An Inference Mechanism for Point-Interval Logic

We present a new inference algorithm for Point-Interval Logic. The mechanism removes the incompleteness of previously reported inference mechanism for PointInterval Logic. We also show how this inference mechanism can be used to prune the search space for an instance in Generalized Point-Interval Logic. Introduction Point-Interval Logic (PIL) is a tractable subclass of Allen’s interval algebra [1] which is used for modeling temporal information. The temporal information given in the form of a set statements in PIL (conjunction of statements), is converted into graph representation called Point Graph (PG), and checked for consistency (there is a mapping on timeline that satisfies all constraints). Once we have a consistent PG, we can answer temporal queries by executing various graph search algorithms on the PG representation. The language of PIL has been shown to capture the temporal aspects of time-sensitive mission planning [5, 9, 10], project management [3], and criminal forensics [4]. It is important to have a complete and efficient inference mechanism to effectively solve problems of interest in the mentioned application domains. In this paper we describe a new inference mechanism for PIL, which removes the incompleteness of the previous mechanism reported in [8, 10]. The paper has been organized as follows: in Section 2 we briefly describe PIL and its PG representation, in Section 3 we present the new inference mechanism; in Section 4 we show how to use the inference mechanism to prune the search space for instances in Generalized Point-Interval Logic; and finally in Section 5 we identify future research directions. Point-Interval Logic and Point Graphs We begin with a brief description of Point-Interval Logic and Point Graphs for making this presentation self-contained; same description can also be found in [3, 8, 9, 10]. PIL is a formal logic for reasoning with temporal events. It has two types of variables: points (events) and intervals (activities with duration). An interval X implicitly defines two points Copyright c © 2008, Association for the Advancement of Artificial Intelligence (www.aaai.org). All rights reserved. sX and eX that represent the start and end of the interval, respectively. The PIL is a pointisable logic [6], i.e. every relation between the temporal variables can be represented in terms of relationships between their start/end points. In Figure 1 we show examples of some temporal relationships between two intervals; for all possible relationships, see [3,9]. PIL also provides constructs to represent quantitative temporal information. In PIL a point variable can be assigned a stamp that represents its occurrence on the timeline. Similarly, an interval can be assigned a length that represents its duration on the timeline.