## 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.

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## 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.

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

- 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). - 5.
A. E. Andreev, “On minimization of disjunctive normal forms,” Dokl. AN SSSR,

__274__, No. 2, 265–269 (1984). - 6.
A. E. Andreev, “On one modification of the gradient algorithm,” Vestn. Mosk. Gos. Univ., Mat. Mekh., No. 3, 29–35 (1985).

- 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).

- 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).

- 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.

- 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).

- 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.

- 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). - 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.

- 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.

- 17.
Yu. L. Vasil'ev, “Difficulties of boolean function minimization by universal approaches,” Dokl. AN SSSR,

__171__, No. 1, 13–16 (1966). - 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.

- 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.

- 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.

- 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.

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

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

- 24.
M. A. Gavrilov, “Minimization of boolean functions characterizing relay circuits,” Avtomat. Telemekh.,

__20__, No. 9, 1217–1238 (1959). - 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.

- 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.

- 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.

- 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.

- 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.

- 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.

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

- 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). - 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.

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

- 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.

- 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).

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

- 39.
P. Ya. Danilenko and V. V. Umenskii, “A method of boolean function minimization,” Soobshch. AN GruzSSR,

__54__, No. 1, 45–48 (1969). - 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.

- 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.

- 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). - 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.

- 45.
Yu. I. Zhuravlev, “Algorithms of simplification of disjunctive normal forms of finite index,” Dokl. AN SSSR,

__139__, No. 6, 1329–1331 (1961). - 46.
Yu. I. Zhuravlev, “Local algorithms for computation of information, I,” Kibernetika, No. 1, 12–19 (1965).

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

- 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.

- 49.
Yu. I. Zhuravlev, “On algorithms of simplification of disjunctive normal forms,” Dokl. AN SSSR,

__132__, No. 2, 260–263 (1960). - 50.
Yu. I. Zhuravlev, “On one class of incompletely specified boolean functions,” in: Discrete Analysis [in Russian], No. 2, Novosibirsk (1964), pp. 23–27.

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

__51__, 143–157 (1958). - 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). - 53.
Yu. I. Zhuravlev, “On inessential variables of incompletely specified boolean functions,” in: Discrete Analysis [in Russian], No. 1, Novosibirsk (1963), 28–31.

- 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). - 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.

- 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). - 57.
Yu. I. Zhuravlev, “On various notions of minimality of d.n.f.s,” Sib. Mat. Zh.,

__1__, No. 4, 609–611 (1960). - 58.
Yu. I. Zhuravlev, “Set-theoretical methods in boolean algebra,” in: Problems of Cybernetics [in Russian], No. 8, Fizmatgiz, Moscow (1962), pp. 5–44.

- 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). - 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.

- 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.

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

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

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

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

- 66.
A. D. Zakrevskii, “On enumeration reduction for the solution of some synthesis problems of digital automata,” Radiofizika,

__7__, No. 1, 166–174 (1964). - 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.

- 68.
V. D. Kazakov, “Minimization of logic functions of many variables,” Avtomat. Telemekh.,

__23__, No. 9, 1237–1242 (1962). - 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.

- 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.

- 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).

- 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).

- 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).

- 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).

- 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).

- 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).

- 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.

- 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).

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

- 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.

- 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.

- 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.

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

- 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).

- 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.

- 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). - 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). - 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). - 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.

- 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.

- 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.

- 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.

- 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.

- 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).

- 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.

- 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.

- 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).

- 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.

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

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

- 109.
Nguen Kim Ang, “On some characteristics of algorithms of boolean function minimization,” Dokl. AN SSSR,

__267__, No. 5, 1058–1062 (1982). - 110.
R. G. Nigmatullin, “The variational principle in boolean algebra,” in: Discrete Analysis [in Russian], No. 10, Novosibirsk (1967), pp. 69–89.

- 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.

- 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.

- 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.

- 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). - 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). - 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).

- 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.

- 124.
G. N. Povarov, “On mathematical theory of synthesis of contact (1, qk)-pole circuits,” Dokl. AN SSSR,

__111__, No. 1, 102–104 (1956). - 125.
G. N. Povarov, “A method of boolean function minimization,” Izv. AN SSSR, Tekh. Kibern., No. 4, 198–201 (1982).

- 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).

- 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.

- 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.

- 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.

- 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). - 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.

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

- 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). - 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).

- 139.
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## Additional information

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

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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

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### Keywords

- Normal Form
- Boolean Function
- Function Minimization
- Minimization Algorithm
- Main Emphasis