Incremental Qualitative Temporal Reasoning: Algorithms for the Point Algebra and the ORD-Horn Class

In many applications of temporal reasoning we are interested in processing temporal information incrementally. In particular, given a set of temporal constraints (a temporal CSP) and a new constraint, we want to maintain certain properties of the extended temporal CSP (e.g., a solution), rather than recomputing them from scratch. The Point Algebra (PA) and the Interval Algebra (IA) are two well-known frameworks for qualitative temporal reasoning. The reasoning algorithms for PA and the tractable fragments of IA, such as Nebel and Bürckert’s maximal tractable class of relations (ORD-Horn), have originally been designed for “static” reasoning. In this paper, we study the incremental version of the fundamental reasoning problems in the context of these tractable classes. We propose a collection of new polynomial algorithms that can amortize their complexity when processing a sequence of input constraints to incrementally decide satisfiability, to maintain a solution, or to update the minimal representation of the CSP. Our incremental algorithms improve the total time complexity of using existing static techniques by a factor of O(n) or O(n^2), where n is the number of the variables involved by the temporal CSP. An experimental analysis focused on constraints over PA confirms the computational advantage of our incremental approach.