Simple Temporal Problems with Preferences and Uncertainty

Simple Temporal Problems (STPs) are a restriction of the framework of Temporal Constraint Satisfaction Problems, tractable in polynomial time. Their expressiveness has been extended independently in two ways. First, to account for uncontrollable events, to Simple Temporal Problems with Uncertainty (STPUs). Second, more recently, to account for soft temporal preferences, to Simple Temporal Problems with Preferences (STPPs). The motivation for both extensions is from real-life problems; and indeed such problems may well necessitate both preferences and uncertainty. Our research proposes the study of Simple Temporal Problems with Preferences and Uncertainty (STPPUs), and puts forward two notions of controllability for their resolution. Motivation Research on temporal reasoning, once exposed to the difficulties of real-life problems, can be found lacking both expressiveness and flexibility. Planning and scheduling for satellite observations, for example, involves not only quantitative temporal constraints between events and qualitative temporal ordering of events, but also soft temporal preferences and contingent events over which the agent has no control. For example, slewing and scanning activities should not overlap, but may if necessary. On the other hand, the duration of failure recovery procedures is not under the direct control of the satellite executive. To address the lack of expressiveness of hard constraints, preferences can be added to the framework; to address the lack of flexibility to contingency, handling of uncertainty can be added. Some real-world problems, however, have need for both. It is this requirement that motivates us here. Background Temporal Constraint Satisfaction Problems In a temporal constraint problem, variables denote timepoints or intervals, and constraints represent the possible temporal relations between them. Temporal constraints can be quantitative (distance between timepoints) or qualitative (relative position of temporal objects). Dechter, Meiri, & Pearl (1991) introduced the quantitative Temporal CSP (TCSP) and its restricted subclass, the Simple Temporal Problem (STP). Variables represent timepoints (events) and constraints represent the relations between them. The restriction to at most one interval in each temporal constraint — hence the constraints have form — entails that a STP can be solved in polynomial time. By solved, we mean that consistency is decided and the minimal network obtained; applying path consistency suffices for this. In contrast, the general TCSP is NP-complete. Simple Temporal Problems with Preferences To address the lack of expressiveness in standard STPs, Khatib et al. (2001) introduced the Simple Temporal Problem with Preferences (STPP). The framework merges temporal CSP with the standard semiring-based soft constraints (Bistarelli, Montanari, & Rossi 1997). In addition to the hard temporal constraints of a STP, soft temporal constraints are specified by means of a preference function on an interval, , where ! and is a set of preference values, part of a semiring