Skip to main content
SpringerLink
Log in

Boolean function minimization in the class of disjunctive normal forms

  • Published:
Journal of Soviet Mathematics Aims and scope Submit manuscript

Abstract

The survey focuses on minimization of boolean functions in the class of disjunctive normal forms (d.n.f.s) and covers the publications from 1953 to 1986. The main emphasis is on the mathematical direction of research in boolean function minimization: bounds of parameters of boolean functions and algorithmic difficulties of minimal d.n.f. synthesis). The survey also presents a classification of minimization algorithms and gives some examples of minimization heuristics with their efficiency bounds.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

Literature cited

Publications in Russian and Russian translations

  1. G. P. Agibalov, “Minimizing the number of arguments of boolean functions,” in: Problems of Synthesis of Digital Automata [in Russian], Nauka, Moscow (1967), pp. 96–100.

    Google Scholar 

  2. F. I. Andon, “A simplification algorithm for d.n.f.s of boolean functions,” Kibernetika, No. 6, 12–14 (1966).

    Google Scholar 

  3. F. I. Andon, “On computer implementation of A-algorithms,” in: Topics of Theoretical Cybernetics [in Russian], Kiev (1965), pp. 49–54.

  4. A. E. Andreev, “On the synthesis of disjunctive normal forms close to minimal,” Dokl. AN SSSR,269, No. 1, 11–15 (1983).

    Google Scholar 

  5. A. E. Andreev, “On minimization of disjunctive normal forms,” Dokl. AN SSSR,274, No. 2, 265–269 (1984).

    Google Scholar 

  6. A. E. Andreev, “On one modification of the gradient algorithm,” Vestn. Mosk. Gos. Univ., Mat. Mekh., No. 3, 29–35 (1985).

    Google Scholar 

  7. L. A. Aslanyan, “On complexity of the reduced d.n.f. of incompletely specified boolean functions in boolean algebra, I,” Uch. Zap. Erevan. Univ., Estestvennye Nauki, No. 1, (125), 11–18 (1974).

    Google Scholar 

  8. L. A. Aslanyan, “On complexity of the reduced d.n.f. of incompletely specified boolean functions in boolean algebra, II,” Uch. Zap. Erevan. Univ., Estestvennye Nauki, No. 3, (127), 16–23 (1974).

    Google Scholar 

  9. A. K. Barinov, “Minimization of switching functions by means of graphs,” in: Instruments and Systems in Automation, Republican Interdepartmental Thematic Scientific-Technical Collection [in Russian], No. 28 (1973), pp. 34–46.

    Google Scholar 

  10. A. M. Bogomolov, “Methods of minimization of boolean functions using graph theory,” On New Achievements of Soviet Science, Abstracts and Communications, Scientific Conferences [in Russian], Donetsk (1966), pp. 205–206.

  11. S. V. Bykova, “A system of shortest covering algorithms,” Trudy Sibirskogo Fiz.-Tekh. Inst. Pri Tomsk. Univ., No. 64, 3–11 (1973).

    Google Scholar 

  12. S. Waligorsky, “Computer algorithms for simplification of boolean expressions,” Proc. Int. Symposium on the Theory of Relay Devices and Finite Automata, 1962, Synthesis of Relay Structures [in Russian], Moscow (1965), pp. 328–331.

  13. Yu. L. Vasil'ev, “On the number of irredundant and minimal d.n.f.s,” in: Discrete Analysis [in Russian], No. 2, Novosibirsk (1964), pp. 3–9.

    Google Scholar 

  14. Yu. L. Vasil'ev, “On the length of a cycle in n-dimensional unit cube,” Dokl. AN SSSR,148, No. 4, 753–756 (1963).

    Google Scholar 

  15. Yu. L. Vasil'ev, “On complexity comparison of irredundant and minimal d.n.f.s,” in: Problems of Cybernetics [in Russian], No. 10, Nauka, Moscow (1963), pp. 5–61.

    Google Scholar 

  16. Yu. L. Vasil'ev, “On ‘superposition’ of reduced d.n.f.s,” in: Problems of Cybernetics [in Russian], No. 12, Nauka, Moscow (1964), pp. 239–242.

    Google Scholar 

  17. Yu. L. Vasil'ev, “Difficulties of boolean function minimization by universal approaches,” Dokl. AN SSSR,171, No. 1, 13–16 (1966).

    Google Scholar 

  18. Yu. L. Vasil'ev and V. V. Glagolev, “Metric properties of disjunctive normal forms,” in: Discrete Mathematics and Mathematical Topics in Cybernetics [in Russian], Nauka, Moscow (1974), pp. 99–206.

    Google Scholar 

  19. Yu. L. Vasil'ev and V. K. Korobkov, “Metric studies in discrete analysis,” in: Problems of Cybernetics [in Russian], No. 27, Nauka, Moscow (1973), pp. 63–73.

    Google Scholar 

  20. K. Weber, “On various notions of minimality of disjunctive normal forms,” in: Problems of Cybernetics [in Russian], No. 36, Nauka, Moscow (1979), pp. 129–158.

    Google Scholar 

  21. A. P. Vikulin, “A bound of the number of conjunctions in a reduced d.n.f.,” in: Problems of Cybernetics [in Russian], No. 29, Nauka, Moscow (1974), pp. 151–156.

    Google Scholar 

  22. V. V. Voitishek, “On one approach to boolean function classification,” in: Discrete Analysis [in Russian], No. 8, Novosibirsk (1966), pp. 35–41.

    Google Scholar 

  23. E. K. Voishvillo, “A method of simplification of truth function expressions,” Nauchnye Doklady Vysshei Shkoly, Filos. Nauki, No. 2, 120–135 (1958).

    Google Scholar 

  24. M. A. Gavrilov, “Minimization of boolean functions characterizing relay circuits,” Avtomat. Telemekh.,20, No. 9, 1217–1238 (1959).

