Abstract
We explore the concept of local transformations of monotone switching circuits, i.e. what kind of local changes in a network leave the input/output behavior invariant. We obtain several general theorems in this direction. We apply these results to boolean matrix product and prove that the school-method for matrix multiplication yields the unique monotone circuit.
Zusammenfassung
Wir untersuchen den Effekt lokaler Änderungen auf das Eingabe/Ausgabe-Verhalten monotoner Netzwerke. Wir wenden die Ergebnisse auf die Multiplikation Boolescher Matrizen an und zeigen, daß die Schulmethode für die Matrizenmultiplikation den alleinigen optimalen Schaltkreis liefert.
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References
Lamagna, Savage: Combinational Complexity of Some Monotone Functions. 15th SWAT Conference, New Orleans, 1974, 140–144.
Mehlhorn: On the Complexity of Monotone Realizations of Matrix Multiplication. Universität des Saarlandes, TR 74-11 (1974).
Paterson: Complexity of Monotone Networks for Boolean Matrix Product. University of Warwick, TR 2, 1974. (To appear in Theoretical Computer Science.)
Pratt: The Power of Negative Thinking in Multiplying Boolean Matrices. 6th ACM Conference on Theory of Computing, Seattle, 1974, 80–83.
Schnorr: Lower Bounds on the Complexity of Monotone Networks (unpublished memo).
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The second author was partially supported by ONR Grant N00014-67-A-0077-0021
Related results were obtained independently by Mike Paterson.
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Mehlhorn, K., Galil, Z. Monotone switching circuits and boolean matrix product. Computing 16, 99–111 (1976). https://doi.org/10.1007/BF02241983
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DOI: https://doi.org/10.1007/BF02241983