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Combinatorica

, Volume 10, Issue 1, pp 81–93 | Cite as

Applications of matrix methods to the theory of lower bounds in computational complexity

  • A. A. Razborov
Article

Abstract

We present some criteria for obtaining lower bounds for the formula size of Boolean functions. In the monotone case we get the boundnΩ(logn) for the function “MINIMUM COVER” using methods considerably simpler than all previously known. In the general case we are only able to prove that the criteria yield an exponential lower bound when applied to almost all functions. Some connections with graph complexity and communication complexity are also given.

AMS subject classification (1980)

68Q30 

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References

  1. [1]
    A. E. АНДРЕЕВ, ОБ ОДНОМ МЕтОДЕ пОлУЧЕНИь НИжНИх ОцЕНОк слОж НОстИ ИНДИВИДУАльН ых МОНОтОННых ФУНкцИИ — ДАН сссР,1985, т. 282, N5, 1033–1037. (Engl. transl. in:Sov. Math. Dokl. 31, 530–534.)Google Scholar
  2. [2]
    A. E. АНДРЕЕВ, ОБ ОДНОМ МЕтОДЕ пОлУЧЕНИь ЁФФЕктИВНых НИжНИх ОцЕНОк МОНОтОННОИ слОжНОстИ, АлгЕБРА И лОгИкА,1987, т. 26, 1, с. 3–26.Google Scholar
  3. [3]
    А. А. РАжБОРОВ, НИжНИЕ ОцЕНкИ МОНОтОННОИ слОжНОстИ НЕкОтОРых БУлЕВых ФУНкцИИ — ДАН сссР,1985, т. 281, N4, с. 798–801. (Engl. transl. in:Sov. Math. Dokl. 31, 354–357.)Google Scholar
  4. [4]
    А. А. РАжБОРОВ, НИжНИЕ ОцЕНкИ МОНОтОННОИ слОжНОстИ лОгИЧЕс кОгО пЕРМАНЕНтА — “МАтЕМ. жАМ.”,1985, т. 37, Вып. 6, с. 887–900. (Engl. transl. in:Mathem. Notes of the Academy of Sci. of the USSR 37, 485–493.)Google Scholar
  5. [5]
    А. А. РАжБОРОВ, НИж НИЕ ОцЕНкИ РАжМЕРА сх ЕМ ОгРАНИЧЕННОИ глУБИНы В пОлНОМ БАжИсЕ, сОДЕРжАЩЕМ ФУНкцИУ лОгИЧЕскОгО слОжЕНИь —МАтЕМ. жАМ.,1987, т. 41, Вып. 4, с. 598–607. (Engl. transl. in:Mathem. Notes of the Academy of Sci. of the USSR. 41:4, 333–338.)Google Scholar
  6. [6]
    А. А.РАжБОРОВ, ФОРМУлы ОгРАНИЧЕННОИ глУБИНы В БАжИсЕ &, ⊕ И НЕкОтОРыЕ кОМБИНАтОРНыЕ жАДАЧИ — В сБ. «ВОпРОсы кИБЕРНЕтИкИ. слОж НОсть ВыЧИслЕНИИ И пРИклАДНАь МАтЕМАтИЧЕскАь лОг ИкА», М.:1988, с. 149–166.Google Scholar
  7. [7]
    к. л.РыНкОВ, МОДИФИ кАцИь МЕтОДА В. М. хРАпЧЕНкО И пРИМЕНЕНИЕ ЕЕ к ОцЕНк АМ слОжНОстИ п-схЕМ Дль кОДОВых ФУНкцИИ — В сБ. «МЕтОДы ДИскРЕтНОгО АНАлИжА В тЕОРИИ гРАФОВ И схЕМ», Вып. 42, НОВОсИБИРск,1985, с. 91–98.Google Scholar
  8. [8]
    В. М. хРАпЧЕНкО, О слОжНОстИ РЕАлИжАцИИ лИНЕИНОИ ФУНкцИИ В клАссЕ п-схЕМ,МАтЕМ. жАМ.,1971, т. 9, Вып. 1, с. 35–40. (Engl. transl. in:Mathem. Notes of the Academy of Sci. of the USSR 11 (1972), 474–479.)Google Scholar
  9. [9]
    A. V.Aho, J. D.Ullman, M.Yannakakis, On Notions of Information Transfer in VLSI Circuits —Proc. 15th ACM STOC,1983, 133–139.Google Scholar
  10. [10]
    N. Alon, R. B. Boppana, The monotone circuit complexity of Boolean functions,Combinatorica,1987, v. 7, N1, 1–22.Google Scholar
  11. [11]
    L.Babai, P.Frankl, J.Simon, Complexity classes in communication complexity theory,Proc. 27th IEEE FOCS,1986, 337–347.Google Scholar
  12. [12]
    A. Borodin, von zur Gathen, J. Hopcroft, Fast parallel matrix and GCD computations,Information and Control,52 (1982), 241–256.Google Scholar
  13. [13]
    B.Halsenberg, R.Reischuk, On Different Modes of Communication, 1988, 20th ACM STOC. 162–172.Google Scholar
  14. [14]
    M.Karchmer, A.Wigderson, Monotone Circuits for Connectivity Require Super-logarithmic Depth,Proc. 20th ACM STOC,1988, 539–550.Google Scholar
  15. [15]
    B. Lindström, H. O. Zetterström, A combinatorial problem in thek-adic number system,Proc. of the Amer. Math. Soc., 1967,18, 1, 166–170.Google Scholar
  16. [16]
    R. J.Lipton, R.Sedgewick, Lower Bounds for VLSI,Proc. 13th ACM STOC,1981, 300–307.Google Scholar
  17. [17]
    K.Mehlhorn, E. M.Schmidt, Las Vegas is better than determinism in VLSI and distributive computing,Proc. 14th ACM STOC,1982, 330–337.Google Scholar
  18. [18]
    P.Pudlak, V.Rödl,A combinatorial approach to complexity—unpublished manuscript,1989.Google Scholar
  19. [19]
    P. Pudlak, V. Rödl, P. Savicky, Graph Complexity,Acta Informatica,25 (1988), 515–535.Google Scholar
  20. [20]
    P. M. Spira, On time-hardware complexity tradeoffs for Boolean functions,Proceedings of 4th Hawaii Symposium on System Sciences,1971, Western Periodicals Company, North Hollywood, 525–527.Google Scholar
  21. [21]
    é. Tardos, The gap between monotone and non-monotone circuit complexity is exponential,Combinatorica,1988, v. 8,1, 141–142.Google Scholar
  22. [22]
    I. Wegeher, Relating monotone formula size and monotone depth of Boolean functions,Information Processing Letters,16, (1983), 41–42.Google Scholar
  23. [23]
    A. C.Yao, Some Complexity Questions Related to Distributed Computing,Proc. 11th ACM STOC,1979, 209–213.Google Scholar

Copyright information

© Akadémiai Kiadó 1990

Authors and Affiliations

  • A. A. Razborov
    • 1
  1. 1.Steklov Mathematical InstituteMoscowUSSR

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