ISSN 1997-7670 (Print)
ISSN 2541-8785 (Online)

List of issues > Series «Mathematics». 2017. Vol. 22

On Complexity of Standard Forms for Multifunctions

A. S. Kazimirov

Consider discrete functions defined on set A. In this case we define multifunctions as functions on set 2A. Values of a multifunction for inputs equal to one-element sets are given and values for other sets are calculated as a union of values on one-element sets. Superposition of multifunctions is defined in the same way.

Multifunction is a generalization of different models of uncertainty, incomplete and partial functions and hyperfunctions. These models can be useful for processing incomplete and contradictional information in intelligent systems.

Standard forms representing multifunctions are defined using intersection multifunction. Standard form representation of a multifunction is not unique. It is natural to define complexity of a standard form as the number of its components.

This paper introduces exact bounds on complexity of n-ary multifunctions and proposes an algorithm for minimization of 4-argument multifunctions.

This paper considers the relationship between multifunctions that have only two output values, and Boolean functions. It is shown that the complexity of the standard forms of any such multifunction coincides with the length of the disjunctive normal form of the corresponding Boolean function. The article gives an upper bound for the complexity of the standard forms of multifunctions, and also introduces a sequence of multifunctions whose complexity coincides with this upper bound. Thus, the complexity of the class of n-ary multifunctions is obtained. Also, an algorithm is proposed for minimizing multifunctions of rank 2, based on a sequential search of formulas of increasing complexity. This algorithm allows us to find the complexities of all 4-ary multifunctions of rank 2.

For citation:

Kazimirov A.S. On Complexity of Standard Forms for Multifunctions. The Bulletin of Irkutsk State University. Series Mathematics, 2017, vol. 22, pp. 63-70. (In Russian). https://doi.org/10.26516/1997-7670.2017.22.63

multifunction, minimization, complexity, disjunctive normal form

1. Lupanov O.B. On logic functions realization with formulas of finite classes (of finite depth) in basis of AND, OR, NOT Problemy kibernetiki, vol. 6, Moscow, Fizmatgiz, 1961, pp. 5-14 (in Russian).

2. Panteleyev V.I. Completeness criteria for incompletely defined partial Boolean functions Vestn. Novosib. Gos. Univ., Ser. Mat. Mekh. Inform., 2009, vol. 9, no 3, pp. 95-114 (in Russian).

3. Peryazev N.A. Clones, co-clones, hyperclones and superclones Uchenye zapiski Kazanskogo gosudarstvennogo universiteta. Seriya: Fiziko-matematicheskiye nauki, Kazan, 2009, no 151, vol. 2, pp. 120-125 (in Russian).

4. Peryazev N.A., Yakovchuk I.A. Minimization of multioperations in the class of standard forms Izv. Irkutsk. Gos. Univ., Ser. Mat., 2009, vol. 2, no 2, pp. 117-126 (in Russian).

5. Peryazev N.A. Standard forms of multioperations in superclones Izv. Irkutsk. Gos. Univ., Ser. Mat., 2010, vol. 3, no 4, pp. 88–95 (in Russian).

6. Peryazev N.A. Superclones of multioperations Trudy VIII Mezhdunarodnoy konferentsii ¾Diskretnye sistemy v teorii upravlyayuschih sistem¿, Moscow, MAIS Press, 2009, pp. 233-238 (in Russian).

7. Sharankhaev I.K. On decomposition method for multifunctions Diskretnye modeli v teorii upravlyayuschih sistem IX Mezhdunarodnaya konferentsiya: Trudy, Moscow, MAKS Press, 2015, pp. 266-267 (in Russian).

8. Kazimirov A., Panteleyev V., Riabets L., Vinokurov S. Decision support system based on 4-valued logic with multi-interpretations Proceedings of International Conference on Soft Computing and Measurements SCM 2015, May 19–21, St.Petersburg, Russia, 2015, pp. 198–199.

9. Peryazev N.A., Sharankhaev I.K. Galois theory for clones and superclones Discrete Mathematics and Applications, 2016, vol. 26, no 4, pp. 227–238. https://doi.org/10.1515/dma-2016-0020

10. Romov B. A. The completeness problem in partial hyperclones Discrete Mathematics, 2006, pp. 1405–1414. https://doi.org/10.1016/j.disc.2005.11.033

11. Vinokurov S., Kazimirov A., Pustovoytov N., Frantseva A. Decision Support System for Medical Prescriptions Based on 4-Valued Logic Proceedings of the XIX International Conference on Soft Computing and Measurement, SCM 2016, May25–27, St. Petersburg, Russia, 2016, pp. 307-308.

Full text (russian)