    Google Scholar 

  25. M. A. Gavrilov, “State of the art in the theory of relay circuits,” in: Structural Theory of Relay Devices [in Russian], Izd. AN SSSR, Moscow (1963), pp. 5–73.

    Google Scholar 

  26. M. M. Gadzhiev, “Maximum length of the reduced d.n.f. for boolean functions of five and six variables,” in: Discrete Analysis [in Russian], No. 18, Novosibirsk (1971), pp. 3–24.

    Google Scholar 

  27. V. V. Glagolev, “An upper bound of cycle length in n-dimensional unit cube,” in: Discrete Analysis [in Russian], No. 6, Novosibirsk (1966), pp. 3–7.

    Google Scholar 

  28. V. V. Glagolev, “An upper bound of the complexity of minimal d.n.f. for almost all boolean functions,” in: Discrete Analysis [in Russian], No. 5, Novosibirsk (1965), pp. 3–8.

    Google Scholar 

  29. V. V. Glagolev, “Some bounds of d.n.f.s of boolean functions,” in: Problems of Cybernetics [in Russian], No. 19, Nauka, Moscow (1967), pp. 75–94.

    Google Scholar 

  30. V. V. Glagolev, “On the length of the reduced d.n.f. for boolean functions of dimension 1,” in: Discrete Analysis [in Russian], No. 22, Novosibirsk (1973), pp. 29–33.

    Google Scholar 

  31. V. V. Glagolev, “On the length of an irredundant d.n.f.,” Mat. Zametki, No. 6, 665–672 (1967).

    Google Scholar 

  32. V. V. Glagolev, “A bound of complexity of the reduced d.n.f. for almost all boolean functions,” Dokl. AN SSSR,158, No. 4, 770–773 (1964).

    Google Scholar 

  33. V. V. Glagolev, “A lower bound of the number of irredundant d.n.f.s for almost all boolean functions,” in: Discrete Analysis [in Russian], No. 3, Novosibirsk (1964), pp. 31–40.

    Google Scholar 

  34. V. M. Glushkov, Synthesis of Digital Automata [in Russian], Fizmatgiz, Moscow (1962).

    Google Scholar 

  35. V. I. Gorodetskii and V. A. Boyar, “A method of boolean function minimization,” in: Stability and Reliability of Information Devices and Systems [in Russian], Tekhnika, Kiev (1975), pp. 43–50.

    Google Scholar 

  36. Yu. G. Grigor'yan, “An arithmetic method of boolean function minimization,” Seminar on Automata Theory [in Russian], Kiev (1964).

  37. I. B. Gurevich, “On noncomputability in the class of local algorithms of some predicates connected with the problem of boolean function minimization,” Kibernetika, No. 2, 24–30 (1974).

    Google Scholar 

  38. I. B. Gurevich and Yu. I. Zhuravlev, “Boolean function minimization and effective decision algorithms,” Kibernetika, No. 3, 16–20 (1974).

    Google Scholar 

  39. P. Ya. Danilenko and V. V. Umenskii, “A method of boolean function minimization,” Soobshch. AN GruzSSR,54, No. 1, 45–48 (1969).

    Google Scholar 

  40. V. P. Didenko, “Some methods of boolean function minimization,” in: Structural Theory of Relay Devices [in Russian], AN SSSR, Moscow (1963), pp. 148–162.

    Google Scholar 

  41. V. P. Didenko and K. P. Pokudin, “Boolean function minimization by metacore extraction,” in: Automatic Devices in Accounting and Control [in Russian], No. 6, Izhevsk (1970), pp. 50–71.

    Google Scholar 

  42. A. A. Evdokimov, “On maximum chain length in the n-dimensional unit cube,” Mat. Zametki,6, No. 3 (1969).

  43. É. V. Egiazaryan, “Quantitiative characteristics of systems of equations with partial boolean functions,” Dokl. AN ArmSSR,68, No. 2, 74–78 (1979).

    Google Scholar 

  44. Yu. I. Zhuravlev, “Algorithms of construction of minimal d.n.f.s for boolean functions,” in: Discrete Mathematics and Mathematical Topics in Cybernetics [in Russian], Nauka, Moscow (1974), pp. 67–82.

    Google Scholar 

  45. Yu. I. Zhuravlev, “Algorithms of simplification of disjunctive normal forms of finite index,” Dokl. AN SSSR,139, No. 6, 1329–1331 (1961).

    Google Scholar 

  46. Yu. I. Zhuravlev, “Local algorithms for computation of information, I,” Kibernetika, No. 1, 12–19 (1965).

    Google Scholar 

  47. Yu. I. Zhuravlev, “Local algorithms for computation of information, II,” Kibernetika, No. 2, 1–11 (1966).

    Google Scholar 

  48. Yu. I. Zhuravlev, “On algorithms identifying collections of essential variables of incompletely specified boolean functions,” in: Problems of Cybernetics [in Russian], No. 11, Nauka, Moscow (1964), pp. 271–275.

    Google Scholar 

  49. Yu. I. Zhuravlev, “On algorithms of simplification of disjunctive normal forms,” Dokl. AN SSSR,132, No. 2, 260–263 (1960).

    Google Scholar 

  50. Yu. I. Zhuravlev, “On one class of incompletely specified boolean functions,” in: Discrete Analysis [in Russian], No. 2, Novosibirsk (1964), pp. 23–27.

    Google Scholar 

  51. Yu. I. Zhuravlev, “On separability of vertex subsets of the n-dimensional unit cube,” Trudy Mat. Inst. AN SSSR,51, 143–157 (1958).

    Google Scholar 

  52. Yu. I. Zhuravlev, “On impossibility of construction of minimal disjunctive normal forms of boolean functions in one class of algorithms,” Dokl. AN SSSR,132, No. 3, 504–506 (1960).

