Replacement rules have played an important role in the study of monotone boolean function complexity. In this paper, notions of replaceability and computational equivalence are formulated in an abstract algebraic setting, and examined in detail for finite distributive lattices — the appropriate algebraic context for monotone boolean functions. It is shown that when computing an element f of a finite distributive lattice D, the elements of D partition into classes of computationally equivalent elements, and define a quotient of D in which all intervals of the form [t ∧ f, t ∨ f] are boolean. This quotient is an abstract simplicial complex with respect to ordering by replaceability. Other results include generalisations and extensions of known theorems concerning replacement rules for monotone boolean networks. Possible applications of computational equivalence in developing upper and lower bounds on monotone boolean function complexity are indicated, and new directions of research both abstract mathematical and computational, are suggested.
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Beynon, W.M.: Replacement in monotone boolean networks: an algebraic perspective. In: Proc. 4th. FST & TCS, Bangalore. Lect. Notes Comput. Sci. 181, 165–178 (1984)
Beynon, W.M.: Replaceability and computational equivalence in finite distributive lattices. Univ. of Warwick, Theory of Computation Report 61, 1984
Beynon, W.M.: Geometric aspects of partially-ordered systems. King's College, Univ. of London, Ph.D. thesis, 1973
Beynon, W.M.: Duality theorems for finitely-generated vector lattices. Proc. Lond. Math. Soc. 3, 114–128 (1975)
Beynon, W.M.: Vector lattices freely generated by distributive lattices. Math. Proc. Camb. Philos. Soc. 81, 193–220 (1977)
Beynon, W.M., Buckle, J.F.: Computational equivalence and replaceability in finite algebras. Univ. of Warwick, Theory of Computation Report 72, 1985
Birkhoff, G.A.: Lattice Theory, 3rd ed. AMS Colloquium Publications, Vol. XXV, 1967
Dunne, P.E.: A 2.5n lower bound on the monotone network complexity of T 3 n. Univ. of Warwick, Theory of Computation Report 62, 1984
Dunne, P.E.: Some results on replacement rules in monotone boolean networks. Univ. of Warwick, Theory of Computation Report 64, 1984
Grätzer, G.: Lattice Theory: first concepts and distributive lattices. San Francisco: Freeman 1971
Mehlhorn, K., Galil, Z.: Monotone switching networks and boolean matrix product. Computing 16, 99–111 (1976)
Paterson, M.S.: Complexity of monotone networks for boolean matrix product. Theor. Comput. Sci. 1, 13–20 (1975)
Wegener, I.: On the complexity of slice-functions. Univ. of Frankfurt, Internal Report, 1983
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Beynon, M. Replaceability and computational equivalence for monotone boolean functions. Acta Informatica 22, 433–449 (1985). https://doi.org/10.1007/BF00288777
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