Comparator Circuits over Finite Bounded Posets

Comparator circuit model was originally introduced in [

4

] (and further studied in [

2

]) 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

NLOG

$$\subseteq $$

⊆

CC

$$\subseteq $$

⊆

P

. Cook

et al

[

2

] 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.

LOG

). Complementing this, we show that general comparator circuits of polynomial size over arbitrary fixed finite lattices exactly characterizes

P

and that when the comparator circuit is skew they characterize

NLOG

. In addition, we show a characterization of the class

NP

by a family of polynomial sized comparator circuits over fixed

finite bounded posets

. These results generalize the results in [

2

] regarding the power of comparator circuits. As an aside, we consider generalizations of Boolean formulae over arbitrary lattices. We show that Spira’s theorem[

5

] can be extended to this setting as well and show that polynomial sized Boolean formulae over finite fixed lattices capture exactly

NC

$$^1$$

1

.

Our techniques involve design of comparator circuits and finite posets. We then use known results from lattice theory to show that the posets that we obtain can be embedded into appropriate lattices. Our results gives new methods to establish

CC

upper bound for problems also indicate potential new approaches towards the problems

P

vs

CC

and

NLOG

vs

LOG

using lattice theoretic methods.