GSTP: A Temporal Reasoning System Supporting Multi-Granularity Temporal Constraints

Claudio Bettini, Sergio Mascetti, Vincenzo Pupillo

Abstract: The GSTP system has been developed at the University of Milan with the objective of providing universal access to the implementation of a set of algorithms for multi-granularity temporal constraint satisfaction. The many formalisms and algorithms proposed in the literature for Temporal Constraint Satisfaction Problems (TCSP) have essentially ignored the subtleties involved in the presence of multiple time granularities in the temporal constraints. Examples of simple constraints specified in terms of a time granularity are the following: "package shipment must occur the next business day after check clearance" and "package delivery should occur during working hours". More technically, the GSTP system allows the user to specify binary constraints of the form Y X € [m^n]G, where m and n are the minimum and maximum values of the distance between X and Y in terms of granularity G. Variables take values in the positive integers, and unary constraints can be applied on their domains. This can be considered the extension of STP [Dechter et al. 1991] to multiple and arbitrary granularities. A first issue in the representation and processing of these constraints is the need for a clear semantics for time granularities. Business days, for example, may really have different meanings in different countries or even in different companies. In this respect GSTP adopts a formalism, first introduced in [Wang et al. 1997; Bettini et al. 1998], which can model arbitrary user-defined time granularities and has a clear set-theoretic semantics. In order to guarantee a finite representation, granularities in GSTP are limited to those that can be defined in terms of periodic sets. Hours, days, weeks, business days, business weeks, fiscal years, and academic semesters are common examples. A second issue is related to the difficulty to reduce a network of constraints given in terms of different granularities into an equivalent one with all constraints in terms of the same granularity, so that some of the standard algorithms for CSP, like consistency checking through arcor path-consistency [Bessiere 1994; Dechter et al. 1991], could be successfully applied. Indeed, any conversion necessarily introduces an approximation; For example, the above constraint imposing delivery to start the next business day may be translated in