On properties of multiaffine predicates on a finite set
-
Svetlana N. Selezneva
Abstract
We consider predicates on a finite set that are invariant with respect to an affine operation fG, where G is some Abelian group. Such predicates are said to be multiaffine for the group G. Special attention is paid to predicates that are affine for a group Gq of addition modulo q=ps, where p is a prime number and s=1. We establish the predicate multiaffinity criterion for a group Gq. Then we introduce disjunctive normal forms (DNF) for predicates on a finite set and obtain properties of DNFs of predicates that are multiaffine for a group Gq. Finally we show how these properties can be used to design a polynomial algorithm that decides satisfiability of a system of predicates which are multiaffine for a group Gq, if predicates are specified by DNF.
Note
Originally published in Diskretnaya Matematika (2021) 33, №4, 141–152 (in Russian).
References
[1] Schaefer T., “Complexity of satisfiability problems”, Proc. 10th ACM Symp. Theory of Computing, ACM Press, 1978, 216–226.10.1145/800133.804350Search in Google Scholar
[2] Jeavons P., Coher D., Gyssens M., “Closure properties of constraints”, J. ACM, 44 (1997), 527–548.10.1145/263867.263489Search in Google Scholar
[3] Gorshkov S. P., “On the complexity of recognizing multiaffinity, bijunctivity, weak positivity and weak negativity”, Obozr. promyshl. i prikl. matem., 4:2 (1997), 216–237 (in Russian).Search in Google Scholar
[4] Selezneva S.N., “Multiaffine polynomials over a finite field”, Discrete Math. Appl., 31:6 (2021), 421–430.10.1515/dma-2021-0038Search in Google Scholar
[5] Selezneva S.N., “On bijunctive predicates over a finite set”, Discrete Math. Appl., 29:1 (2019), 49–58.10.1515/dma-2019-0006Search in Google Scholar
[6] Selezneva S.N., “On 𝑚-junctive predicates on a finite set”, J. Appl. Industr. Math., 13:3 (2019), 528–535.10.1134/S199047891903013XSearch in Google Scholar
[7] Selezneva S.N., “On weak positive predicates over a finite set”, Discrete Math. Appl., 30:3 (2020), 203–213.10.1515/dma-2020-0019Search in Google Scholar
[8] Tyrtyshnikov E. E., Basics of Algebra, M.: Fizmatlit, 2017 (in Russian), 464 pp.Search in Google Scholar
[9] Goldmann M., Russell A., “The complexity of solving equations over finite groups”, Inf. and Comput., 178 (2002), 253–262.10.1006/inco.2002.3173Search in Google Scholar
[10] Vasilenko O.N., Number-theoretic algorithms in cryptography, M.: MCCME, 2003 (in Russian), 328 pp.Search in Google Scholar
[11] Elizarov V. P., “Systems of linear equations over finite rings”, Trudy po diskretnoy matematike, 6 (2002), 31–47 (in Russian).Search in Google Scholar
[12] Elizarov V. P., “On the algorithm of sequential solution of a system of linear equations over a residue ring”, Trudy po diskretnoy matematike, 11:2 (2008), 31–42 (in Russian).Search in Google Scholar
[13] Elizarov V. P., Finite rings, M.: Gelios ARV, 2006 (in Russian), 303 pp.Search in Google Scholar
[14] Yablonskiy S. V, “Functional constructions in a 𝑘-valued logic”, Trudy Matem. Inst. Steklov, 51, M.: Acad. Sci. USSR, 1958, 5–142 (in Russian).Search in Google Scholar
[15] Gorshkov S. P., Tarasov A. V., The complexity of solving systems of Boolean equations, M.: Kurs, 2017 (in Russian), 192 pp.Search in Google Scholar
© 2023 Walter de Gruyter GmbH, Berlin/Boston
Articles in the same Issue
- Frontmatter
- On distance-regular graphs Γ of diameter 3 for which Γ3 is a triangle-free graph
- Limit theorems for the maximal tree size of a Galton – Watson forest in the critical case
- Short complete diagnostic tests for circuits with two additional inputs in some basis
- Nonlinearity of functions over finite fields
- The limit joint distributions of statistics of three tests of the NIST package
- On properties of multiaffine predicates on a finite set
- On the universality of product for classes of linear functions of two variables
Articles in the same Issue
- Frontmatter
- On distance-regular graphs Γ of diameter 3 for which Γ3 is a triangle-free graph
- Limit theorems for the maximal tree size of a Galton – Watson forest in the critical case
- Short complete diagnostic tests for circuits with two additional inputs in some basis
- Nonlinearity of functions over finite fields
- The limit joint distributions of statistics of three tests of the NIST package
- On properties of multiaffine predicates on a finite set
- On the universality of product for classes of linear functions of two variables
