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On complexity of standard forms for multifunctions
A. S. Kazimirov Irkutsk State University, 1, K. Marx st., Irkutsk, 664003,
Russian Federation
Abstract:
Consider discrete functions defined on set $A$.
In this case we define multifunctions as functions on set $2^A$.
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 $2$.
Keywords:
multifunction, minimization, complexity, disjunctive normal form.
Citation:
A. S. Kazimirov, “On complexity of standard forms for multifunctions”, Bulletin of Irkutsk State University. Series Mathematics, 22 (2017), 63–70
Linking options:
https://www.mathnet.ru/eng/iigum323 https://www.mathnet.ru/eng/iigum/v22/p63
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Abstract page: | 155 | Full-text PDF : | 65 | References: | 25 |
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