    Google Scholar 

  53. Yu. I. Zhuravlev, “On inessential variables of incompletely specified boolean functions,” in: Discrete Analysis [in Russian], No. 1, Novosibirsk (1963), 28–31.

    Google Scholar 

  54. Yu. I. Zhuravlev, “Bounds of the number of irredundant disjunctive normal forms of boolean functions,” Sib. Mat. Zh.,3, No. 5, 802–804 (1962).

    Google Scholar 

  55. Yu. I. Zhuravlev, “Complexity bounds of algorithms for construction of minimal d.n.f.s for boolean functions,” in: Discrete Analysis [in Russian], No. 3, Novosibirsk (1964), pp. 41–47.

    Google Scholar 

  56. Yu. I. Zhuravlev, “Complexity bounds of local algorithms for some extremal problems on finite sets,” Dokl. AN SSSR,158, No. 5, 1018–1021 (1964).

    Google Scholar 

  57. Yu. I. Zhuravlev, “On various notions of minimality of d.n.f.s,” Sib. Mat. Zh.,1, No. 4, 609–611 (1960).

    Google Scholar 

  58. Yu. I. Zhuravlev, “Set-theoretical methods in boolean algebra,” in: Problems of Cybernetics [in Russian], No. 8, Fizmatgiz, Moscow (1962), pp. 5–44.

    Google Scholar 

  59. Yu. I. Zhuravlev and A. Yu. Kogan, “Realization of boolean functions with few zeros by disjunctivenormal forms and related problems,” Dokl. AN SSSR,285, No. 4, 795–799 (1985).

    Google Scholar 

  60. L. A. Zhuravleva, “On some algorithms of constructions of reduced d.n.f.s for boolean functions of n variables,” in: Some Topics of Automated Programming [in Russian], Nauka, Novosibirsk (1970), pp. 109–124.

    Google Scholar 

  61. L. A. Zhuravleva, “Construction of the reduced d.n.f. by Blake's method,” in: Discrete Analysis [in Russian], No. 10, Novosibirsk (1967), pp. 23–38.

    Google Scholar 

  62. A. D. Zakrevskii, Logic Design of Cascade Circuits [in Russian], Nauka, Moscow (1981).

    Google Scholar 

  63. A. D. Zakrevskii, “Minimization of disjunctive normal forms of boolean functions,” Izv. AN SSSR, Tekh. Kibern., No. 4, 102–104 (1970).

    Google Scholar 

  64. A. D. Zakrevskii, Synthesis Algorithms of Discrete Automata [in Russian], Nauka, Moscow (1971).

    Google Scholar 

  65. A. D. Zakrevskii, “New minimization algorithms for weakly specified boolean functions,” Kibernetika, No. 5, 21–28 (1969).

    Google Scholar 

  66. A. D. Zakrevskii, “On enumeration reduction for the solution of some synthesis problems of digital automata,” Radiofizika,7, No. 1, 166–174 (1964).

    Google Scholar 

  67. A. D. Zakrevskii and V. I. Ostrovskii, “Optimization of the search for shortest covering,” in: Problems of Synthesis of Digital Automata [in Russian], Nauka, Moscow (1967), pp. 84–95.

    Google Scholar 

  68. V. D. Kazakov, “Minimization of logic functions of many variables,” Avtomat. Telemekh.,23, No. 9, 1237–1242 (1962).

    Google Scholar 

  69. V. D. Kazakov, “Finding the maximum number of prime implicants of an arbitrary logic function of n variables,” in: Automatic Control [in Russian], AN SSSR, Moscow (1960), pp. 330–338.

    Google Scholar 

  70. V. D. Kazakov, “Finding minimal normal forms of logic functions by the method of restricted enumeration,” in: Structural Theory of Relay Devices [in Russian], AN SSSR, Moscow (1963), pp. 145–147.

    Google Scholar 

  71. V. I. Kaznacheev, N. P. Murav'ev, and N. V. Murav'eva, “An analytical method for boolean function minization,” Trudy NII Grazhd. Aviatsii, No. 83, 31–41 (1973).

    Google Scholar 

  72. L. M. Karakhanyan, “On efficiency of extraction of prime implicants of minimum rank,” in: Combinatorial-Algebraic Methods in Applied Mathematics [in Russian], Gor'kii (1982), pp. 66–75.

  73. L. M. Karakhanyan, “Relative complexity of uniquely specified d.n.f.s of partial boolean functions,” in: Optimization and Control [in Russian], Moscow State Univ. (1983), pp. 85–97.

  74. L. M. Karakhanyan, “On relative efficiency of the gradient algorithms of minimization of partial boolean functions,” in: Software of Computerized Information Systems [in Russian], Moscow State Univ. (1984), pp. 97–116.

  75. L. M. Karakhanyan, “Metric comparison of various minimal d.n.f.s of partial boolean functions,” Sbornik Trudov Inst. Mat. SO AN SSSR, No. 38, 19–36 (1982).

    Google Scholar 

  76. L. M. Karakhanyan and A. A. Sapozhenko, “Parameter bounds of incompletely specified (partial) boolean functions,” in: Combinatorial-Algebraic Methods in Applied Mathematics [in Russian], Gor'kii (1979), pp. 48–56.

  77. A. D. Korshunov, “Upper bound of complexity of shortest d.n.f.s for almost all boolean functions,” Kibernetika, No. 6, 1–8 (1969).

    Google Scholar 

  78. A. D. Korshunov, “Comparison of complexity of longest and shortest d.n.f.s and lower bound of the number of irredundant d.n.f.s for almost all boolean functions,” Kibernetika, No. 4, 1–11 (1969).

    Google Scholar 

  79. A. D. Korshunov, “On complexity of shortest disjunctive normal forms of boolean functions,” Sb. Trudov Inst. Mat. SO AN SSSR, No. 37, 9–41 (1981).

