252 research outputs found
Fuzzy Bigraphs: An Exercise in Fuzzy Communicating Agents
Bigraphs and their algebra is a model of concurrency. Fuzzy bigraphs are a
generalization of birgraphs intended to be a model of concurrency that
incorporates vagueness. More specifically, this model assumes that agents are
similar, communication is not perfect, and, in general, everything is or
happens to some degree.Comment: 11 pages, 3 figure
Causal Fermion Systems: A Quantum Space-Time Emerging from an Action Principle
Causal fermion systems are introduced as a general mathematical framework for
formulating relativistic quantum theory. By specializing, we recover earlier
notions like fermion systems in discrete space-time, the fermionic projector
and causal variational principles. We review how an effect of spontaneous
structure formation gives rise to a topology and a causal structure in
space-time. Moreover, we outline how to construct a spin connection and
curvature, leading to a proposal for a "quantum geometry" in the Lorentzian
setting. We review recent numerical and analytical results on the support of
minimizers of causal variational principles which reveal a "quantization
effect" resulting in a discreteness of space-time. A brief survey is given on
the correspondence to quantum field theory and gauge theories.Comment: 23 pages, LaTeX, 2 figures, footnote added on page
Generalized Satisfiability Problems via Operator Assignments
Schaefer introduced a framework for generalized satisfiability problems on
the Boolean domain and characterized the computational complexity of such
problems. We investigate an algebraization of Schaefer's framework in which the
Fourier transform is used to represent constraints by multilinear polynomials
in a unique way. The polynomial representation of constraints gives rise to a
relaxation of the notion of satisfiability in which the values to variables are
linear operators on some Hilbert space. For the case of constraints given by a
system of linear equations over the two-element field, this relaxation has
received considerable attention in the foundations of quantum mechanics, where
such constructions as the Mermin-Peres magic square show that there are systems
that have no solutions in the Boolean domain, but have solutions via operator
assignments on some finite-dimensional Hilbert space. We obtain a complete
characterization of the classes of Boolean relations for which there is a gap
between satisfiability in the Boolean domain and the relaxation of
satisfiability via operator assignments. To establish our main result, we adapt
the notion of primitive-positive definability (pp-definability) to our setting,
a notion that has been used extensively in the study of constraint satisfaction
problems. Here, we show that pp-definability gives rise to gadget reductions
that preserve satisfiability gaps. We also present several additional
applications of this method. In particular and perhaps surprisingly, we show
that the relaxed notion of pp-definability in which the quantified variables
are allowed to range over operator assignments gives no additional expressive
power in defining Boolean relations
The Category of Node-and-Choice Forms, with Subcategories for Choice-Sequence Forms and Choice-Set Forms
The literature specifies extensive-form games in many styles, and eventually
I hope to formally translate games across those styles. Toward that end, this
paper defines , the category of node-and-choice forms. The
category's objects are extensive forms in essentially any style, and the
category's isomorphisms are made to accord with the literature's small handful
of ad hoc style equivalences.
Further, this paper develops two full subcategories: for
forms whose nodes are choice-sequences, and for forms whose
nodes are choice-sets. I show that is "isomorphically enclosed"
in in the sense that each form is isomorphic to
a form. Similarly, I show that is
isomorphically enclosed in in the sense that each
form with no-absentmindedness is isomorphic to a
form. The converses are found to be almost immediate, and the
resulting equivalences unify and simplify two ad hoc style equivalences in
Kline and Luckraz 2016 and Streufert 2019.
Aside from the larger agenda, this paper already makes three practical
contributions. Style equivalences are made easier to derive by [1] a natural
concept of isomorphic invariance and [2] the composability of isomorphic
enclosures. In addition, [3] some new consequences of equivalence are
systematically deduced.Comment: 43 pages, 9 figure
On Unbounded Composition Operators in -Spaces
Fundamental properties of unbounded composition operators in -spaces are
studied. Characterizations of normal and quasinormal composition operators are
provided. Formally normal composition operators are shown to be normal.
Composition operators generating Stieltjes moment sequences are completely
characterized. The unbounded counterparts of the celebrated Lambert's
characterizations of subnormality of bounded composition operators are shown to
be false. Various illustrative examples are supplied
Parameter-Independent Strategies for pMDPs via POMDPs
Markov Decision Processes (MDPs) are a popular class of models suitable for
solving control decision problems in probabilistic reactive systems. We
consider parametric MDPs (pMDPs) that include parameters in some of the
transition probabilities to account for stochastic uncertainties of the
environment such as noise or input disturbances.
We study pMDPs with reachability objectives where the parameter values are
unknown and impossible to measure directly during execution, but there is a
probability distribution known over the parameter values. We study for the
first time computing parameter-independent strategies that are expectation
optimal, i.e., optimize the expected reachability probability under the
probability distribution over the parameters. We present an encoding of our
problem to partially observable MDPs (POMDPs), i.e., a reduction of our problem
to computing optimal strategies in POMDPs.
We evaluate our method experimentally on several benchmarks: a motivating
(repeated) learner model; a series of benchmarks of varying configurations of a
robot moving on a grid; and a consensus protocol.Comment: Extended version of a QEST 2018 pape
Functor of continuation in Hilbert cube and Hilbert space
A -set in a metric space is a closed subset of such that each
map of the Hilbert cube into can uniformly be approximated by maps of
into . The aim of the paper is to show that there exists a
functor of extension of maps between -sets of [or ] to maps acting
on the whole space [resp. ]. Special properties of the functor are
proved.Comment: 9 page
Measuring processes and the Heisenberg picture
In this paper, we attempt to establish quantum measurement theory in the
Heisenberg picture. First, we review foundations of quantum measurement theory,
that is usually based on the Schr\"{o}dinger picture. The concept of instrument
is introduced there. Next, we define the concept of system of measurement
correlations and that of measuring process. The former is the exact counterpart
of instrument in the (generalized) Heisenberg picture. In quantum mechanical
systems, we then show a one-to-one correspondence between systems of
measurement correlations and measuring processes up to complete equivalence.
This is nothing but a unitary dilation theorem of systems of measurement
correlations. Furthermore, from the viewpoint of the statistical approach to
quantum measurement theory, we focus on the extendability of instruments to
systems of measurement correlations. It is shown that all completely positive
(CP) instruments are extended into systems of measurement correlations. Lastly,
we study the approximate realizability of CP instruments by measuring processes
within arbitrarily given error limits.Comment: v
de Branges-Rovnyak spaces: basics and theory
For a contractive analytic operator-valued function on the unit disk
, de Branges and Rovnyak associate a Hilbert space of analytic
functions and related extension space
consisting of pairs of analytic functions on the unit disk . This
survey describes three equivalent formulations (the original geometric de
Branges-Rovnyak definition, the Toeplitz operator characterization, and the
characterization as a reproducing kernel Hilbert space) of the de
Branges-Rovnyak space , as well as its role as the underlying
Hilbert space for the modeling of completely non-isometric Hilbert-space
contraction operators. Also examined is the extension of these ideas to handle
the modeling of the more general class of completely nonunitary contraction
operators, where the more general two-component de Branges-Rovnyak model space
and associated overlapping spaces play key roles. Connections
with other function theory problems and applications are also discussed. More
recent applications to a variety of subsequent applications are given in a
companion survey article
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