Abstract
In the present survey, we consider all logical extensions of the parametric closure operator. Such extensions are obtained by augmenting the parametric closure language with arbitrary logical connectives or the universal quantifier. This gives, in addition to the parametric closure operator, the operators of positive and implicative closure, and also the closure operator with complete system of logical connectives, and the conjunctive-quantifier closure operator. We present basic facts on classifications of the sets Pk generated by these operators.
Originally published in Diskretnaya Matematika (2022) 34, №3, 52–62 (in Russian).
References
[1] Golunkov Yu. V., “Completeness of function systems in the operator algorithms that realize the functions of a k-valuedlogic”, Veroyatnostnye metody i kibernetika, 17 (1980), 23–34 (in Russian).Search in Google Scholar
[2] Danil’chenko A. F., “Parametric expressibility of functions of three-valued logic”, Algebra and Logic, 16 (1977), 266–280.10.1007/BF01669278Search in Google Scholar
[3] Danil’chenko A. F., “Parametrically closed classes of functions of three-valued logic”, Izvestiya AN MSSR, 2 (1978), 13–20 (in Russian).Search in Google Scholar
[4] Danil’chenko A. F., Problems of parametric expressibility of functions of three-valued logic, diss kand. fiz.-matem. nauk, Kishinev, 1979 (in Russian), 141 pp.Search in Google Scholar
[5] Kuznetsov A. V., “About the means for detecting non-deducibility and inexpressibility”, Logicheskiy vyvod, M.: Nauka, 1979, 5–33 (in Russian).Search in Google Scholar
[6] Marchenkov S. S., “Basic relations of the S-classification of functions of the multi-valued logic”, Discrete Math. Appl., 6:2 (1996), 149–178.10.1515/dma.1996.6.2.149Search in Google Scholar
[7] Marchenkov S. S., “The S-classification of functions of many-valued logic”, Discrete Math. Appl., 7:4 (1997), 353–381.10.1515/dma.1997.7.4.353Search in Google Scholar
[8] Marchenkov S. S., “On expressibility of functions of many-valued logic in some logical-functional languages”, Discrete Math. Appl., 9:6 (1999), 563–581.10.1515/dma.1999.9.6.563Search in Google Scholar
[9] Marchenkov S. S., Closed classes of Boolean functions, M.: Fizmatlit, 2000 (in Russian), 126 pp.Search in Google Scholar
[10] Marchenkov S. S., S-classification of functions of three-valued logic, M.: Fizmatlit, 2001 (in Russian), 79 pp.Search in Google Scholar
[11] Marchenkov S. S., “Closure operators with predicate branching”, Vestnik Mosk. univ. Seriya 1: Matematika, mekhanika, 6 (2003), 37–39 (in Russian).Search in Google Scholar
[12] Marchenkov S. S., “Equational closure”, Discrete Math. Appl., 15:3 (2005), 289–298.10.1515/156939205774464495Search in Google Scholar
[13] Marchenkov S. S., “Criterion for positive completeness in three-valued logic”, Diskret. analiz i issled. operatsiy. Seriya 1, 13:3 (2006), 27–39 (in Russian).Search in Google Scholar
[14] Marchenkov S. S., “On the structure of equationally closed classes”, Discrete Math. Appl., 16:6 (2006), 563–576.10.1515/156939206779217961Search in Google Scholar
[15] Marchenkov S. S., “Discriminator positively complete classes of ternary logic”, J. Appl. Industr. Math., 2:4 (2008), 542–549.10.1134/S1990478908040108Search in Google Scholar
[16] Marchenkov S. S., “On closed classes of k-valued logic functions defined by a single endomorphism”, Diskretn. Anal. Issled. Oper., 16:6 (2009), 52–67 (in Russian).Search in Google Scholar
[17] Marchenkov S. S., “Positively closed classes of three-valued logic generated by one-place functions”, Discrete Math. Appl., 19:4 (2009), 375–382.10.1515/DMA.2009.026Search in Google Scholar
[18] Marchenkov S. S., “The closure operator in a multivalued logic based on functional equations”, J. Appl. Indust. Math., 5:3 (2011), 383–390.10.1134/S1990478911030100Search in Google Scholar
[19] Marchenkov S. S., “On classificatons of many-valued logic functions by means of automorphism groups”, Diskretn. Anal. Issled. Oper., 18:4 (2011), 66–76 (in Russian).Search in Google Scholar
[20] Marchenkov S. S., “FE classification of functions of many-valued logic”, Moscow Univ. Comput. Math. and Cybern., 35:2 (2011), 89–96.10.3103/S0278641911020063Search in Google Scholar
[21] Marchenkov S. S., “Atoms of the lattice of positively closed classes of three-valued logic”, Discrete Math. Appl., 22:2 (2012), 123–137.10.1515/dma-2012-009Search in Google Scholar
[22] Marchenkov S. S., “Operator of positive closure”, Doklady Mathematics, 85 (2012), 102–103.10.1134/S1064562412010346Search in Google Scholar
[23] Marchenkov S. S., Functional equations of discrete mathematics, M.: Fizmatlit, 2013 (in Russian), 58 pp.Search in Google Scholar
[24] Marchenkov S. S., Fundamentals of the theory of Boolean functions, M.: Fizmatlit, 2014 (in Russian), 135 pp.Search in Google Scholar
