Completeness of uniformly delayed operations

  • Teruo Hikita
  • Ivo G. Rosenberg
Conference paper
Part of the NATO Science Series II: Mathematics, Physics and Chemistry book series (NAII, volume 207)

Abstract

The paper reports on progress toward an effective completeness criterion for uniformly delayed multiple-valued combinatorial circuits. In view of previous work by Hikita and Nozaki, and Hikita it suffices to study periodic closed spectra. The main tool is the use of polyrelations and certain constructions on polyrelations developed by Hikita and Miličié. We were able to restrict the search to unary polyrelations and three types of binary polyrelations ρ = (ρ0, ρ1, …):
  1. (1)

    period 2m, ρ0 bounded order, ρ2m−1 its converse and ρi = l2: = (a, a) | a Є k otherwise,

     
  2. (2)

    every nonempty ρi = (a, si(a)) | a Є k where si is a permutation of k; the permutations are interrelated,

     
  3. (3)

    components are either (i) all equivalence relations on k or (ii) all central or equal to k2. In both cases they have strong properties in terms of intersecting cliques.

     

Keywords

Equivalence Relation Binary Relation Prime Divisor Switching Circuit Galois Connection 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
This is a preview of subscription content, log in to check access.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [Ba67]
    R. A. Baĭramov, On the question of functional completeness in many-valued logics, Diskret. Analiz 11 (1967), 3–20 (Russian).Google Scholar
  2. [BW79]
    N. L. Biggs and A. T. White, Permutation Groups and Combinatorial Structures, London Math. Soc. Lecture Note Series 33, Cambridge Univ. Press, 1979.Google Scholar
  3. [BK70]
    L. A. Biryukova and V. B. Kudryavtsev, On completeness of functions with delays, Problemy Kibernet. 23 (1970), 5–25 (Russian); Systems Theory Research 23 (1973), 3–24.Google Scholar
  4. [BKKR69]
    V. G. Bodnarchuk, L. A. Kaluzhnin, V. N. Kotov, and B. A. Romov, The Galois theory for Post algebras I–II, Kibernetika (Kiev) 3 (1969), 1–10, 5 (1969), 1–9 (Russian); Systems Theory Research 3.Google Scholar
  5. [Bu80]
    V. A. Buevich, On the t-completeness in the class of automata maps, Dokl. Akad. Nauk. SSSR 252 (1980), 1037–1041 (Russian).MATHMathSciNetGoogle Scholar
  6. [Da81]
    J. Dassow, Completeness Problems in the Structural Theory of Automata, Akademie-Verlag, Berlin, 1981.Google Scholar
  7. [Ge68]
    D. Geiger, Closed systems of functions and predicates, Pacific J. Math. 27(1) (1968), 95–100.MATHMathSciNetGoogle Scholar
  8. [Go68]
    D. Gorenstein, Finite Groups, second ed., Chelsea Pub., 1968, 1980.Google Scholar
  9. [Hi78]
    T. Hikita, Completeness criterion for functions with delay defined over a domain of three elements, Proc. Japan Acad. 54 (1978), 335–339.MATHMathSciNetGoogle Scholar
  10. [Hi81a]
    T. Hikita, Completeness properties of k-valued functions with delays: Inclusions among closed spectra, Math. Nachr. 103 (1981), 5–19.MATHMathSciNetGoogle Scholar
  11. [Hi81b]
    T. Hikita, On completeness for k-valued functions with delay, in: Finite Algebra and Multiple-Valued Logic (B. Csákány and I. Rosenberg, eds.), Coll. Math. Soc. János Bolyai 28, North-Holland, 1981, 345–371.Google Scholar
  12. [HN77]
