Clones on infinite sets

A clone on a set X is a set of finitary functions on X which contains the projections and which is closed under composition. The set of all clones on X forms a complete algebraic lattice Cl(X). We obtain several results on the structure of Cl(X) for infinite X. In the first chapter we prove the combinatorial result that if X is linearly ordered, then the median functions of different arity defined by that order all generate the same clone. The second chapter deals with clones containing the almost unary functions, that is, all functions whose value is determined by one of its variables up to a small set. We show that on X of regular cardinality, the set of such clones is always a countably infinite descending chain. The third chapter generalizes a result due to L. Heindorf from the countable to all uncountable X of regular cardinality, resulting in an explicit list of all clones containing the permutations but not all unary functions of X. Moreover, all maximal submonoids of the full transformation monoid which contain the permutations of X are determined, on all infinite X; this is an extension of a theorem by G. Gavrilov for countable base sets.Comment: 70 pages; Dissertation written at the Vienna University of Technology under the supervision of Martin Goldstern; essentially consists of the author's papers "The clone generated by the median functions", "Clones containing all almost unary functions, "Maximal clones on uncountable sets that include all permutations" which are all available from arXi

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

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

Galois correspondence for counting quantifiers

We introduce a new type of closure operator on the set of relations, max-implementation, and its weaker analog max-quantification. Then we show that approximation preserving reductions between counting constraint satisfaction problems (#CSPs) are preserved by these two types of closure operators. Together with some previous results this means that the approximation complexity of counting CSPs is determined by partial clones of relations that additionally closed under these new types of closure operators. Galois correspondence of various kind have proved to be quite helpful in the study of the complexity of the CSP. While we were unable to identify a Galois correspondence for partial clones closed under max-implementation and max-quantification, we obtain such results for slightly different type of closure operators, k-existential quantification. This type of quantifiers are known as counting quantifiers in model theory, and often used to enhance first order logic languages. We characterize partial clones of relations closed under k-existential quantification as sets of relations invariant under a set of partial functions that satisfy the condition of k-subset surjectivity. Finally, we give a description of Boolean max-co-clones, that is, sets of relations on {0,1} closed under max-implementations.Comment: 28 pages, 2 figure

Anergy in self-directed B lymphocytes from a statistical mechanics perspective

The ability of the adaptive immune system to discriminate between self and non-self mainly stems from the ontogenic clonal-deletion of lymphocytes expressing strong binding affinity with self-peptides. However, some self-directed lymphocytes may evade selection and still be harmless due to a mechanism called clonal anergy. As for B lymphocytes, two major explanations for anergy developed over three decades: according to "Varela theory", it stems from a proper orchestration of the whole B-repertoire, in such a way that self-reactive clones, due to intensive interactions and feed-back from other clones, display more inertia to mount a response. On the other hand, according to the `two-signal model", which has prevailed nowadays, self-reacting cells are not stimulated by helper lymphocytes and the absence of such signaling yields anergy. The first result we present, achieved through disordered statistical mechanics, shows that helper cells do not prompt the activation and proliferation of a certain sub-group of B cells, which turn out to be just those broadly interacting, hence it merges the two approaches as a whole (in particular, Varela theory is then contained into the two-signal model). As a second result, we outline a minimal topological architecture for the B-world, where highly connected clones are self-directed as a natural consequence of an ontogenetic learning; this provides a mathematical framework to Varela perspective. As a consequence of these two achievements, clonal deletion and clonal anergy can be seen as two inter-playing aspects of the same phenomenon too

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