Comparator Circuits over Finite Bounded Posets

TL;DR

This paper generalizes comparator circuit models to finite bounded posets, characterizing complexity classes like CC, L, P, and NP, and extends classical results such as Spira's theorem to lattice-based Boolean formulas.

Contribution

It introduces comparator circuits over finite bounded posets, establishing their computational power and characterizations of various complexity classes, extending prior work on comparator circuits.

Findings

Comparator circuits over finite distributive lattices characterize CC and L.

Comparator circuits over arbitrary finite lattices characterize P and NP.

Boolean formulas over finite lattices capture NC^1.

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 P-complete but still not known to admit efficient parallel algorithms. The class CC is the complexity class of problems many-one 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.