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.