    Google Scholar 

  80. A. D. Korshunov, “On complexity of shortest disjunctive normal forms of random boolean functions,” Sbornik Trudov Mat. Inst. SO AN SSSR, No. 40, 25–53 (1983).

    Google Scholar 

  81. É. Sh. Kospanov, “On the product of shortest d.n.f.s,” in: Discrete Analysis [in Russian], No. 18, Novosibirsk (1971), pp. 35–40.

    Google Scholar 

  82. O. S. Krakovskaya and T. L. Maistrova, “On a method of minimization of normal forms of boolean functions,” in: Formal Logic and Methodology of Science [in Russian], Moscow (1964), pp. 269–300.

  83. S. E. Kuznetsov, “On lower bound of length of shortest d.n.f. of almost all boolean functions,” Veroyatn. Metody i Kibernet. (Kazan'), No. 19, 44–47 (1983).

    Google Scholar 

  84. S. A. Kuk, “Complexity of theorem proving procedures,” Kibern. Sb. (new series), No. 12, 5–15 (1975).

    Google Scholar 

  85. A. A. Levin, “On relative complexity of the reduced d.n.f.,” in: Discrete Analysis [in Russian], No. 15, Novosibirsk (1969), pp. 25–34.

    Google Scholar 

  86. A. A. Levin, “On complexity ratio of the d.n.f. of a function to the d.n.f. of its negation,” in: Discrete Analysis [in Russian], No. 16, Novosibirsk (1970), pp. 77–81.

    Google Scholar 

  87. A. A. Levin, “Complexity of the d.n.f. of the sum of irredundant d.n.f.s relative to the d.n.f. of the sum of minimal d.n.f.s,” in: Discrete Analysis, No. 24, Novosibirsk (1974), pp. 50–68.

    Google Scholar 

  88. A. A. Levin, “Comparative complexity of d.n.f.s,” Sbornik Trudov Inst. Mat. SO AN SSSR, No. 36, 23–38 (1981).

    Google Scholar 

  89. A. A. Levin, “Join of boolean functions and its application for bounds of comparative complexity of d.n.f.s,” Sbornik Trudov Inst. Mat. SO AN SSSR, Novosibirsk, No. 40, 54–71 (1983).

    Google Scholar 

  90. Lin Sin-Lyan, “On comparison of complexities of minimal and shortest disjunctive normal forms for boolean functions,” in: Problems of Cybernetics [in Russian], No. 18, Nauka, Moscow (1967), pp. 11–44.

    Google Scholar 

  91. G. F. Losev, “The best local algorithm of index 1 for constructing the sum of irredundant d.n.f.s of a boolean function,” Dokl. AN SSSR,212, No. 4, 816–817 (1973).

    Google Scholar 

  92. G. F. Losev, “The best local algorithm for constructing the sum of irredundant d.n.f.s of a boolean function using neighborhods of minimum order,” Zh. Vychisl. Mat. Mat. Fiz.,14, No. 2, 470–478 (1974).

    Google Scholar 

  93. G. F. Losev, “Constrained-majorizing local algorithms with arbitrary memory for problems of minimum covering synthesis,” Zh. Vychisl. Mat. Mat. Fiz.,14, No. 3, 744–745 (1974).

    Google Scholar 

  94. I. R. Losev, “Minimization of weakly specified boolean functions by extracting families of combinations,” in: Automatic Devices in Accounting and Control [in Russian], No. 6, Izhevsk (1970), pp. 3–10.

    Google Scholar 

  95. I. R. Losev and Yu. L. Teptin, “On some minimization algorithms for weakly specified boolean functions,” in: Automatic Devices in Accounting and Control [in Russian], No. 6, Izhevsk (1970), pp. 32–49.

    Google Scholar 

  96. O. B. Lupanov, “On realization of boolean functions by formulas of finite classes (formulas of bounded depth) in the basis AND, OR, NOT,” in: Problems of Cybernetics [in Russian], No. 6, Fizmatgiz, Moscow (1961), pp. 5–14.

    Google Scholar 

  97. O. B. Lupanov, “On synthesis of some classes of control systems,” in: Problems of Cybernetics [in Russian], No. 10, Fizmatgiz, Moscow (1963), pp. 63–97.

    Google Scholar 

  98. G. G. Lyubchenko and V. S. Podlipenskii, “Minimization algorithms of many-place switching functions,” in: Techniques of Automatic Data Transmission and Processing [in Russian], Tekhnika, Kiev (1974), pp. 35–43.

    Google Scholar 

  99. G. G. Lyubchenko and V. S. Podlipenskii, “Method of finding minimal normal forms of boolean functions,” Vestnik Kiev. Politekh. Inst., ser. Avtomatika i Élektropriborostr., No. 11, 184–189 (1974).

    Google Scholar 

  100. T. L. Maistrova, “Linear programming and the problem of minimization of normal forms of boolean functions,” in: Problems of Information Transmission [in Russian], No. 12, AN SSSR, Moscow (1962), 5–15.

    Google Scholar 

  101. S. V. Makarenkov, “Joint minimization of boolean functions in the class of lattice d.n.f.s using the relation frequency matrix,” in: Mathematical-Economic Methods and Programming of Planning and Economic Problems [in Russian], Moscow (1972), pp. 40–48.

  102. A. V. Markovskii, “Minimization of systems of boolean functions in the class of d.n.f.s,” in: Abstract and Structural Theory of Relay Devices [in Russian], Nauka, Moscow (1972), pp. 167–181.

    Google Scholar 

  103. A. V. Markovskii, “Approximate minimization of systems of weakly specified boolean functions in the class of d.n.f.s,” Informatsionnye Materialy, Nauchnyi Sovet po Kompleksnoi Probleme “Kibernetika” AN SSSR, No. 7(54), 115–116 (1971).

    Google Scholar 

  104. S. P. Matsyulyavichus, “Matrix method of boolean function minimization,” in: Problems of Computer Logic Design [in Russian], Part 1, Vil'nyus (1974), pp. 86–91.