    T. Hikita and A. Nozaki, A completeness criterion for spectra, SIAM J. Comput. 6 (1977), 285–297. Corrigenda, ibid., 8 (1979), 656.CrossRefMathSciNetGoogle Scholar
  13. [HR98]
    T. Hikita and I. G. Rosenberg, Completeness for uniformly delayed circuits, a survey, Acta Applic. Math. 52 (1998), 49–61.MathSciNetGoogle Scholar
  14. [Ia58]
    S. V. Iablonskiĭ, Functional constructions in the k-valued logic, Trudy Mat. Inst. Steklov. 51 (1958), 5–142 (Russian).MathSciNetGoogle Scholar
  15. [Ib68]
    K. Ibuki, A study on universal logical elements, NTT — Inst. of Electrocomm. Report, 3747 (1968) (Japanese).Google Scholar
  16. [In82]
    K. Inagaki, t6-completeness of sets of delayed logic elements, Trans. IECE Japan J63-D (1982), 827–834 (Japanese).Google Scholar
  17. [Kr59]
    R. E. Krichevskiĭ, The realization of functions by superpositions, Problemy Kibernet. 2 (1959), 123–138 (Russian); Probleme der Kybernetik 2 (1963), 139–159 (German).Google Scholar
  18. [Ku60]
    V. B. Kudryavtsev, Completeness theorem for a class of automata without feedback couplings, Dokl. Akad. Nauk SSSR 132 (1960), 272–274 (Russian); Soviet Math. Dokl. 1 (1960), 537–539.MATHGoogle Scholar
  19. [Ku62]
    V. B. Kudryavtsev, Completeness theorem for a class of automata without feedback couplings, Problemy Kibernet. 8 (1962), 91–115 (Russian); Probleme der Kybernetik 8 (1965), 105–136 (German).Google Scholar
  20. [Ku65]
    V. B. Kudryavtsev, The power of sets of precomplete sets for certain functional systems connected with automata, Problemy Kibernet. 13 (1965), 45–74 (Russian).Google Scholar
  21. [Ku73]
    V. B. Kudryavtsev, On functional properties of logical nets, Math. Nachr. 55 (1973), 187–211 (Russian).MathSciNetGoogle Scholar
  22. [Ku82]
    V. B. Kudryavtsev, Functional Systems, Monograph, I.M.U., Moscow, 1982 (Russian).Google Scholar
  23. [Lo65]
    H. H. Loomis, Jr., Completeness of sets of delayed-logic devices, IEEE Trans. Electron. Comput. EC-14 (1965), 157–172.Google Scholar
  24. [MRR78]
    L. Martin, C. Reischer, and I. G. Rosenberg, Completeness problems for switching circuits constructed from delayed gates, Proc. 8th Internat. Symp. Multiple-Valued Logic, 1978, 142–148; Elektron. Informationsverarb. Kybernet., 19 (1983), 415, 171–186 (in French); An expanded French version appeared in: Réseaux modulaires, Monographies de l.U.Q.T.R. 11 1980.Google Scholar
  25. [Ma59]
    N. M. Martin, On the completeness of sets of gating functions with delay, in: Sympos. on Gating Algebra, Proc. Internat. Congr. Information Processing, Paris, 1959.Google Scholar
  26. [Mi84]
    M. Miličić, Galois connections for closed classes of functions with delays Publ. de l’Inst. Mathématique Nouvelle Série 36(50) (1984), Part I: 119–124; Part II: 125–136 (Russian).Google Scholar
  27. [Mi88]
    M. Miličić, On Galois connections for one place functions with delays, Publ. de l’Inst. Mathématique Nouvelle Série 44(58) (1988), 147–150 (Russian).Google Scholar
  28. [MSHMF88]
    M. Miyakawa, I. Stojmenović, T. Hikita, H. Machida, and R. Freivalds, Sheffer and symmetric Sheffer Boolean functions under various functional constructions, J. Inform. Process. Cybernet. EIK, (formerly: Elektron. Informationsverarb. Kybernet.) 24(6) (19881), 251–266.Google Scholar
  29. [No70a]
    A. Nozaki, Functional studies of automata I–II, Sci. Papers College Gen. Ed. Univ. Tokyo 20 (1970), 21–36, 109–121.MATHMathSciNetGoogle Scholar