[25] Marchenkov S. S., “Positive closed classes of three-valued logic”, J. Appl. Indust. Math., 8:2 (2014), 256–266.10.1134/S1990478914020124Search in Google Scholar
[26] Marchenkov S. S., “On the enumeration closure operator in multivalued logic”, Moscow Univ. Comput. Math. and Cybern., 39:2 (2015), 81–87.10.3103/S0278641915020053Search in Google Scholar
[27] Marchenkov S. S., “Closure operators with positive connectives and quantifiers”, Moscow Univ. Comput. Math. Cybern., 41:1 (2017), 32–37.10.3103/S027864191701006XSearch in Google Scholar
[28] Marchenkov S. S., “On extensions of the parametric closure operator using logical connectives”, Izvestiya vysshikh uchebnykh zavedeniy. Povolzhskiy region. Fiz.-mat. nauki, 1 (2017), 22–31 (in Russian).10.21685/2072-3040-2017-1-3Search in Google Scholar
[29] Marchenkov S. S., “Completeness criterion for the enumeration closure operator in three-valued logic”, Discrete Math. Appl., 30:1 (2020), 1–6.10.1515/dma-2020-0001Search in Google Scholar
[30] Marchenkov S. S., “Extensions of the positive closure operator by means of logical connectives”, J. Appl. Indust. Math., 12:4 (2018), 678–683.10.1134/S1990478918040087Search in Google Scholar
[31] Marchenkov S. S., Strong Closure Operators, M.: Fizmatlit, 2017 (in Russian), 94 pp.Search in Google Scholar
[32] Marchenkov S. S., Chernyshev A. V., “Calculating the number of functions with a given endomorphism”, Moscow Univ. Comput. Math. Cybern., 42:4 (2018), 171–176.10.3103/S0278641918040052Search in Google Scholar
[33] Nguyen Van Hoa, “On the structure of self-dual closed classes of the three-valued logic P3”, Diskretnaya matematika, 4:4 (1992), 82–95 (in Russian).Search in Google Scholar
[34] Nguyen Van Hoa, “On families of closed classes of k-valued logic that are preserved by all automorphism”, Diskretnaya matematika, 5:4 (1993), 87–108 (in Russian).Search in Google Scholar
[35] Podol’ko D. K. A family of classes of functions closed with respect to a strengthened superposition operation, Moscow Univ. Math. Bull., 70:2 (2015), 79–83.10.3103/S0027132215020059Search in Google Scholar
[36] Podol’ko D. K., “On classes of multivalued logic functions closed under the strong superposition operation”, Matematicheskie voprosy kibernetiki, 19, M.: Fizmatlit, 2019, 199–278 (in Russian).Search in Google Scholar
[37] Solovyev V. D., “Closed classes in k-valued logic with an operation of branching by predicates”, Diskretnaya matematika, 2:4 (1990), 82–95.Search in Google Scholar
[38] Taimanov V. A., “On function systems in k-valued logic with closure operations of program type”, Sov. Math., Dokl., 27 (1983), 255–258.Search in Google Scholar
[39] Tarasova O. S., “Classes of the k-valued logic closed with respect to extended superposition operation”, Mosc. Univ. Math. Bull., 56:6 (2001), 35–38.Search in Google Scholar
[40] Yanov Yu. I., Muchnik A. A., “On the existence of k-valued closed classes without a finite basis”, Proc. Acad. Sci. USSR, 127:1 (1959), 44–46 (in Russian).Search in Google Scholar
[41] Barris S., Willard R., “Finitely many primitive positive clones”, Proc. Amer. Math. Soc., 101:3 (1987), 427–430.10.1090/S0002-9939-1987-0908642-5Search in Google Scholar
[42] Danil’čenko A. F., “On parametrical expressibility of the functions of k-valued logic”, Colloq. Math. Soc. J. Bolyai, 28 (1981), 147–159.Search in Google Scholar
[43] Hermann M., “On Boolean primitive positive clones”, Discrete Mathematics, 308 (2008), 3151–3162.10.1016/j.disc.2007.06.018Search in Google Scholar
[44] Snow J. W., “Generating primitive positive clones”, Algebra universalis, 44 (2000), 169–185.10.1007/s000120050179Search in Google Scholar
[45] Szabó L., “Concrete representation of structures of universal algebras I”, Acta Sci. Math. (Szeged), 40 (1978), 175–184.Search in Google Scholar
[46] Szabó L., “On the lattice of clones acting bicentrally”, Acta Cybernet., 6 (1984), 381–388.Search in Google Scholar
© 2023 Walter de Gruyter GmbH, Berlin/Boston
Articles in the same Issue
- Frontmatter
- On the relationship between local affinities of a Boolean function and some types of its degeneracy
- A generalized model of the Colonel Blotto stochastic game
- Logical extensions of the parametric closure operator
- Short conditional complete diagnostic tests for circuits under one-type constant faults of gates
- Approximation of vectorial functions over finite fields and their restrictions to linear manifolds by affine analogues
Articles in the same Issue
- Frontmatter
- On the relationship between local affinities of a Boolean function and some types of its degeneracy
- A generalized model of the Colonel Blotto stochastic game
- Logical extensions of the parametric closure operator
- Short conditional complete diagnostic tests for circuits under one-type constant faults of gates
- Approximation of vectorial functions over finite fields and their restrictions to linear manifolds by affine analogues