  105. S. S. Medvedev, “8-ary charts for boolean function minimization,” Automata Theory, Seminar [in Russian], No. 1, Kiev (1966), pp. 54–69.

    Google Scholar 

  106. L. Mikheeva and Kh. Salum, “Construction of reduced d.n.f. by the method of masks,” Izv. AN EstSSR, Fiz., Mat.,18, No. 4, 458–460 (1969).

    Google Scholar 

  107. R. V. Mozharov, “On statistical study of minimization of boolean functions,” in: Discrete Analysis [in Russian], No. 5, Novosibirsk (1965), pp. 31–33.

    Google Scholar 

  108. R. V. Mozharov, “A study of minimization of random boolean functions,” Trudy Moskovsk. Inst. Inzh. Zh.-D. Transporta, No. 209, 182–183 (1965).

    Google Scholar 

  109. Nguen Kim Ang, “On some characteristics of algorithms of boolean function minimization,” Dokl. AN SSSR,267, No. 5, 1058–1062 (1982).

    Google Scholar 

  110. R. G. Nigmatullin, “The variational principle in boolean algebra,” in: Discrete Analysis [in Russian], No. 10, Novosibirsk (1967), pp. 69–89.

    Google Scholar 

  111. R. G. Nigmatullin, “Fastest descent method in covering problems,” in: Topics of Accuracy and Efficiency of Algorithms [in Russian], Proc. of a Symposium, No. 5, Kiev (1969), pp. 116–126.

    Google Scholar 

  112. R. G. Nigmatullin, “On variation of complexity of shortest disjunctive normal form on unit sphere,” in: Discrete Analysis [in Russian], No. 6, Novosibirsk (1966), pp. 69–80.

    Google Scholar 

  113. R. G. Nigmatullin, “Method of successive simplification of a boolean function,” Abstracts of Ukrainian Republican Conf. of Young Researchers in Theoretical Cybernetics [in Russian], Kiev (1965), pp. 27–39.

  114. V. G. Novoselov, “Minimization of boolean functions (a survey),” in: Progress of Research in Cybernetics [in Russian], Tomsk (1968), pp. 40–60.

  115. V. G. Novoselov, “Statistical efficiency bounds of methods of boolean function minimization,” in: Automata, Hybrid and Control Computers [in Russian], Nauka, Moscow (1972), pp. 96–104.

    Google Scholar 

  116. A. N. Nurlybaev, “On simplification of normal forms of k-valued logic,” Collected Papers in Mathematical Cybernetics [in Russian], No. 1, VTs AN SSSR (1976), pp. 56–68.

  117. V. A. Orlov, “Algorithmic undecidability of the problem of finding the asymptotic behavior of Shannon function in realization of bounded deterministic operators by circuits in an arbitrary basis,” Dokl. AN SSSR,196, No. 5, 1036–1039 (1971).

    Google Scholar 

  118. A. V. Pastukhov, “On minimization of boolean functions by decoding,” in: Topics of Computer Science [in Russian], Moscow (1969), pp. 45–54.

  119. S. Pashev, “A method of boolean function minimization in synthesis of digital devices,” Fiz.-Mat. Spisanie,7, No. 2, 134–143 (1964).

    Google Scholar 

  120. I. I. Percheklii, “Finding the prime implicants entering some minimal d.n.f.s of incompletely specified boolean functions,” in: Proc. of 5th Sci.-Tech. Conf. of Kishinev Polytech. Inst. [in Russian], Kishinev (1969), pp. 237–238.

  121. I. I. Percheklii, “Polynomial decidability of some algorithmic problems relating to d.n.f.s,” Abstracts of Papers of 3rd All-Union Conf. on Problems of Theoretical Cybernetics [in Russian], Novosibirsk (1974), pp. 115–116.

  122. V. G. Persheev, “Minimization of boolean functions of many variables,” Trudy Moskovsk. Inst. Zh.-D. Transporta, No. 367, 293–299 (1970).

    Google Scholar 

  123. A. F. Petrenko and G. F. Fritsnovich, “Construction of shortest irredundant d.n.f.s of boolean functions,” in: Discrete Systems [in Russian], Vol. 1, Zinatne, Riga (1974), pp. 193–204.

    Google Scholar 

  124. G. N. Povarov, “On mathematical theory of synthesis of contact (1, qk)-pole circuits,” Dokl. AN SSSR,111, No. 1, 102–104 (1956).

    Google Scholar 

  125. G. N. Povarov, “A method of boolean function minimization,” Izv. AN SSSR, Tekh. Kibern., No. 4, 198–201 (1982).

    Google Scholar 

  126. V. A. Popov, N. G. Moklyak, and I. T. Skibenko, “On a method of boolean function minimization,” Samoletostr. Tekhn. Vozd. Flota, Resp. Mezhved. Temat. Nauchno-Tekhn. Sb., No. 33, 34–39 (1974).

    Google Scholar 

  127. P. S. Poretskii, “On a method of solution of logic equalities and on the inverse method of mathematical logic,” Collection of Protocols of Sessions of the Section of Physico-Mathematical Sciences of the Society of Natural Scientists, Kazan Univ. [in Russian], Vol. 2 (1884).

  128. D. A. Pospelov, “Questions of minimization of normal forms of boolean functions on modern digital computers,” Proc. Int. Symp. on Theory of Relay Structures [in Russian], Moscow (1965), pp. 328–331.

  129. D. A. Pospelov, Logical Methods of Circuit Analysis and Synthesis [in Russian], Énergiya (1974).

  130. Yu. N. Potepalov and A. V. Sazonov, “Minimization of a system of weakly specified boolean functions in the class of d.n.f.s,” in: Computer Software [in Russian], No. 1, Moscow (1973), pp. 116–119.

