Comparator Circuits over Finite Bounded Posets
Abstract
Comparator circuit model was originally introduced by Mayr and Subramanian (1992) (and further studied by Cook, Filmus and Le (2012)) to capture problems which are not known to be Pcomplete but still not known to admit efficient parallel algorithms. The class CC is the complexity class of problems manyone logspace reducible to the Comparator Circuit Value Problem and we know that NL is contained in CC which is inturn contained in P. Cook, Filmus and Le (2012) showed that CC is also the class of languages decided by polynomial size comparator circuits. We study generalizations of the comparator circuit model that work over fixed finite bounded posets. We observe that there are universal comparator circuits even over arbitrary fixed finite bounded posets. Building on this, we show that general (resp. skew) comparator circuits of polynomial size over fixed finite distributive lattices characterizes CC (resp. L). Complementing this, we show that general comparator circuits of polynomial size over arbitrary fixed finite lattices exactly characterizes P even when the comparator circuit is skew. In addition, we show a characterization of the class NP by a family of polynomial sized comparator circuits over fixed {\em finite bounded posets}. These results generalize the results by Cook, Filmus and Le (2012) regarding the power of comparator circuits. As an aside, we consider generalizations of Boolean formulae over arbitrary lattices. We show that Spira's theorem (1971) can be extended to this setting as well and show that polynomial sized Boolean formulae over finite fixed lattices capture exactly NC^1.
 Publication:

arXiv eprints
 Pub Date:
 March 2015
 DOI:
 10.48550/arXiv.1503.00275
 arXiv:
 arXiv:1503.00275
 Bibcode:
 2015arXiv150300275K
 Keywords:

 Computer Science  Computational Complexity
 EPrint:
 21 pages, previous version incorrectly claimed NL = LSkewCC when in fact P = LSkewCC