Abstract
We survey algebraic results on combinatorial circuits constructed from many-valued uniformly delayed gates. These involve the composition of uniformly delayed operations, the lattice of uniform clones, the corresponding relational theory, uniform completeness and the search for an effective completeness criterion.
Similar content being viewed by others
References
Baramov, R. A.: On the question of functional completeness in many-valued logics (Russian), Diskret. Anal. 11(1967), 3-20.
Bodnarchuk, V. G., Kaluzhnin, L. A., Kotov, V. N. and Romov, B. A.: The Galois theory for Post algebras I-II (Russian), Kibernetika(Kiev) 3(1969), 1-10; 5(1969), 1-9; English translation: Systems Theory Res. 3.
Birjukova, L. A. and Kudrjavcev, V. B.: On completeness of functions with delays (Russian), Probl. Kibernet. 23(1970), 5-25; English translation: Systems Theory Res. 23(1973), 3-24.
Dassow, J.: Completeness Problems in the Structural Theory of Automata, Akademie-Verlag, Berlin, 1981.
Geiger, D.: Closed systems of functions and predicates, Pacific J. Math. 27(1) (1968), 95-100.
Hikita, T.: Completeness criterion for functions with delay defined over a domain of three elements, Proc. Japan Acad. 54(1978), 335-339.
Hikita, T.: Completeness properties of k-valued functions with delays: Inclusions among closed spectra, Math. Nachr. 103(1981), 5-19.
Hikita, T.: Completeness criterion for delayed logic devices, Preprint, Univ. of Tokyo, 1972.
Hikita, T.: On completeness for k-valued functions with delay, in: B. Csákány and I. Rosenberg (eds), Coll. Math. Soc. János Bolyai28, Finite Algebra and Multiple-Valued Logic, North-Holland, Amsterdam, 1981, pp. 345-371.
Hikita, T. and Nozaki, A.: A completeness criterion for spectra, SIAM J. Comput. 6(1977), 285-297. Corrigenda, ibid. 8(1979), 656.
Hikita, T. and Rosenberg, I. G.: Completeness for uniformly delayed circuits, Proceedings 13th Internat. Sympos. Multiple-Valued Logic, Kyoto, May 1983, IEEE (1983), pp. 2-10. Proofs: Preprint, Tokyo Metropolitan Univ., 1983.
Iablonski, S. V.: Functions constructions in the k-valued logic (Russian), Trudy Mat. Inst. Steklov. 51(1958), 5-142.
Ibuki, K.: Study on the universal logic elements (Japanese), Electric Communication Laboratory Tech. Report 3747 (1968).
Ibuki, K., Namura, K. and Nozaki, A.: (Japanese) Trans. IECE 46(1963), 934.
Kritshevskii, R. E.: The realization of functions by superpositions (Russian), Probl. Kibernet. 2(1959), 123-138; German translation: Probl. Kybernet. 2(1963), 139-159.
Kudryavtsev, V. B.: Completeness theorem for a class of automata without feedback couplings (Russian), Dokl. Akad. Nauk SSSR 132(1960), 272-274; English translation: Soviet Math. Dokl. 1(1960), 537-539.
Kudryavtsev, V. B.: Completeness theorem for a class of automata without feedback couplings (Russian), Probl. Kibernet. 8(1962), 91-115; German translation: Probl. Kybernet. 8(1965), 105-136.
Kudryavtsev, V. B.: The power of sets of precomplete sets for certain functional systems connected with automata (Russian), Probl. Kibernet. 13(1965), 45-74.
Kudryavtsev, V. B.: On functional properties of logical nets (Russian), Math. Nachr. 55(1973), 187-211.
Kudryavtsev, V. B.: Functional Systems(Russian), Monograph, I.M.U., Moscow, 1982.
Loomis, H. H., Jr.: Completeness of sets of delayed-logic devices, IEEE Trans. Elektron. Comput. EC-14(1965), 157-172.
Martin, L., Reischer, C. and Rosenberg, I. G.: Completeness problems for switching circuits constructed from delayed gates, Proc. 8th Int. Symp. Multiple-Valued Logic, 1978, pp. 142-148. Electron. Informationsverarb. Kybernet.(in French) 19(1983), 415, 171-186. An expanded French version appeared in Reseaux modulaires, Monographies de l.U.Q.T.R. No. 11, 1980
Martin, N. M.: On the completeness of sets of gating functions with delay, in: Sympos. on Gating Algebra, Proc. Internat. Congr. Information Processing, Paris, 1959.
Miličić, M.: Galois connections for closed classes of functions with delays (Russian), Publ. de l'Inst. Mathématique Nouvelle Série 36(50) (1984), Part I: 119-124, Part II: 125-136.
Miličić, M.: On Galois connections for one place functions with delays (Russian), Publ. de l'Inst. Mathématique Nouvelle Série 44(58) (1988), 147-150.
Nozaki, A.: Réalisation des fonctions definies dans un ensemble fini à l'aide des organes élémentaires d'entrée-sortie, Proc. Japan Acad. 46(1970), 478-482.
Nozaki, A.: Functional studies of automata I-II, Sci. Papers College Gen. Ed. Univ. Tokyo 20(1970), 21-36, 109-121.
Nozaki, A.: Complete sets of switching elements and related topics, First USA-Japan Computer Conference, 1972, pp. 393-396.
Nozaki, A.: Functional completeness of multi-valued logical functions under uniform compositions, Rep. Fac. Eng. Yamanashi Univ. 29(1978), 61-67.
Nozaki, A.: Completeness criterion for a set of delayed functions with or without non-uniform compositions, in: B. Csákány and I. Rosenberg (eds), Coll. Math. Soc. János Bolyai28, Finite Algebra and Multiple-Valued Logic, North-Holland, Amsterdam, 1981, pp. 489-519.
Nozaki, A.: Completeness of logical gates based on sequential circuits (Japanese), Trans. IECE Japan J65-D(1982).
Pöschel, R. and Kalužnin, L. A.: Funktionen-space und Relationenalgebren(German), VEB Deutscher Verlag der Wissenschaften, Berlin, 1979.
Post, E. L.: Introduction to a general theory of elementary propositions, Amer. J. Math. 43(1921), 163-185.
Rosenberg, I. G.: La structure des fonctions de plusieurs variables sur un ensemble fini, C.R. Acad. Sci. ParisSér. A-B 260(1965), 3817-3819.
Rosenberg, I. G.: Über die funktionale Vollständigkeit in dem mehrwertigen Logiken (Struktur der Funktionen von mehreren Veränderlichen auf endlichen Mengen), RozpravyČeskoslovenskéAkad. Vied., Ser. Math. Nat. Sci. 80(1970), 3-93.
Rosenberg, I. G.: Completeness properties of multiple-valued logic algebras, in: D. C. Rine (ed.), Computer Science and Multiple-Valued Logic, Theory and Applications, North-Holland, Amsterdam, 1977, pp. 144-186. 2nd edition: ibid., 1984, pp. 150-194.
Rosenberg, I. G.: The multifaceted completeness problem of the structural theory of automata, Preprint CRMA-1197, Université de Montréal, 1983.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Hikita, T., Rosenberg, I.G. Completeness for Uniformly Delayed Circuits, A Survey. Acta Applicandae Mathematicae 52, 49–61 (1998). https://doi.org/10.1023/A:1005946705819
Issue Date:
DOI: https://doi.org/10.1023/A:1005946705819