Abstract

Pattern matching is an important operation used in many applications such as functional programming, rewriting, and rule-based expert systems. By preprocessing the patterns into a deterministic finite state automaton, we can rapidly select the matching pattern(s) in a single scan of the relevant portions of the input term. This automaton is typically based on left-to-right traversal of the patterns. By adapting the traversal order to suit the set of input patterns, it is possible to considerably reduce the space and matching time requirements of the automaton. The design of such adaptive automata is the focus of this paper. We first formalize the notion of an adaptive traversal. We then present several strategies for synthesizing adaptive traversal orders aimed at reducing space and matching time complexity. In the worst case, however, the space requirements can be exponential in the size of the patterns. We show this by establishing an exponential lower bound on space that is independent of the traversal order used. We then discuss an orthogonal approach to space minimization based on direct construction of optimal directed acyclic graph (dag) automata. Finally, our work stresses the impact of typing in pattern matching. In particular, we show that several important problems (e.g., lazy pattern matching in ML) are computationally difficult in the presence of type disciplines, whereas they can be solved efficiently in the untyped setting.