Abstract
We investigate the Boolean functions that combine various properties: the extremal values of complexity characteristics ofminimization, the inapplicability of local methods for reducing the complexity of the exhaustion, and the impossibility to efficiently use sufficient minimality conditions. Some quasicyclic functions are constructed that possess the properties of cyclic and zone functions, the dominance of vertex sets, and the validity of sufficient minimality conditions based on independent families of sets. For such functions, we obtain the exponential lower bounds for the extent and special sets and also a twice exponential lower bound for the number of shortest and minimal complexes of faces with distinct sets of proper vertices.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Yu. L. Vasil’ev and V. V. Glagolev, “Metric Properties of Disjunctive Normal Forms,” in DiscreteMathematics and Mathematical Problems of Cybernetics, Vol. 1 (Nauka, Moscow, 1974), pp. 99–148.
M. R. Garey and D. S. Johnson, Computers and Intractability: A Guide to the Theory of NP-Completeness [Freeman, San Francisco, 1979; Mir, Moscow, 1982].
A. A. Evdokimov, “Maximal Length of Circuit in a Unitary n-Dimensional Cube,” Mat. Zametki 6 (3), 309–319 (1969) [Math. Notes Acad. Sci. USSR 6 (3), 642–648 (1969)].
Yu. I. Zhuravlyov, “Algorithms for Constructing Minimal Disjunctive Normal Forms of Boolean Functions,” in Discrete Mathematics and Mathematical Problems of Cybernetics, Vol. 1 (Nauka, Moscow, 1974), pp. 67–98.
V. B. Kudryavtsev and A. E. Andreev, “On Algorithm Complexity,” Fundam. Prikl. Mat. 15 (3), 135–169 (2009) [J. Math. Sci. 168 (1), 89–122 (2010)].
Yu. V. Maksimov, “Realization of Boolean Functions with a Bounded Number of Zeros in the Class of Disjunctive Normal Forms,” Zh. Vychisl.Mat. Mat. Fiz. 53 (9), 1569–1588 (2013) [Comput. Math. Math. Phys. 53 (9), 1391–1409 (2013)].
A. V. Panov, “Algorithms Using First-Order Neighborhoods for Minimization of Boolean Functions,” Zh. Vychisl. Mat. Mat. Fiz. 53 (9), 1589–1600 (2013) [Comput. Math. Math. Phys. 53 (9), 1410–1420 (2013)].
A. A. Sapozhenko and I. P. Chukhrov, “Boolean Function Minimization in the class of Disjunctive Normal Forms,” Itogi Nauki Tekh., Ser. Teor. Veroyatnost., Mat. Statist., Teor. Kibern. 25, 68–116 (1987) [J. Sov. Math. 46 (4), 2021–2052 (1989)].
I. P. Chukhrov, “Estimates of the Number of Minimal Disjunctive Normal Forms for a Zone Function,” in Methods of Discrete Analysis in Studies of Functional Systems (Inst. Mat., Novosibirsk, 1981), Vol. 36, pp. 74–92.
I. P. Chukhrov, “On Complexity Measures of Complexes of Faces in the Unit Cube,” Diskretn. Anal. Issled. Oper. 20 (6), 77–94 (2013) [J. Appl. Indust. Math. 8 (1), 9–19 (2014)].
I. P. Chukhrov, “Proof of CoveringMinimality by Generalizing the Notion of Independence,” Diskretn. Anal. Issled. Oper. 24 (2), 87–106 (2017) [J. Appl. Indust. Math. 11 (2), 193–203 (2017)].
O. Coudert and T. Sasao, “Two-Level Logic Minimization,” in Logic Synthesis and Verification (Kluwer Acad. Publ., Netherlands, 2002) pp. 1–27.
C. Umans, T. Villa, and A. L. Sangiovanni-Vincentelli, “Complexity of Two-Level Logic Minimization,” IEEE Trans. CAD Integr. Circuits Syst. 25 (7), 1230–1246 (2006).
Author information
Authors and Affiliations
Corresponding author
Additional information
Original Russian Text © I.P. Chukhrov, 2018, published in Diskretnyi Analiz i Issledovanie Operatsii, 2018, Vol. 25, No. 3, pp. 126–151.
Rights and permissions
About this article
Cite this article
Chukhrov, I.P. On the Complexity of Minimizing Quasicyclic Boolean Functions. J. Appl. Ind. Math. 12, 426–441 (2018). https://doi.org/10.1134/S1990478918030043
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1134/S1990478918030043