NDLI: On the Computational Complexity of Algebra on Lattices

Content Provider	Society for Industrial and Applied Mathematics (SIAM)
Author	Rosenkrantz, D. J. Bloniarz, P. A. Hunt, III, H. B.
Copyright Year	1987
Abstract	We study the computational complexity of equivalence and minimization problems for expressions on many different lattices including each finite lattice and each distributive lattice. A general efficient expressibility condition C on a lattice is presented such that 1. The equivalence problem is co$NP$ hard for constant-free expressions on any lattice with at least two elements that satisfies condition C.Each finite or distributive lattice is shown to satisfy condition C. Moreover, if a lattice $\mathcal{L}$ satisfies condition C and $ \equiv $ is a congruence relation on $\mathcal{L}$, then ${\mathcal{L} / \equiv }$ also satisfies condition C. Several additional results are also presented. These results include the following: 2. In contrast to 1, the equivalence and operator minimization problems are solvable deterministically in polynomial time for disjunctive normal form and conjunctive normal form expressions on any lattice and for constant-free expressions on any free lattice with at least three generators: 3. Let $\mathcal{L}$ be a lattice. Then, the operator minimization problem and various approximate operator minimization problems for expressions on $\mathcal{L}$ are as hard as the problem of determining, for expressions F and G on $\mathcal{L}$, if $F \leqq G$.
Starting Page	129
Ending Page	148
Page Count	20
File Format	PDF
ISSN	00975397
DOI	10.1137/0216011
e-ISSN	10957111
Journal	SIAM Journal on Computing (SMJCAT)
Issue Number	1
Volume Number	16
Language	English
Publisher	Society for Industrial and Applied Mathematics
Publisher Date	2006-07-13
Access Restriction	Subscribed
Subject Keyword	NP-hardness deterministic polynomial time Analysis of algorithms and problem complexity lattice formula equivalence distributive lattice Lattices and related structures free lattice finite lattice Free lattices, projective lattices, word problems formula minimization computational complexity
Content Type	Text
Resource Type	Article
Subject	Mathematics Computer Science

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