793,729 research outputs found
Natural extension of choice functions
International Conference on Information Processing and Management of Uncertainty in Knowledge-Based Systems, IPMU (17 th, 2018, Cádiz, Spain
Irrelevant natural extension for choice functions
We consider coherent choice functions under the recent axiomatisation proposed by De Bock and De Cooman that guarantees a representation in terms of binary preferences, and we discuss how to define conditioning in this framework. In a multivariate context, we propose a notion of marginalisation, and its inverse operation called weak (cylindrical) extension. We combine this with our definition of conditioning to define a notion of irrelevance, and we obtain the irrelevant natural extension in this framework: the least informative choice function that satisfies a given irrelevance assessment
Restrictions and extensions of semibounded operators
We study restriction and extension theory for semibounded Hermitian operators
in the Hardy space of analytic functions on the disk D. Starting with the
operator zd/dz, we show that, for every choice of a closed subset F in T=bd(D)
of measure zero, there is a densely defined Hermitian restriction of zd/dz
corresponding to boundary functions vanishing on F. For every such restriction
operator, we classify all its selfadjoint extension, and for each we present a
complete spectral picture.
We prove that different sets F with the same cardinality can lead to quite
different boundary-value problems, inequivalent selfadjoint extension
operators, and quite different spectral configurations. As a tool in our
analysis, we prove that the von Neumann deficiency spaces, for a fixed set F,
have a natural presentation as reproducing kernel Hilbert spaces, with a
Hurwitz zeta-function, restricted to FxF, as reproducing kernel.Comment: 63 pages, 11 figure
Independent Natural Extension for Choice Functions
We investigate epistemic independence for choice functions in a multivariate setting. This work is a continuation of earlier work of one of the authors [23], and our results build on the characterization of choice functions in terms of sets of binary preferences recently established by De Bock and De Cooman [7]. We obtain the independent natural extension in this framework. Given the generality of choice functions, our expression for the independent natural extension is the most general one we are aware of, and we show how it implies the independent natural extension for sets of desirable gambles, and therefore also for less informative imprecise-probabilistic models. Once this is in place, we compare this concept of epistemic independence to another independence concept for choice functions proposed by Seidenfeld [22], which De Bock and De Cooman [1] have called S-independence. We show that neither is more general than the other
Independent natural extension for choice functions
We introduce an independence notion for choice functions, which we call ‘epistemic independence’ following the work by De Cooman et al. [17] for lower previsions, and study it in a multivariate setting. This work is a continuation of earlier work of one of the authors [29], and our results build on the characterization of choice functions in terms of sets of binary preferences recently established by De Bock and De Cooman [11]. We obtain the many-to-one independent natural extension in this framework. Given the generality of choice functions, our expression for the independent natural extension is the most general one we are aware of, and we show how it implies the independent natural extension for sets of desirable gambles, and therefore also for less expressive imprecise-probabilistic models. Once this is in place, we compare this concept of epistemic independence to another independence concept for choice functions proposed by Seidenfeld [28], which De Bock and De Cooman [2] have called S-independence. We show that neither is more general than the other
Repeated Nash implementation
We study the repeated implementation of social choice functions in environments with complete information and changing preferences. We define dynamic monotonicity, a natural but non-trivial dynamic extension of Maskin monotonicity, and show that it is necessary and almost sufficient for repeated Nash implementation, regardless of whether the horizon is finite or
infinite and whether the discount factor is "large" or "small.
Fair Allocation based on Diminishing Differences
Ranking alternatives is a natural way for humans to explain their
preferences. It is being used in many settings, such as school choice, course
allocations and residency matches. In some cases, several `items' are given to
each participant. Without having any information on the underlying cardinal
utilities, arguing about fairness of allocation requires extending the ordinal
item ranking to ordinal bundle ranking. The most commonly used such extension
is stochastic dominance (SD), where a bundle X is preferred over a bundle Y if
its score is better according to all additive score functions. SD is a very
conservative extension, by which few allocations are necessarily fair while
many allocations are possibly fair. We propose to make a natural assumption on
the underlying cardinal utilities of the players, namely that the difference
between two items at the top is larger than the difference between two items at
the bottom. This assumption implies a preference extension which we call
diminishing differences (DD), where X is preferred over Y if its score is
better according to all additive score functions satisfying the DD assumption.
We give a full characterization of allocations that are
necessarily-proportional or possibly-proportional according to this assumption.
Based on this characterization, we present a polynomial-time algorithm for
finding a necessarily-DD-proportional allocation if it exists. Using
simulations, we show that with high probability, a necessarily-proportional
allocation does not exist but a necessarily-DD-proportional allocation exists,
and moreover, that allocation is proportional according to the underlying
cardinal utilities. We also consider chore allocation under the analogous
condition --- increasing-differences.Comment: Revised version, based on very helpful suggestions of JAIR referees.
Gaps in some proofs were filled, more experiments were done, and mor
The hidden geometric character of relativistic quantum mechanics
The presentation makes use of geometric algebra, also known as Clifford
algebra, in 5-dimensional spacetime. The choice of this space is given the
character of first principle, justified solely by the consequences that can be
derived from such choice and their consistency with experimental results. Given
a metric space of any dimension, one can define monogenic functions, the
natural extension of analytic functions to higher dimensions; such functions
have null vector derivative and have previously been shown by other authors to
play a decisive role in lower dimensional spaces. All monogenic functions have
null Laplacian by consequence; in an hyperbolic space this fact leads
inevitably to a wave equation with plane-like solutions. This is also true for
5-dimensional spacetime and we will explore those solutions, establishing a
parallel with the solutions of the Dirac equation. For this purpose we will
invoke the isomorphism between the complex algebra of 4x4 matrices, also known
as Dirac's matrices. There is one problem with this isomorphism, because the
solutions to Dirac's equation are usually known as spinors (column matrices)
that don't belong to the 4x4 matrix algebra and as such are excluded from the
isomorphism. We will show that a solution in terms of Dirac spinors is
equivalent to a plane wave solution. Just as one finds in the standard
formulation, monogenic functions can be naturally split into positive/negative
energy together with left/right ones. This split is provided by geometric
projectors and we will show that there is a second set of projectors providing
an alternate 4-fold split. The possible implications of this alternate split
are not yet fully understood and are presently the subject of profound
research.Comment: 29 pages. Small changes in V3 suggested by refere
Semantics for first-order superposition logic
We investigate how the sentence choice semantics (SCS) for propositional
superposition logic (PLS) developed in \cite{Tz17} could be extended so as to
successfully apply to first-order superposition logic(FOLS). There are two
options for such an extension. The apparently more natural one is the formula
choice semantics (FCS) based on choice functions for pairs of arbitrary
formulas of the basis language. It is proved however that the universal
instantiation scheme of FOL, , is
false, as a scheme of tautologies, with respect to FCS. This causes the total
failure of FCS as a candidate semantics. Then we turn to the other option which
is a variant of SCS, since it uses again choice functions for pairs of
sentences only. This semantics however presupposes that the applicability of
the connective is restricted to quantifier-free sentences, and thus the
class of well-formed formulas and sentences of the language is restricted too.
Granted these syntactic restrictions, the usual axiomatizations of FOLS turn
out to be sound and conditionally complete with respect to this second
semantics, just like the corresponding systems of PLS.Comment: 35 page
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