Completeness of logical functions on a square and in the system Pk�P?

Romov, B. A.

doi:10.1007/bf01078898

Cited by 5 publications

(23 citation statements)

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“…which, therefore, is incomplete in view of incompleteness of relation (6). Since σ\ and (7) have no common variables and σ\ is central, the relation π' is incomplete (and is not a diagonal).…”

Section: Minimal Relations Over Arbitrary Sets £* E Kmmentioning

confidence: 92%

“…If / tl = ... = l is = 2, then the minimality of the relation is proved in [4]. It is easy to see (see also [6,7]) that for t > 2 we cannot obtain an incomplete diagonal from this relation by an (3, &)-formula. Therefore, by choosing appropriate epimorphisms % , .…”

Section: Description Of the Minimal Relations Over The Sets £* mentioning

confidence: 93%

“…The minimality of m-based relations of item 1 follows from the minimality of corresponding one-based relations by Lemma 1. The minimality of relations of item 2 was shown in [6]. The minimality of central multi-based relation was proved in [3, Theorem 1].…”

Section: Description Of the Minimal Relations Over The Sets £* mentioning

confidence: 95%

“…. x PI all precomplete classes were found in [9] (an alternative description is given in [6,7]). The case P 2 x ... x PI x Pk is considered in [8] and the case P 3 x P 3 is completely investigated in [3].…”

Section: Introductionmentioning

confidence: 99%

“…The case P 2 x ... x PI x Pk is considered in [8] and the case P 3 x P 3 is completely investigated in [3]. In addition, the papers [3,4,[6][7][8][9] contain a series of assertions on precomplete classes in systems P^ x . .…”

Section: Introductionmentioning

confidence: 99%

See 4 more Smart Citations

On precomplete classes in Cartesian products of P 2 and P 3

Марченков¹

1994

Discrete Mathematics and Applications

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Section: Minimal Relations Over Arbitrary Sets £* E Kmmentioning

confidence: 92%

Section: Description Of the Minimal Relations Over The Sets £* mentioning

confidence: 93%

Section: Description Of the Minimal Relations Over The Sets £* mentioning

confidence: 95%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

On precomplete classes in Cartesian products of P 2 and P 3

Марченков¹

1994

Discrete Mathematics and Applications

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Polynomial completeness criteria in finite Boolean algebras

Romov

Proceedings of 26th IEEE International Symposium on Multiple-Valued Logic (ISMVL'96)

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For a given finite Boolean algebra with r (r 2 2) atoms we consider the set BF(r) of all polynomials produced bysuperpositions of the main operations and r atomic constants. Using the isomoFhism between BF(r) and P; , the arity-calibrated product of r two-valued logic algebras Pz , and also the description of all maximal subalgebras of P i , we establish a general completeness criterion in BF(r), a Sheer criterion for a single Boolean function to be a generating element in BF(r), and Slupecki type criterion in BF(r) as well.We consider finite Boolean algebras as finite distributive lattices with complement. Due to a well-known result ( see e.g.[l],Ch.V) any algebra of such kind is isomorphic to the 2' -element Boolean e e b r a on the set of all subsets of r-element set of its atoms. So, in what follows, Rr) is the set of all subsets of r = {e,, ..., e, } ( r 2 2 ) and C wr); 0 , el,.*., e,, U, n, 1 > is the Boolean algebra with r atoms having in its signature the empty set, the atoms, the union, the intersection and the complement.The polynomials in this algebra (i.e. operations constructed from the main operations, the projections, and the atomic constants by superposition) are called Boolean functions or Boolean operations of rank r. Denote by BF(r) the set of all Boolean functions. Functions from BF(r), r 2 2 , play an essential role as building blocks of set-valued logic networks which serve as a mathematical apparatus for the wave-parallel (wire-free) computing architecture design (see e.g. [2]).Example. Let r=2. Hence, we have 4-element set B (2) with atoms e, and e,. Assume #l(X)=(ie, n X)U(e, n l x ) , MN= ( l e , n x)u(~, n W, then hm,, x,, 262 0-8186-7392-3/96 $05.00 0 1996 IEEE

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On the completeness in the System P 3 x P 3

Марченков¹

1992

Discrete Mathematics and Applications

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Completeness of logical functions on a square and in the system Pk�P?

Cited by 5 publications

References 1 publication

On precomplete classes in Cartesian products of P 2 and P 3

On precomplete classes in Cartesian products of P 2 and P 3

Polynomial completeness criteria in finite Boolean algebras

On the completeness in the System P 3 x P 3

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