Combined traces (i.e., comtraces), a generalization of Mazurkiewicz traces,
were introduced by Janicki and Koutny in 1995 as congruence classes of step
sequences, where the congruence relation is induced by two relations
simultaneity and serializability on events. They also showed that comtraces
corresponds to some class of labeled stratified order structures, but the
question "what class of labeled stratified orders represent comtraces?" was
left open. In this work, we proposed a class of labeled stratified order
structures that captures exactly the comtrace notion.
This paper defines a class of labeled stratified order structures that
characterizes exactly the notion of combined traces (i.e., comtraces) proposed
by Janicki and Koutny in 1995. Our main technical contributions are the
representation theorems showing that comtrace quotient monoid, combined
dependency graph (Kleijn and Koutny 2008) and our labeled stratified order
structure characterization are three different and yet equivalent ways to
represent comtraces.
In this paper, I proposed to utilize partial-order alignment technique as a
heuristic method to cope with the state-space explosion problem in progressive
near-optimal alignment. The key idea of my approach is a formal treatment of
progressive partial order alignment based on the graph product construction.