    Google Scholar 

  131. A. A. Sapozhenko, “Geometrical construction of almost all boolean functions,” in: Problems of Cybernetics [in Russian], No. 30, Nauka, Moscow (1975), pp. 227–261.

    Google Scholar 

  132. A. A. Sapozhenko, Disjunctive Normal Forms, Metric Theory [in Russian], Lecture Notes, Moscow State Univ. (1975).

  133. A. A. Sapozhenko, “Metric properties of almost all boolean functions,” in: Discrete Analysis [in Russian], No. 10, Novosibirsk (1967), pp. 91–119.

    Google Scholar 

  134. A. A. Sapozhenko, “On maximum length of irredundant disjunctive normal form of almost all boolean functions,” Mat. Zametki,4, No. 6, 649–658 (1968).

    Google Scholar 

  135. A. A. Sapozhenko, “On complexity of disjunctive normal forms obtained by the gradient algorithm,” in: Discrete Analysis [in Russian], No. 21, Novosibirsk (1972), pp. 62–71.

    Google Scholar 

  136. A. A. Sapozhenko, “On the number of classes of boolean functions,” in: Discrete Analysis [in Russian], No. 9, Novosibirsk (1967), pp. 59–66.

    Google Scholar 

  137. A. A. Sapozhenko, “Bounds of length and number of irredundant d.n.f.s for almost all incompletely specified functions,” Mat. Zametki,28, No. 2, 279–300 (1980).

    Google Scholar 

  138. A. A. Sapozhenko and L. M. Karakhanyan, “On negative effects associated with elimination of inessential variables,” Avtomat. Vychisl. Tekhn., No. 3, 28–35 (1981).

    Google Scholar 

  139. I. É. Strazdin', “Affine classification of boolean functions of five variables,” Avtomat. Vychisl. Tekh., No. 1, 1–9 (1975).

    Google Scholar 

  140. I. É. Strazdin' and D. P. Strazdinya, “On possibilities of thickening the Harvard classification of boolean functions,” in: Theory of Finite Automata and Its Applications [in Russian], No. 2, Zinatne, Riga (1973), pp. 24–30.

    Google Scholar 

  141. É. Toman, “Geometrical construction of random boolean functions,” in: Problems of Cybernetics [in Russian], No. 35, Nauka, Moscow (1979), pp. 111–132.

    Google Scholar 

  142. I. A. Chegis and S. V. Yablonskii, “Logic methods of monitoring the operation of electrical circuits,” Trudy Mat. Inst. AN SSSR,51, 270–360 (1958).

    Google Scholar 

  143. Yu. O. Chernyshev and V. Ya. Nasekin, “On the solution of covering problems by the gradient algorithm,” Kibernetika, No. 4, 85–88 (1976).

    Google Scholar 

  144. Yu. O. Chernyshev and V. Ya. Nasekin, “Reduction of the problem of choosing maximum intervals to finding the shortest path,” Izv. Vyssh. Uchebn. Zaved., Élektromekh., No. 3, 235–238 (1974).

    Google Scholar 

  145. I. P. Chukhrov, “Bounds of numerical characteristics of search functions,” 5th All-Union Conf. on Problems of Theoretical Cybernetics, abstracts of papers [in Russian], Novosibirsk (1980), pp. 179–180.

  146. I. P. Chukhrov, “Bounds of the number of minimal disjunctive normal forms for search function, I,” Sb. Trudov Inst. Mat. SO AN SSSR, No. 36, 74–92 (1981).

    Google Scholar 

  147. I. P. Chukhrov, “On the number of irredundant disjunctive normal forms,” Dokl. AN SSSR,262, No. 6, 1329–1332 (1982).

    Google Scholar 

  148. I. P. Chukhrov, “On the number of minimal disjunctive normal forms,” Dokl. AN SSSR,276, No. 6, 1335–1339 (1984).

    Google Scholar 

  149. N. A. Shimko, “On bounds of complexity of irredundant d.n.f.s,” in: Problems of Cybernetics [in Russian], No. 19, Nauka, Moscow (1967), pp. 311–315.

    Google Scholar 

  150. S. V. Yablonskii, “On impossibility of eliminating the enumeration of all functions from P2 when solving some problems of circuit theory,” Dokl. AN SSSR,124, No. 1, 44–47 (1959).

    Google Scholar 

  151. S. V. Yablonskii, “On bounds of the length of irredundant d.n.f.s,” in: Problems of Cybernetics [in Russian], No. 7, Moscow (1962), pp. 229–232.

    Google Scholar 

  152. S. V. Yablonskii, “On algorithmic difficulties of synthesis of minimal contact circuits,” in: Problems of Cybernetics [in Russian], No. 2, Moscow (1959), pp. 75–121.

    Google Scholar 

  153. S. V. Yabslonskii, “Function constructions in k-valued logic,” Trudy Mat. Inst. AN SSSR,51, 5–142 (1958).

    Google Scholar 

  154. S. V. Yablonskii and I. A. Chegis, “On tests for electric circuits,” Usp. Mat. Nauk,10, No. 4, 182–184 (1955).

    Google Scholar 

Publications in other languages

  1. H. H. Aiken and Staff of the Computational Laboratory, Harvard Univ., Synthesis of Electronic Computing and Control Circuits, Harvard Univ. Press, Cambridge, Mass. (1951).

    Google Scholar 

  2. S. B. Akers, “A truth table method for the synthesis of combinational logic,” IRE Trans., Electron. Comput.,10, No. 4, 604–615 (1961).

    Google Scholar 

  3. A. Blake, Canonical Expressions in Boolean Algebra, Dissertation, Chicago (1937).

  4. P. M. Bowman and E. S. McVey, “A method for fast approximate solution of large prime implicant charts,” IEEE Trans., Comput.,19, No. 2, 169–173 (1970).

    Google Scholar 

  5. H. Burwick, “Erregung der irredundanten Normalformenaussagen logischer Ausdrücke,” Elektron. Rechenanlag.,5, No. 3, 108–117 (1963).

