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### Counting Problems and Clones of Functions

Andrei A. Bulatov
2009 2009 39th International Symposium on Multiple-Valued Logic
In other cases, such as approximate counting, the connection is weaker, allows one to employ only partial functions and co-clones of the corresponding type. In those cases many problems remain open.  ...  The central motif of this survey is the connection between the complexity of counting CSPs and clones of functions of multi-valued logic and co-clones of relations.  ...  An n-ary partial function f on a set D is just a partial mapping f : D n → D.  ...

### Minimal partial clones

1991 Bulletin of the Australian Mathematical Society
Let A be a finite Set. A partial clone on A is a composition closed set of operations containing all projections. It is well known that the partial clones on A , ordered by inclusion, form a lattice.  ...  A clone is a superposition closed set of finitary operations on a fixed finite set A containing aii projections.  ...  Let A be a finite set and C be a proper partial done on A.  ...

### Polynomial Functions on a Central Relation [article]

Dietmar Schweigert
2003 arXiv   pre-print
We show that the algebra R = (R; ∧, , 0, f̅_i(x) (i ∈ I)) is central polynomially complete and finite.  ...  It is clear that maximal clone presents a prepolynomially complete algebra on finite set A. 6.3. On the other hand, we may consider the clone generated by the projections.  ...  A clone C of functions on set A is called maximal clone if, for any f ∈ O A C follows (C ∪ {f }) = O A . That is, the set C ∪ {f } generates the clone of all functions on A.  ...

### Galois correspondence for counting quantifiers [article]

Andrei A. Bulatov, Amir Hedayaty
2012 arXiv   pre-print
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  ...  We introduce a new type of closure operator on the set of relations, max-implementation, and its weaker analog max-quantification.  ...  APPROXIMATE COUNTING AND MAX-IMPLEMENTATION Let D be a set, and let Γ be a finite set of relations over D.  ...

### Page 2247 of Mathematical Reviews Vol. , Issue 88e [page]

1988 Mathematical Reviews
If the number m of H-maximal clones is finite, then for f € H its characteristic vector is x = (@),---, 4m), where a; = 0 if the function f is an element of the ith H-maximal clone and a; = 1 otherwise  ...  F is an H- maximal clone if F c H (proper inclusion) and there is no clone G such that F Cc GC H.  ...

### Page 1659 of Mathematical Reviews Vol. , Issue 81E [page]

1981 Mathematical Reviews
For the proofs a geometric lemma is used. B. Zelinka (Liberec) Szendrei, Agnes 81e:08006 On closed sets of linear operations over a finite set of squarefree cardinality.  ...  For a cardinal number m, k(A, B, m) denotes the cardinality of the set of all maximal systems of disjoint isomorphic copies of A in B which have cardinality m.  ...

### Partial Sheffer Operations

1991 European journal of combinatorics (Print)
A partial operation f on a finite set A is called Sheffer if the partial algebra (A, t> is complete, i.e. if all partial operations on A are compositions off (or definable in terms off).  ...  In this paper we describe all partial Sheffer operations for IA I = 2 and all binary Sheffer operations for IAI = 3. 9  ...  ACKNOWLEDGEMENTS The authors are grateful to the referee for his fruitful suggestions. This research was partially supported by NSERC Canada and FCAR Quebec Eq-0539.  ...

### Optimal cloning of arbitrary mirror-symmetric distributions on the Bloch sphere: a proposal for practical photonic realization

2012 Physica Scripta
information on the set of qubits to be cloned.  ...  detectors, (iv) dark counts and finite efficiency of the detectors.  ...  information on the set of qubits to be cloned.  ...

### Page 6556 of Mathematical Reviews Vol. , Issue 93m [page]

1993 Mathematical Reviews
Summary: “The generation and counting of order ideals in a partially ordered set is one of the basic, frequently occurring themes in combinatorial optimization and operations research, e.g., in such problems  ...  The authors investigate under what conditions the set [(P,, P)) of functions isotone with respect to two isomorphic posets P;, P, on a finite set A is trivial, that is, contains no functions distinct from  ...

### The submaximal clones on the three-element set with finitely many relative R-classes [article]

Erkko Lehtonen, Agnes Szendrei
2009 arXiv   pre-print
We study the maximal and submaximal clones on a three-element set and determine which of them have only finitely many relative R-classes.  ...  For each clone C on a set A there is an associated equivalence relation analogous to Green's R-relation, which relates two operations on A if and only if each one is a substitution instance of the other  ...  Acknowledgements We would like to thank Miguel Couceiro for helpful discussions concerning the clones in lines 34 and 35 of Table 2 .  ...

### Clones on infinite sets [article]

Michael Pinsker
2004 arXiv   pre-print
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 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.  ...  A clone is called maximal iff it is a dual atom in Cl(X). On finite X there exist finitely many maximal clones and an explicit list of those clones has been provided by I.  ...

### Boolean Max-Co-Clones

A. A. Bulatov
2013 2013 IEEE 43rd International Symposium on Multiple-Valued Logic
In our ISMVL 2012 paper we introduced the notion of max-co-clone as a set of relations closed under a new type of quantification, max-quantification.  ...  In this paper we go beyond scattered examples of max-co-clones and describe all max-co-clones on a 2-elements set (Boolean maxco-clones).  ...  Sets of functions of the form Pol(Γ) are exactly clones of functions  . APPROXIMATE COUNTING AND MAX-QUANTIFICATION Let D be a set, and let Γ be a finite set of relations over D.  ...

### Boolean max-co-clones

Andrei A. Bulatov
2015 Algebra Universalis
Each of the Boolean max-co-clones is also a regular co-clone, although this is a coincidence, since there are max-clones on larger sets that are not co-clones.  ...  Another connection exists for partial co-clones (closed only under intersections) and the approximation complexity of counting CSPs [8, 12] .  ...  Sets of functions of the form Pol(Γ) are exactly clones of functions  . APPROXIMATE COUNTING AND MAX-QUANTIFICATION Let D be a set, and let Γ be a finite set of relations over D.  ...

### Clones containing all almost unary functions

Michael Pinsker
2004 Algebra Universalis
Let X be an infinite set of regular cardinality. We determine all clones on X which contain all almost unary functions.  ...  It turns out that independently of the size of X, these clones form a countably infinite descending chain. Moreover, all such clones are finitely generated over the unary functions.  ...  A clone is called maximal iff it is a dual atom in Clone(X). For finite X, the number of maximal clones is finite and all of them are known (a result due to I. Rosenberg  , see also  ).  ...

### Information Processing in Convex Operational Theories

Howard Barnum, Alexander Wilce
2011 Electronical Notes in Theoretical Computer Science
and possibilities, e.g., the impossibility of cloning in the latter, and the possibility of teleportation in both.  ...  In order to understand the source and extent of the greater-than-classical information processing power of quantum systems, one wants to characterize both classical and quantum mechanics as points in a  ...  The proof of the no-cloning result is essentially to show that if one can clone a set of states, one can distinguish them by repeatedly cloning to create many independent copies, performing an informationally  ...
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