11,922 research outputs found
A survey of clones on infinite sets
A clone on a set X is a set of finitary operations on X which contains all
projections and which is moreover closed under functional composition. Ordering
all clones on X by inclusion, one obtains a complete algebraic lattice, called
the clone lattice. We summarize what we know about the clone lattice on an
infinite base set X and formulate what we consider the most important open
problems.Comment: 37 page
A Generalization of the AL method for Fair Allocation of Indivisible Objects
We consider the assignment problem in which agents express ordinal
preferences over objects and the objects are allocated to the agents based
on the preferences. In a recent paper, Brams, Kilgour, and Klamler (2014)
presented the AL method to compute an envy-free assignment for two agents. The
AL method crucially depends on the assumption that agents have strict
preferences over objects. We generalize the AL method to the case where agents
may express indifferences and prove the axiomatic properties satisfied by the
algorithm. As a result of the generalization, we also get a speedup on
previous algorithms to check whether a complete envy-free assignment exists or
not. Finally, we show that unless P=NP, there can be no polynomial-time
extension of GAL to the case of arbitrary number of agents
Asymmetric quantum telecloning of d-level systems and broadcasting of entanglement to different locations using the "many-to-many" communication protocol
We propose a generalization of quantum teleportation: the so-called
many-to-many quantum communication of the information of a d-level system from
N spatially separated senders to M>N receivers situated at different locations.
We extend the concept of asymmetric telecloning from qubits to d-dimensional
systems. We investigate the broadcasting of entanglement by using local 1->2
optimal universal asymmetric Pauli machines and show that the maximal
fidelities of the two final entangled states are obtained when symmetric
machines are applied. Cloning of entanglement is studied using a nonlocal
optimal universal asymmetric cloning machine and we show that the symmetric
machine optimally copies the entanglement. The "many-to-many" teleportation
scheme is applied in order to distribute entanglement shared between two
observers to two pairs of spatially separated observers.Comment: 17 pages, 1 figur
Equivalence of operations with respect to discriminator clones
For each clone C on a set A there is an associated equivalence relation,
called C-equivalence, on the set of all operations on A, which relates two
operations iff each one is a substitution instance of the other using
operations from C. In this paper we prove that if C is a discriminator clone on
a finite set, then there are only finitely many C-equivalence classes.
Moreover, we show that the smallest discriminator clone is minimal with respect
to this finiteness property. For discriminator clones of Boolean functions we
explicitly describe the associated equivalence relations.Comment: 17 page
General properties of Nonsignaling Theories
This article identifies a series of properties common to all theories that do
not allow for superluminal signaling and predict the violation of Bell
inequalities. Intrinsic randomness, uncertainty due to the incompatibility of
two observables, monogamy of correlations, impossibility of perfect cloning,
privacy of correlations, bounds in the shareability of some states; all these
phenomena are solely a consequence of the no-signaling principle and
nonlocality. In particular, it is shown that for any distribution, the
properties of (i) nonlocal, (ii) no arbitrarily shareable and (iii) positive
secrecy content are equivalent.Comment: 10 page
Clones from Creatures
A clone on a set X is a set of finitary operations on X which contains all
the projections and is closed under composition.
The set of all clones forms a complete lattice Cl(X) with greatest element O,
the set of all finitary operations. For finite sets X the lattice is "dually
atomic": every clone other than O is below a coatom of Cl(X).
It was open whether Cl(X) is also dually atomic for infinite X. Assuming the
continuum hypothesis, we show that there is a clone C on a countable set such
that the interval of clones above C is linearly ordered, uncountable, and has
no coatoms.Comment: LaTeX2e, 20 pages. Revised version: some concepts simplified, proof
details adde
The complexity of counting locally maximal satisfying assignments of Boolean CSPs
We investigate the computational complexity of the problem of counting the
maximal satisfying assignments of a Constraint Satisfaction Problem (CSP) over
the Boolean domain {0,1}. A satisfying assignment is maximal if any new
assignment which is obtained from it by changing a 0 to a 1 is unsatisfying.
For each constraint language Gamma, #MaximalCSP(Gamma) denotes the problem of
counting the maximal satisfying assignments, given an input CSP with
constraints in Gamma. We give a complexity dichotomy for the problem of exactly
counting the maximal satisfying assignments and a complexity trichotomy for the
problem of approximately counting them. Relative to the problem #CSP(Gamma),
which is the problem of counting all satisfying assignments, the maximal
version can sometimes be easier but never harder. This finding contrasts with
the recent discovery that approximately counting maximal independent sets in a
bipartite graph is harder (under the usual complexity-theoretic assumptions)
than counting all independent sets.Comment: V2 adds contextual material relating the results obtained here to
earlier work in a different but related setting. The technical content is
unchanged. V3 (this version) incorporates minor revisions. The title has been
changed to better reflect what is novel in this work. This version has been
accepted for publication in Theoretical Computer Science. 19 page
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