    Google Scholar 

  6. G. Centkowsky, “Minimizacia form boolowskich metoda liezb strukturalnych,” Pr. Nauk. P. Wars., No. 19, 47–63 (1975).

    Google Scholar 

  7. A. K. Chandra and G. Markowsky, “On the number of prime implicants,” Discrete Math.,24, No. 1, 7–11 (1978).

    Google Scholar 

  8. D. M. Chang and T. N. Mott, “Computing irredundant normal forms from abbreviated presence functions,” IEEE Trans., Comput.,EC-14, No. 3, (1965).

  9. A. K. Choudhury and M. S. Basu, “A mechnized chart for simplification of switching functions,” IRE Trans., Electron. Comput.,11, No. 5, 713–714 (1962).

    Google Scholar 

  10. A. K. Choudhury and M. S. Basu, “On the minimization of Boolean functions,” Indian J. Phys.,36, No. 1, 1–12 (1962).

    Google Scholar 

  11. A. K. Choudhury and S. R. Das, “Direct determination of all the minimal prime implicant covers of switching functions,” I. Electr. Contr.,17, No. 5, 553–576 (1964).

    Google Scholar 

  12. J. T. Chu, “Some methods for simplifying switching circuits using ‘don't care’ conditions,” J. ACM,8, No. 4, 497–512 (1961).

    Google Scholar 

  13. S. R. Das and A. K. Choudhury, “Maxterm type expressions of switching functions and their prime implicants,” IEEE Trans., Comput.,EC-14, No. 6, 920–923 (1965).

    Google Scholar 

  14. S. R. Das, A. K. Choudhury, and K. K. Roy, “Simpification of switching functions, involving a very large number of don't care states,” Intern. J. Control,3, No. 1, 17–18 (1966).

    Google Scholar 

  15. B. Dunham and R. Fridshal, “The problem of simplifying logical expressions,” J. Symbol. Logic,24, No. 1, 17–19 (1959).

    Google Scholar 

  16. S. Even, “On correspondence between simplest disjunctive and conjunctive forms and simplest stroke and dagger functions,” IRE Trans., Circuit Theory,CT-8, No. 4, 489 (1961).

    Google Scholar 

  17. M. J. Gazala, “Irredundant disjunctive and conjunctive forms of a Boolean function,” IBM J. Res. Dev.,1, No. 2, 171–176 (1957).

    Google Scholar 

  18. J. Gimpel, “A reduction technique for prime implicant tables,” IEEE Trans., Comput.,EC-14, No. 4, 534–541 (1965).

    Google Scholar 

  19. R. Hockney, “An intersection algorithm dividing all irredundant normal forms from a prime implicant list,” IRE Trans., Electron. Comput.,11, No. 2, 289–290 (1962).

    Google Scholar 

  20. H. R. Hwa, “A method for generating prime implicants of a boolean expression,” IEEE Trans., Comput.,EC-23, No. 6, 637–641 (1974).

    Google Scholar 

  21. A. Jakobschuk, “Graphical simplification of switching functions with more than four variables,” Electron. Lett., No. 22, 566 (1969).

    Google Scholar 

  22. M. Karnaugh, “The map method for synthesis of combinatorial logic circuits,” Trans. Am. Inst. Electron. Eng.,72, No. 1, 593–599 (1953).

    Google Scholar 

  23. J. B. Klahn and C. H. Roth, “Evaluation of Akers method for simplification of switching functions,” IEEE Trans., Comput.,ED-18, No. 10, 959–960 (1969).

    Google Scholar 

  24. V. Kudielka, “Ein Verfahren zur Ermittlung aller nicht redundanten zweistufigen Darstellungen einer logischen Funktion,” Electron. Rechenanlag.,5, No. 1, 11–21 (1963).

    Google Scholar 

  25. E. J. McCluskey, Jr., “Minimization of Boolean functions,” Bell Syst. Techn. J.,35, No. 6, 1417–1444 (1956).

    Google Scholar 

  26. E. J. McCluskey and H. Schorr, “Essential multiple-output prime implicants,” Microwave Res. Inst. Symp. Ser. 12 (1962), pp. 437–457.

    Google Scholar 

  27. F. Mileto and G. Putzolu, “Average values of quantities appearing in Boolean function minimization,” IEEE Trans., Comput.,EC-13, No. 2, 87–92 (1964).

    Google Scholar 

  28. F. Mileto and G. Putzolu, “Average values of quantities appearing in multiple-output Boolean minimization,” IEEE Trans., Comput.,EC-14, No. 4, 542–552 (1965).

    Google Scholar 

  29. F. Mileto and G. Putzolu, “Statistical complexity of algorithms for Boolean function minimization,” J. ACM,12, No. 3, 364–375 (1965).

    Google Scholar 

  30. E. Morreale, “Partitioned list algorithms for prime implicant determination from canonical forms,” IEEE Trans., Comput.,EC-16, No. 5, 611–620 (1967).

    Google Scholar 

  31. E. Morreale, “Recursive operators for prime implicant and irredundant normal form determination,” IEEE Trans., Comput.,EC-19, No. 6, 504–509 (1970).

    Google Scholar 

  32. T. H. Mott, “Determination of the irredundant normal forms of a truth function by iterated consensus of the prime implicants,” IRE Trans., Electron. Comput.,9, No. 2, 245–252 (1960).

    Google Scholar 

  33. T. H. Mott, “An algorithm for determining minimal forms of an incomplete truth function,” Commun. Electron., No. 53, 73–76 (1961).

    Google Scholar 

  34. T. H. Mott and C. C. Carrol, “Numerical procedures for Boolean function minimization,” IEEE Trans., Comput.,EC-13, No. 4, 470–471 (1964).

