On the positive side, we derive the decidability of the corresponding
problems for monadic relational languages --- equality- and function-free
languages with finitely many unary and zero other predicates. This result holds
for all three of the unrestricted, the countable, and the finite model cases.
Applications in computational learning theory, weighted graphs, and neural
networks are discussed in the context of these decidability and undecidability
results.Comment: 47 pages, 4 tables. Comments welcome. Fixed errors found by Rutger
Kuype
Logical Omnipotence and Two notions of Implicit Belief
The most widespread models of rational reasoners (the model based on modal epistemic logic and the model based on probability theory) exhibit the problem of logical omniscience. The most common strategy for avoiding this problem is to interpret the models as describing the explicit beliefs of an ideal reasoner, but only the implicit beliefs of a real reasoner. I argue that this strategy faces serious normative issues. In this paper, I present the more fundamental problem of logical omnipotence, which highlights the normative content of the problem of logical omniscience. I introduce two developments of the notion of implicit belief (accessible and stable belief ) and use them in two versions of the most common strategy applied to the problem of logical omnipotence
On Decidability Properties of Local Sentences
Local (first order) sentences, introduced by Ressayre, enjoy very nice
decidability properties, following from some stretching theorems stating some
remarkable links between the finite and the infinite model theory of these
sentences. We prove here several additional results on local sentences. The
first one is a new decidability result in the case of local sentences whose
function symbols are at most unary: one can decide, for every regular cardinal
k whether a local sentence phi has a model of order type k. Secondly we show
that this result can not be extended to the general case. Assuming the
consistency of an inaccessible cardinal we prove that the set of local
sentences having a model of order type omega_2 is not determined by the
axiomatic system ZFC + GCH, where GCH is the generalized continuum hypothesi
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