Until now, we have treated biological sequences as one-dimensional strings of independent, uncorrelated symbols. This assumption is computationally convenient but not structurally realistic. The three-dimensional folding of proteins and nucleic acids involves extensive physical interactions between residues that are not adjacent in primary sequence. Can probabilistic models of proteins and nucleic acid sequences be developed that allow for longer range interactions? Can we compute efficiently with such models? In this chapter, we will step back from models of particular sequence problems and address these more theoretical issues. We will see how many of the methods described in previous chapters fit into a more general view of modelling sequences

A general theory for modelling strings of symbols has been developed by computational linguists [Chomsky 1956; 1959]. This theory is known as the Chomsky hierarchy of transformational grammars. In the Chomsky hierarchy, most of the models we have used so far in this book are the lowest of four types of model of increasing complexity and descriptive power. Transformational grammars were developed in an attempt to understand the structure of natural languages. They became important in theoretical computer science [Hopcroft & Ullman 1979; Gersting 1993] because computer languages, unlike natural languages, can be precisely specified as formal grammars. Recently, transformational grammars have been applied to sequence analysis problems in molecular biology [Searls 1992; Dong & Searls 1994; Rosenblueth et al. 1996].