    Google Scholar 

  35. T. Mukherjee and P. K. Sarkar, “On a method of finding the irredundant forms of a Boolean function,” J. Electron. Control,14, No. 5, 563–580 (1963).

    Google Scholar 

  36. A. Mukhopadhyay, “A method of determination of all the minimal forms of Boolean functions,” Proc. Inst. Electr. Eng., C, No. 487E, 6 (1961).

    Google Scholar 

  37. A. Mukhopadhyay, “A method of determination of all the minimal forms of Boolean functions,” Proc. Inst. Electr. Eng.,C109, No. 15, 250–255 (1962).

    Google Scholar 

  38. R. J. Nelson, “Simplest normal truth functions,” J. Symbol. Logic,20, No. 2, 105–108 (1955).

    Google Scholar 

  39. B. Papp, “Procede pour determiner les formes normales minimales des fonctions booleenes en utilisant les regles de minimisation de la fonction de cout,” Acta Cybernet.,1, No. 4, 241–250 (1972).

    Google Scholar 

  40. K. Pasztor-Barga, “On some minimizing algorithm of Boolean functions,” Acta Techn. Acad. Sci. Hung.,73, No. 3–4, 349–362 (1972).

    Google Scholar 

  41. S. R. Petrick, “A direct determination of the irredundant forms of a Boolean function from the prime implicants,” AFCR 56-10 (1956).

  42. I. B. Pyne and E. J. McCluskey, “The reduction of redundancy in solving prime implicant tables,” IRE Trans., Electron. Comput.,11, No. 3, 473–482 (1962).

    Google Scholar 

  43. W. V. Quine, “A way to simplify truth functions,” Am. Math. Mon.,62, No. 9, 627–631 (1955).

    Google Scholar 

  44. W. V. Quine, “The problem of simplifying truth functions,” Am. Math. Mon.,59, 521–531 (1952).

    Google Scholar 

  45. W. V. Quine, “On cores and prime implicants of truth functions,” Am. Math. Mon.,66, No. 9, 755–760 (1959).

    Google Scholar 

  46. E. W. Samson and L. Calabi, “On the theory of Boolean formulas: minimal including sums. I,” J. SIAM,11, No. 2, 212–223 (1963).

    Google Scholar 

  47. E. W. Samson and L. Calabi, “On the theory of Boolean formulas: minimal including sums. II,” J. SIAM,11, No. 2, 224–234 (1963).

    Google Scholar 

  48. J. Spencer, “Asymptotically good coverings,” Pac. J. Math.,118, No. 2, 575–586 (1985).

    Google Scholar 

  49. A. Svoboda, “Graphicomechanical aids for the synthesis of relay circuits,” Ber. Intern. Math.-Kolloq. 1955 (1957), pp. 43–50.

  50. A. Svoboda, “Ordering of implicants,” IEEE Trans., Comput.,EC-16, No. 1, 100–105 (1967).

    Google Scholar 

  51. N. C. Troye, “Classification and minimization of switching functions,” Philips Res. Repts.,14, No. 2, 151–193, No. 3, 250–292 (1959).

    Google Scholar 

  52. R. H. Urbano and R. K. Mueller, “A topological method for the determination of the minimal forms of a Boolean function,” IRE Trans., Electron. Comput.,5, No. 3, 126–132 (1956).

    Google Scholar 

  53. E. W. Veitch, “A chart method for simplifying truth functions,” Proc. ACM, 127–133 (1952).

  54. S. Waligorsky, “Calculation of prime implicants of truth functions,” Algorithmy,1, No. 2, 127–133 (1963).

    Google Scholar 

  55. S. Waligorsky, “Calculation of the Quine's table for truth functions,” Prace ZAM PAN,A2, No. 15, 5 (1961).

    Google Scholar 

  56. K. Weber, “The length of random Boolean functions,” Elektron. Inf. Kybern.,18, No. 12, 659–668 (1982).

    Google Scholar 

  57. K. Weber, “Subcubes of random Boolean functions,” Elektron. Inf. Kybern.,19, No. 7–8, 365–374 (1983).

    Google Scholar 

  58. K. Weber, “Prime implicants of random Boolean functions,” Elektron Inf. Kybern.,19, No. 9, 449–458 (1983).

    Google Scholar 

  59. K. Weber, “Irredundant disjunctive normal forms of random Boolean functions,” Elektron. Inf. Kybern.,19, No. 10–11, 529–534 (1983).

    Google Scholar 

  60. K. Weber and B. Klipps, “Boolean functions of maximum length and Sperner type conditions about the set of faces of the n-cube,” Elektron. Inf. Kybern.,19, No. 3, 187–193 (1983).

    Google Scholar 

  61. F. Weitrsch, “Minimalisierung einiger Verknupfungsklassen der logischen Algebra,” Arch. Elektr. Übertrag.,17, No. 10, 453–460 (1963).

    Google Scholar 

  62. H. J. Zander, “Zur Minimisierung von partiellen Booleschen Funktionen bei gegebener nichtkanonischer Normalform,” Elektron. Inf. Kybern.,9, No. 1–2, 67–79 (1973).

    Google Scholar 

  63. H. J. Zander and W. Wagner, “Methode zur Berechnung der Primkonjunktionen unvollstandig bestimmter Boolescher Funktionen,” Elektron. Inf. Kybern.,8, No. 2–3, 115–127 (1972).

    Google Scholar 

Download references

Authors
  1. A. A. Sapozhenko
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. I. P. Chukhrov
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

Translated from Itogi Nauki i Tekhniki, Seriya Teoriya Veroyatnostei, Matematicheskaya Statistika, Teoreticheskaya Kibernetika, Vol. 25, pp. 68–116, 1987.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Sapozhenko, A.A., Chukhrov, I.P. Boolean function minimization in the class of disjunctive normal forms. J Math Sci 46, 2021–2052 (1989). https://doi.org/10.1007/BF01096022

Download citation

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF01096022

Keywords

Search

Navigation