Alignments as Compositional Structures

TL;DR

This paper explores the formal mathematical structure of alignments, emphasizing their compositional aspects and generalizations to partial orders, which underpins many algorithms used in computational biology and related fields.

Contribution

It introduces a formal, compositional framework for alignments and extends these concepts to finite partially ordered sets and partial maps, providing a new theoretical perspective.

Findings

Alignments can be viewed as compositional structures.

The concepts generalize to finite partially ordered sets.

This formalization underpins many existing algorithms.

Abstract

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 prac- tical 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 algo- rithmic 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.