From exponential to almost linear decomposability of finite or infinite trees

First-order constraints are first-order formulas built on a set of function and relation symbols using the following logical symbols: =, true, false, ¬, ∧, ∨, →, ↔, ∀, ∃, (,). Over the last decade, first-order constraints have been efficiently used in the artificial intelligence world to model many kinds of complex problems such as: scheduling, resource allocation, configuration, temporal and spatial reasoning, computer graphics, bio-informatics. While theory of finite or infinite trees T has played a fundamental role for both modeling and solving these problems, the complexity of solving first-order constraints with nested quantifiers and negations in T has been proved to be inherently huge (a tower of powers of two). However, a new property called decomposability has been recently introduced and used as a black-box to build many efficient first-order constraint solvers over T. We show in this paper that the algorithm which is used in this black-box (i.e. the algorithm which performs decomposability) has an exponential time and space complexity. We then present a much more efficient algorithm in the form of four rewriting rules which can perform the same decomposability in an almost-linear time and space complexity.