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Counting Problems and Clones of Functions
2009
2009 39th International Symposium on Multiple-Valued Logic
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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. ...
doi:10.1109/ismvl.2009.47
dblp:conf/ismvl/Bulatov09
fatcat:4xtyqeumsvbr3lcnfu4ukzfrsy
Minimal partial clones
1991
Bulletin of the Australian Mathematical Society
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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. ...
doi:10.1017/s0004972700029919
fatcat:d3p44hrmavh3he7qh32eebu3q4
Polynomial Functions on a Central Relation
[article]
2003
arXiv
pre-print
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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. ...
arXiv:math/0310131v1
fatcat:5axk53wanze5vf5r7uryhalx2m
Galois correspondence for counting quantifiers
[article]
2012
arXiv
pre-print
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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. ...
arXiv:1210.3344v1
fatcat:bli7tjoyivcmhidf2e3yt7icum
Page 2247 of Mathematical Reviews Vol. , Issue 88e
[page]
1988
Mathematical Reviews
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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
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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)
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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. ...
doi:10.1016/s0195-6698(13)80003-0
fatcat:p5yz2eugibhnvggk6ipxdxhovq
Optimal cloning of arbitrary mirror-symmetric distributions on the Bloch sphere: a proposal for practical photonic realization
2012
Physica Scripta
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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. ...
doi:10.1088/0031-8949/2012/t147/014003
fatcat:6ubkvbaqbrbopnbb7ooauqi4cm
Page 6556 of Mathematical Reviews Vol. , Issue 93m
[page]
1993
Mathematical Reviews
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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]
2009
arXiv
pre-print
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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 . ...
arXiv:0905.1614v1
fatcat:dkwdwbba6remtnpips4qg6mz3m
Clones on infinite sets
[article]
2004
arXiv
pre-print
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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. ...
arXiv:math/0407173v1
fatcat:pouhjs2c2jdrzl5luwxuxvchiu
Boolean Max-Co-Clones
2013
2013 IEEE 43rd International Symposium on Multiple-Valued Logic
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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 [21] .
APPROXIMATE COUNTING AND MAX-QUANTIFICATION Let D be a set, and let Γ be a finite set of relations over D. ...
doi:10.1109/ismvl.2013.16
dblp:conf/ismvl/Bulatov13
fatcat:bapbyst6bbclfbs7bsuk2q7k4q
Boolean max-co-clones
2015
Algebra Universalis
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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 [21] .
APPROXIMATE COUNTING AND MAX-QUANTIFICATION Let D be a set, and let Γ be a finite set of relations over D. ...
doi:10.1007/s00012-015-0336-1
fatcat:g33htzffdjfhxky7w5pk3v3yvq
Clones containing all almost unary functions
2004
Algebra Universalis
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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 [9] , see also [8] ). ...
doi:10.1007/s00012-004-1860-6
fatcat:iikncolfkrgjzh36muw74jsla4
Information Processing in Convex Operational Theories
2011
Electronical Notes in Theoretical Computer Science
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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 ...
doi:10.1016/j.entcs.2011.01.002
fatcat:nwubkcln35gihhxvsquklygiia
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