15,910 research outputs found
Some Examples of Minimal Groupoids on a Finite Set (Algebraic System, Logic, Language and Related Areas in Computer Science)
A minimal clone is an atom in the lattice of clones. The classification of minimal clones on a finite set still remains unsolved. A minimal groupoid is a minimal clone generated by a binary idempotent function. In this paper we report some examples of minimal groupoids generated by binary functions which resemble projections
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
Necessary conditions for tractability of valued CSPs
The connection between constraint languages and clone theory has been a
fruitful line of research on the complexity of constraint satisfaction
problems. In a recent result, Cohen et al. [SICOMP'13] have characterised a
Galois connection between valued constraint languages and so-called weighted
clones. In this paper, we study the structure of weighted clones. We extend the
results of Creed and Zivny from [CP'11/SICOMP'13] on types of weightings
necessarily contained in every nontrivial weighted clone. This result has
immediate computational complexity consequences as it provides necessary
conditions for tractability of weighted clones and thus valued constraint
languages. We demonstrate that some of the necessary conditions are also
sufficient for tractability, while others are provably not.Comment: To appear in SIAM Journal on Discrete Mathematics (SIDMA
Efficient enumeration of solutions produced by closure operations
In this paper we address the problem of generating all elements obtained by
the saturation of an initial set by some operations. More precisely, we prove
that we can generate the closure of a boolean relation (a set of boolean
vectors) by polymorphisms with a polynomial delay. Therefore we can compute
with polynomial delay the closure of a family of sets by any set of "set
operations": union, intersection, symmetric difference, subsets, supersets
). To do so, we study the problem: for a set
of operations , decide whether an element belongs to the closure
by of a family of elements. In the boolean case, we prove that
is in P for any set of boolean operations
. When the input vectors are over a domain larger than two
elements, we prove that the generic enumeration method fails, since
is NP-hard for some . We also study the
problem of generating minimal or maximal elements of closures and prove that
some of them are related to well known enumeration problems such as the
enumeration of the circuits of a matroid or the enumeration of maximal
independent sets of a hypergraph. This article improves on previous works of
the same authors.Comment: 30 pages, 1 figure. Long version of the article arXiv:1509.05623 of
the same name which appeared in STACS 2016. Final version for DMTCS journa
Quantum Cloning of Binary Coherent States - Optimal Transformations and Practical Limits
The notions of qubits and coherent states correspond to different physical
systems and are described by specific formalisms. Qubits are associated with a
two-dimensional Hilbert space and can be illustrated on the Bloch sphere. In
contrast, the underlying Hilbert space of coherent states is
infinite-dimensional and the states are typically represented in phase space.
For the particular case of binary coherent state alphabets these otherwise
distinct formalisms can equally be applied. We capitalize this formal
connection to analyse the properties of optimally cloned binary coherent
states. Several practical and near-optimal cloning schemes are discussed and
the associated fidelities are compared to the performance of the optimal
cloner.Comment: 12 pages, 12 figure
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