Compositional Properties of Alignments

Alignments, i.e., position-wise comparisons of two or more strings or ordered lists are of utmost practical importance in computational biology and a host of other fields, including historical linguistics and emerging areas of research in the Digital Humanities. The problem is well-known to be computationally hard as soon as the number of input strings is not bounded. Due to its practical importance, a huge number of heuristics have been devised, which have proved very successful in a wide range of applications. Alignments nevertheless have received hardly any attention as formal, mathematical structures. Here, we focus on the compositional aspects of alignments, which underlie most algorithmic approaches to computing alignments. We also show that the concepts naturally generalize to finite partially ordered sets and partial maps between them that in some sense preserve the partial orders. As a consequence of this discussion we observe that alignments of even more general structure, in particular graphs, are essentially characterized by the fact that the restriction of alignments to a row must coincide with the corresponding input graphs. Pairwise alignments of graphs are therefore determined completely by common induced subgraphs. In this setting alignments of alignments are well-defined, and alignments can be decomposed recursively into subalignments. This provides a general framework within which different classes of alignment algorithms can be explored for objects very different from sequences and other totally ordered data structures.