How to multiply dynamic programming algorithms

Abstract

We develop a theory of algebraic operations over linear grammars that makes it possible to combine simple "atomic" grammars operating on single sequences into complex, multi-dimensional grammars. We demonstrate the utility of this framework by constructing the search spaces of complex alignment problems on multiple input sequences explicitly as algebraic expressions of very simple 1-dimensional grammars. The compiler accompanying our theory makes it easy to experiment with the combination of multiple grammars and different operations. Composite grammars can be written out in LATEX for documentation and as a guide to implementation of dynamic programming algorithms. An embedding in Haskell as a domain-specific language makes the theory directly accessible to writing and using grammar products without the detour of an external compiler. http://www.bioinf.uni-leipzig.de/Software/gramprod/. © 2013 Springer International Publishing.