Weak bases of Boolean co-clones

Lagerkvist, Victor

doi:10.1016/j.ipl.2014.03.011

Cited by 20 publications

(22 citation statements)

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“…For weak bases, Schnoor and Schnoor [28] gave a systematic procedure for obtaining weak bases, which was later rened in Lagerkvist [20] in order to get a complete list of weak bases for all Boolean co-clones of nite order. These relations can be found in Table 2.…”

Section: Theorem 4 ([28]mentioning

confidence: 99%

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The power of primitive positive definitions with polynomially many variables

Lagerkvist

Wahlström

2016

J Logic Computation

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The self-archived version of this journal article is available at Linköping University Institutional Repository (DiVA): http://urn.kb.se/resolve?urn=urn:nbn:se:liu:diva-139605 N.B.: When citing this work, cite the original publication. Lagerkvist, V., Wahlström, M., (2017) ed variables, and say that a co-clone closed under such denitions is polynomially closed, and otherwise superpolynomially closed. We investigate properties of polynomially closed co-clones and prove that if the corresponding clone contains a k-ary near-unanimity operation for k ≥ 3 then the co-clone is polynomially closed, and if the clone does not contain a k-edge operation for any k ≥ 2, then the co-clone is superpolynomially closed. For the Boolean domain we strengthen these results and prove a complete dichotomy theorem separating polynomially closed co-clones from superpolynomially closed co-clones. Using these results, we then proceed to investigate properties of strong partial clones corresponding to superpolynomially closed co-clones. We prove that if Γ is a nite set of relations over an arbitrary nite domain such that the clone corresponding to Γ is essentially unary, then the strong partial clone corresponding to Γ is of innite order and cannot be generated by a nite set of partial functions.

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Section: Theorem 4 ([28]mentioning

confidence: 99%

“…These relations can be found in Table 2. We give a short description of some of the involved relations: for a full description, see Lagerkvist [20,21] and Creignou et al [9]. We write F and T for the constant relations {(0)} and {(1)};…”

Section: Theorem 4 ([28]mentioning

confidence: 99%

The power of primitive positive definitions with polynomially many variables

Lagerkvist

Wahlström

2016

J Logic Computation

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“…Weak bases were first introduced in Schnoor and Schnoor [16] but were in many cases exponentially larger than the plain bases with respect to arity. Weak bases fulfilling additional minimality conditions was given in Lagerkvist [12] using relational descriptions. By construction the weak base of a coclone can always be given as a single relation.…”

Section: Preliminaries and Notationmentioning

confidence: 99%

“…As is easily verified for every n-ary Boolean relation it holds that R ∈ Γ n SAT ∃ . The full description of the relations involved is given in Lagerkvist [12].…”

Section: Theorem 4 ([16])mentioning

confidence: 99%

Polynomially Closed Co-clones

Lagerkvist

Wahlström

2014

2014 IEEE 44th International Symposium on Multiple-Valued Logic

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Two well-studied closure operators for relations are based on primitive positive (p.p.) definitions and quantifier free p.p. definitions. The latter do however have limited expressiveness and the corresponding lattice of strong partial clones is uncountable. We consider implementations allowing polynomially many existentially quantified variables and obtain a dichotomy for co-clones where such implementations are enough to implement any relation and prove (1) that all remaining coclones contain relations requiring a superpolynomial amount of quantified variables and (2) that the strong partial clones corresponding to two of these co-clones are of infinite order whenever the set of invariant relations can be finitely generated.

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“…Therefore we have to resort to weak co-clones that require only closure under conjunction and equality. In this connection, we apply the theory developed in [28,29] as well as the minimal weak bases of Boolean co-clones from [23]. This paper is structured as follows.…”

Section: Introductionmentioning

confidence: 99%

Minimal Distance of Propositional Models

et al. 2018

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We investigate the complexity of three optimization problems in Boolean propositional logic related to information theory: Given a conjunctive formula over a set of relations, find a satisfying assignment with minimal Hamming distance to a given assignment that satisfies the formula (NearestOtherSolution, NOSol) or that does not need to satisfy it (NearestSolution, NSol). The third problem asks for two satisfying assignments with a minimal Hamming distance among all such assignments (MinSolutionDistance, MSD).For all three problems we give complete classifications with respect to the relations admitted in the formula. We give polynomial time algorithms for several classes of constraint languages. For all other cases we prove hardness or completeness regarding APX, poly-APX, or equivalence to well-known hard optimization problems. * This paper is the final version of a trilogy of conference papers Give Me Another One! [6] (ISAAC 2015), As Close As It Gets [7] (WALCOM 2016), and The Next Whisky Bar [8] (CSR 2016).

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Weak bases of Boolean co-clones

Cited by 20 publications

References 9 publications

The power of primitive positive definitions with polynomially many variables

The power of primitive positive definitions with polynomially many variables

Polynomially Closed Co-clones

Minimal Distance of Propositional Models

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