Rational \emph{A}-functions with rational coefficients

N. Ph. Alexiadis

Čebyševskij sbornik, Tome 23 (2022) no. 4, pp. 11-19.

Résumé

A functional system is a set of functions endowed with a set of operations on these functions. The operations allow one to obtain new functions from the existing ones. Functional systems are mathematical models of real and abstract control systems and thus are one of the main objects of discrete mathematics and mathematical cybernetic. The problems in the area of functional systems are extensive. One of the main problems is deciding completeness that consists in the description of all subsets of functions that are complete, i.e. generate the whole set. In our paper we consider the functional system of rational functions with rational coefficients endowed with the superposition operation. We investigate the special case of the completeness problem which is of a particular interest, namely obtaining complete systems of minimum cardinality, i.e. complete systems consisting of a single rational function (such functions are referred to as $A$-functions and are analogues of Schaeffer stroke in Boolean logic). The main results of the paper are the following: there exists an $A$-function; the cardinality of the set of all $A$-functions equals $c_{0}$; a number of examples of $A$-functions are presented explicitly.

Détail

Keywords: functional system, completeness problem, complete system, Schaeffer function, rational function, $A$-function.

Bibliographie

[1] Aleksiadis N. F., “O ratsionalnykh A-funktsiyakh s ratsionalnymi koeffitsientami”, Algebra, teoriya chisel, diskretnaya geometriya i mnogomasshtabnoe modelirovanie: sovremennye problemy, prilozheniya i problemy istorii, Materialy XIX Mezhdunarodnoi konferentsii, posvyaschennoi dvukhsotletiyu so dnya rozhdeniya akademika P. L. Chebysheva (Tula, 18-22 maya 2021 goda), Tula, 2021, 93–97

[2] Aleksiadis N. F., “O suschestvovanii rekursivnykh A-funktsii”, Vestnik MEI, 2011, no. 6, 109–111

[3] Aleksiadis N. F., Tkhan Tun Aung, “Ob odnoi rekursivnoi A-funktsii”, Trudy XXI mezhdunarodnoi nauchno-tekhnicheskoi konferentsii “Informatsionnye sredstva i tekhnologii” (Moskva, 19-21 noyabrya 2013 g.), v. 3, Izdatelskii dom MEI, M., 2013, 96–101

[4] Babin D. N., “O zadache polnoty dlya avtomatov”, Intellektualnye sistemy. Teoriya i prilozheniya, 23:4 (2020), 82–83

[5] Gavrilov G. P., “O funktsionalnoi polnote v schetnoznachnoi logike”, Problemy kibernetiki, 15, Nauka, M., 1965, 5–64

[6] Kudryavtsev V. B., “O moschnostyakh mnozhestv predpolnykh mnozhestv nekotorykh funktsionalnykh sistemakh, svyazannykh s avtomatami”, Problemy kibernetiki, 13, Nauka, M., 1965, 45–74

[7] Kudryavtsev V. B., Funktsionalnye sistemy, Izd–vo MGU, M., 1982, 157 pp.

[8] Maltsev A. I., Izbrannye trudy, v. II, Izd–vo Nauka, M., 1976, 388 pp. | MR

[9] Salomaa A., “Nekotorye kriterii polnoty dlya mnozhestv funktsii mnogoznachnoi logiki”, Kiberneticheskii sbornik, 8, Mir, M., 1964, 7–32

[10] Chasovskikh A. A., “Problema polnoty v klassakh lineinykh avtomatov”, Intellektualnye sistemy. Teoriya i prilozheniya, 22:2 (2018), 151–154

[11] Yablonskii S. V., Vvedenie v diskretnuyu matematiku, Izd–vo Nauka, M., 1986, 384 pp. | MR

[12] Yablonskii S. V., “O funktsionalnoi polnote v trekhznachnom ischislenii”, DAN SSSR, 1954, no. 6, 1153–1156 | MR

[13] Yablonskii S. V., “Funktsionalnye postroeniya v $k$-znachnoi logike”, Tr. MIAN SSSR im. V. A. Steklova, 51, 1958, 5–142 | Zbl

[14] Post E., Two-valued iterative sistems of mathematical logik, Prinston, 1941 | MR

[15] Rosenberg Y., “Uber die functionale Vollständigkeit in den mehrwertigen Logiken”, Praha, Rozpravi Ceskoslovenska Acodemie Ved., 80:4 (1970), 393 pp. | Zbl

[16] Slupecki J., “Kriterium pelnosci wielowar — tosciowych systemow logiki zdan”, Comptes Rendus des Seances de la Societe des Sciences et des Lettres de Varsivie. Cl. III, 32 (1939), 102–128