Abstract
Proving lower bounds on the combinational complexity of concrete functions is a difficult and challenging problem. Previous successes in establishing lower bounds for monotone networks and the known gaps between the monotone and general combinational complexity indicate the key role that negations play in determining combinational complexity.
We have investigated the way in which the complexity of a set of functions F decreases with the use of additional negations beyond the minimum number necessary to realize F. For sets F of maximum inversion complexity, at most a factor of 2 and an additive term of order n2log2n is saved. However, for sets of lower inversion complexity, no interesting bounds are known on the amount of savings possible. Good upper bounds on the amount of such savings would enable lower bounds on combinational complexity to be concluded from lower bounds on the negation-restricted complexity.
This research was supported in part by the National Science Foundation under research grant GJ-43634X to M.I.T. The author did some of this work at the University of Toronto and the University of Frankfurt.
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Fischer, M.J. (1975). The complexity of negation-limited networks — A brief survey. In: Brakhage, H. (eds) Automata Theory and Formal Languages. Lecture Notes in Computer Science, vol 33. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-07407-4_9
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DOI: https://doi.org/10.1007/3-540-07407-4_9
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