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Comtraces (combined traces) are extensions of Mazurkiewicz traces that can model the "not later than" relationship. In this paper, we first introduce the novel notion of generalized comtraces, extensions of comtraces that can additionally model the "non-simultaneously" relationship. Then we study some basic algebraic properties and canonical reprentations of comtraces and generalized comtraces. Finally we analyze the relationship between generalized comtraces and generalized stratified order structures. The major technical contribution of this paper is a proof showing that generalized comtraces can be represented by generalized stratified order structures.

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In 1990, Subramanian [1990] defined the complexity class CC as the set of problems log-space reducible to the comparator circuit value problem (CCV). He and Mayr showed that NL ⊆ CC ⊆ P, and proved that in addition to CCV several other problems are complete for CC, including the stable marriage problem, and finding the lexicographically first maximal matching in a bipartite graph. Although the class has not received much attention since then, we are interested in CC because we conjecture that it is incomparable with the parallel class NC which also satisfies NL ⊆ NC ⊆ P, and note that this conjecture implies that none of the CC-complete problems has an efficient polylog time parallel algorithm. We provide evidence for our conjecture by giving oracle settings in which relativized CC and relativized NC are incomparable.We give several alternative definitions of CC, including (among others) the class of problems computed by uniform polynomial-size families of comparator circuits supplied with copies of the input and its negation, the class of problems AC 0 -reducible to CCV, and the class of problems computed by uniform AC 0 circuits with CCV gates. We also give a machine model for CC, which corresponds to its characterization as logspace uniform polynomial-size families of comparator circuits. These various characterizations show that CC is a robust class. Our techniques also show that the corresponding function class FCC is closed under composition. The main technical tool we employ is universal comparator circuits.Other results include a simpler proof of NL ⊆ CC, a more careful analysis showing the lexicographically first maximal matching problem and its variants are CC-complete under AC 0 many-one reductions, and an explanation of the relation between the Gale-Shapley algorithm and Subramanian's algorithm for stable marriage.This article continues the previous work of Cook et al. [2011], which focused on Cook-Nguyen style uniform proof complexity, answering several open questions raised in that article.

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