Dichotomy on intervals of strong partial Boolean clones

in Algebra Universalis (2015), 73(3), 347-368

The following result has been shown recently in the form of a dichotomy: For every total clone $C$ on $\2 := \{0,1\}$, the set $\intervalD{C}$ of all partial clones on $\2$ whose total component is $C ... [more ▼]

The following result has been shown recently in the form of a dichotomy: For every total clone $C$ on $\2 := \{0,1\}$, the set $\intervalD{C}$ of all partial clones on $\2$ whose total component is $C$, is either finite or of continuum cardinality. In this paper we show that the dichotomy holds, even if only strong partial clones are considered, i.e., partial clones which are closed under taking subfunctions: For every total clone $C$ on $\2$, the set $\intervalStr{C}$ of all strong partial clones on $\2$ whose total component is $C$, is either finite or of continuum cardinality. [less ▲]

Set-reconstructibility of Post classes

in Discrete Applied Mathematics (2015), 187

Relation graphs and partial clones on a 2-element set.

in Multiple-Valued Logic (ISMVL), 2014 IEEE 44rd International Symposium on (2014)

In a recent paper, the authors show that the sublattice of partial clones that preserve the relation $\{(0,0),(0,1),(1,0)\}$ is of continuum cardinality on $\2$. In this paper we give an alternative proof ... [more ▼]

In a recent paper, the authors show that the sublattice of partial clones that preserve the relation $\{(0,0),(0,1),(1,0)\}$ is of continuum cardinality on $\2$. In this paper we give an alternative proof to this result by making use of a representation of relations derived from $\{(0,0),(0,1),(1,0)\}$ in terms of certain types of graphs. As a by-product, this tool brings some light into the understanding of the structure of this uncountable sublattice of strong partial clones. [less ▲]

Hypomorphic Sperner systems and non-reconstructible functions

in Order: A Journal on the Theory of Ordered Sets and its Applications (2014)

A reconstruction problem is formulated for Sperner systems, and infinite families of non-reconstructible Sperner systems are presented. This has an application to a reconstruction problem for functions of ... [more ▼]

A reconstruction problem is formulated for Sperner systems, and infinite families of non-reconstructible Sperner systems are presented. This has an application to a reconstruction problem for functions of several arguments and identification minors. Sperner systems being representations of certain monotone functions, infinite families of non-reconstructible functions are thus obtained. The clones of Boolean functions are completely classified in regard to reconstructibility. [less ▲]

Sur des classes de fonctions à seuil caractérisables par des contraintes relationnelles

in Marichal, Jean-Luc; Essounbouli, Najib; Guelton, Kevin (Eds.) Actes des 22èmes rencontres francophones sur la Logique Floue et ses Applications, 10-11 octobre 2013, Reims, France (2013, October)

Motivated by modal semantics induced by majority games, we consider the class of threshold functions. It was shown by L. Hellerstein that this class is characterizable by relational constraints (or ... [more ▼]

Motivated by modal semantics induced by majority games, we consider the class of threshold functions. It was shown by L. Hellerstein that this class is characterizable by relational constraints (or equivalently, by functional equations), but that there is no characterization by means of finitely many constraints. In this paper, we present a complete classification of classes of threshold functions induced by Boolean clones, into whether they are characterizable by finitely many relational constraints. Moreover we provide sets of constraints characterizing each of such classes. [less ▲]

Upside down with the strong partial clones

E-print/Working paper (2013)

A Solution to a Problem of D. Lau: Complete Classification of Intervals in the Lattice of Partial Boolean Clones

in Multiple-Valued Logic (ISMVL), 2013 IEEE 43rd International Symposium on (2013)

The following natural problem, first considered by D. Lau, has been tackled by several authors recently: Let C be a total clone on 2 := {0, 1}. Describe the interval I(C) of all partial clones on 2 whose ... [more ▼]

The following natural problem, first considered by D. Lau, has been tackled by several authors recently: Let C be a total clone on 2 := {0, 1}. Describe the interval I(C) of all partial clones on 2 whose total component is C. We establish some results in this direction and combine them with previous ones to show the following dichotomy result: For every total clone C on 2, the set I(C) is either finite or of continuum cardinality. [less ▲]

A Sheffer-criterion for partial 4-valued logic

in Journal of Multiple-Valued Logic and Soft Computing (2012), 18(2), 167-199

We determine the minimal covering of maximal partial clones in 4-valued logic. That means a necessary and sufficient condition for partial Sheffer functions in 4-valued logic in terms of the maximal ... [more ▼]

We determine the minimal covering of maximal partial clones in 4-valued logic. That means a necessary and sufficient condition for partial Sheffer functions in 4-valued logic in terms of the maximal partial clones is established. Furthermore several statements about members of the minimal covering of the maximal partial clones for any finite-valued logic are established. [less ▲]

Counting the maximal partial clones on a finite set

in Journal of Multiple-Valued Logic and Soft Computing (2012), 18(2), 153-165

We show that different coherent relations specify different maximal partial clones. Then we describe a computer program to find all coherent relations and thus all maximal partial clones on 4-element, 5 ... [more ▼]

We show that different coherent relations specify different maximal partial clones. Then we describe a computer program to find all coherent relations and thus all maximal partial clones on 4-element, 5-element, and 6-element sets. [less ▲]

Galois theory for partial clones and some relational clones

in Proceedings - 41st IEEE International Symposium on Multiple-Valued Logic, ISMVL 2011 (2011)

A Galois connection between partial clones and a new variant of relation algebras is established. We introduce a new elementary operation on relations which captures the difference between total and ... [more ▼]

A Galois connection between partial clones and a new variant of relation algebras is established. We introduce a new elementary operation on relations which captures the difference between total and partial clones and allows us to adapt the proof of the Galois connection from the total case to the partial case. This Galois connection is able to capture all partial clones and is not restricted to strong partial clones as in previous work. [less ▲]