  30. [No70b]
    A. Nozaki, Réalisation des fonctions definies dans un ensemble fini à l’aide des organes élémentaires d’entrée-sortie, Proc. Japan Acad. 46 (1970), 478–482.MATHMathSciNetGoogle Scholar
  31. [No72]
    A. Nozaki, Complete sets of switching elements and related topics, First USAJapan Computer Conference, 1972, 393–396.Google Scholar
  32. [No78]
    A. Nozaki, Functional completeness of multi-valued logical functions under uniform compositions, Rep. Fac. Eng. Yamanashi Univ. 29 (1978), 61–67.Google Scholar
  33. [No81]
    A. Nozaki, Completeness criteria for a set of delayed functions with or without non-uniform compositions, in: Finite Algebra and Multiple-Valued Logic (B. Csákány and I. Rosenberg, eds.), Coll. Math. Soc. János Bolyai 28, North-Holland, 1981, 489–519.Google Scholar
  34. [No82]
    A. Nozaki, Completeness of logical gates based on sequential circuits, Trans. IECE Japan J65-D (1982), 171–178 (Japanese).Google Scholar
  35. [PK79]
    R. Pöschel and L. A. Kaluzhnin, Funktionen-und Relationenalgebren, VEB Deutscher Verlag der Wissenschaften, Berlin, 1979.Google Scholar
  36. [Po21]
    E. L. Post, Introduction to a general theory of elementary propositions, Amer. J. Math. 43 (1921), 163–185.MATHMathSciNetGoogle Scholar
  37. [Ro65]
    I. G. Rosenberg, La structure des fonctions de plusieurs variables sur un ensemble fini, C.R. Acad. Sci. Paris Sér. A-B, 260 (1965), 3817–3819.MATHMathSciNetGoogle Scholar
  38. [Ro70]
    I. G. Rosenberg, Über die funktionale Vollständigkeit in dem mehrwertigen Logiken (Struktur der Funktionen von mehreren Veränderlichen auf endlichen Mengen), Rozpravy Československé Akad. Věd., Ser. Math. Nat. Sci., 80 (1970), 3–93.Google Scholar
  39. [Ro73]
    I. G. Rosenberg, The number of maximal closed classes in the set of functions over a finite domain, J. Comb. Theory 14 (1973), 1–7.CrossRefMATHGoogle Scholar
  40. [Ro77]
    I. G. Rosenberg, Completeness properties of multiple-valued logic algebras, in: Computer Science and Multiple-Valued Logic, Theory and Applications (D. C. Rine, ed.), North-Holland, 1977, 144–186; 2nd edition: ibid, 1984, 150–194.Google Scholar
  41. [Ro83]
    I. G. Rosenberg, The multi-faceted completeness problem of the structural theory of automata, preprint, CRMA-1197, Université de Montréal, 1983.Google Scholar
  42. [RH83]
    I. G. Rosenberg and T. Hikita, Completeness for uniformly delayed circuits, Proceedings 13th Internat. Sympos. on Multiple-Valued Logic, IEEE, Kyoto, 1983, 2–10; Proofs: Preprint, Tokyo Metropolitan Univ., 1983.Google Scholar
  43. [RS84]
    I. G. Rosenberg and L. Szabó, Local completeness I, Algebra Universalis 18 (1984), 308–326.CrossRefMathSciNetGoogle Scholar
  44. [Sł39]
    J. Słupecki, A completeness criterion in many-valued propositional calculi, C.R. Séance Soc. Sci. Varsovie 32 (1939), 102–109 (Polish); Studia Logica (1972), 153–157.Google Scholar
  45. [Sz86]
    Á. Szendrei, Clones in universal algebras, Les Presses de l’Université de Montréal, SMS, 1986.Google Scholar

Copyright information

© Springer 2005

Authors and Affiliations

  • Teruo Hikita
    • 1
  • Ivo G. Rosenberg
    • 2
  1. 1.Department of Computer ScienceMeiji UniversityKawasakiJapan
  2. 2.Département de mathématiques et de statistiqueUniversité de MontréalMontréalCanada

Personalised recommendations