Formalizing Resources for Planning

In this paper we present a classification scheme which circumscribes a large class of resources found in the real world. Building on the work of others we also define key properties of resources that allow formal expression of the proposed classification. Furthermore, operations that change the state of a resource are formalized. Together, properties and operations go a long way in formalizing the representation and reasoning aspects of resources for planning. Introduction Historically, allocation of tasks to resources has been considered part of the scheduling problem, and largely omitted from the planning literature. In recent times, the importance of such resource allocation decisions in planning has been recognized (Smith, Frank, & Jónsson 2000; Long et al. 2000). Though progress has been made to meet this challenge (Laborie 2001), the state of the art has not yet advanced to a point where we have a comprehensive treatment of resources as an inherent part of a planning framework. More specifically, efforts to characterize the types of resources of interest have been incomplete and largely constrained by the availability of efficient algorithms to reason with them. Furthermore, approaches to natively incorporate resources into domain descriptions have been largely absent. We believe that the absence of explicit types of resources obfuscates the semantics of the model, impedes detection of domain modeling errors, complicates the mapping to efficient implementations that could be tailored to particular resource types, and hinders domain analysis. For example, resources have not yet been incorporated explicitly in the PDDL 2.1 specification (Fox & Long 2003) although they can be represented through the use of functional expressions. This is illustrated in Figure 1 which describes an action fly consuming a resource fuel. In the action definition, the fuel consumption is expressed as an effect decreasing the level of fuel. Fuel is not identified explicitly as a resource. Furthermore, the inherent properties of fuel and the way in which it is allowed to change are not represented. Copyright c 2003, American Association for Artificial Intelligence (www.aaai.org). All rights reserved. (:durative−action fly :parameters (?p − plane ?t − traveller ?a ?b − location) :duration (= ?duration (flight−time ?a ?b)) :condtion (and (at start (at ?p ?a)) (at start (at ?t ?a)) (over all (inflight ?p)) (over all (aboard ?t ?p)) (at start (>= (fuel−level ?p) (* (flight−time ?a ?b) (consumption−rate ?p))))) :effect (and (at start (not (at ?p ?a))) (at start (not (at ?t ?a))) (at end (at ?p ?b)) (at end (at ?t ?b)) (at start (inflight ?p)) (at end (not (inflight ?p))) (at end (not (aboard ?t ?p))) (at end (decrease (fuel−level ?p) (* (flight−time ?a ?b) (consumption−rate ?p)))))) Figure 1: Example of an activity on a resource in PDDL 2.1 In this paper we present a classification scheme that will circumscribe a large class of resource types found in the real world. We first develop, through exploration of real world examples, an ontology for resources from a planning perspective. We define a set of properties that characterize a resource. We then present a classification scheme based on these properties. We go on to propose an epistemology that will identify transactions on resources. We present examples throughout to illustrate the terms introduced. Where possible, we follow the PDDL 2.1 syntax in the hope that it will be familiar to the reader. We then review related work in PDDL and other languages, in terms of their methods and limitations to address the needs outlined. We conclude with a brief synopsis and discussion of future work. Ontology Ontology by Example The following examples are designed to illustrate the various features of resources that are of interest to the modeler. Consider the cargo bay of the space shuttle. Many items of different sizes are placed in the bay and consume volume. The space is used when an item is placed in it. It is made available again when the item is removed. We can consider the space as an example of a reusable resource. In contrast consider the fuel in a fuel tank. Once consumed it is destroyed permanently. Fuel, in this case, is an example of a consumable resource. If the fuel tank can be refueled then it is an example of a replenishable resource. Process byproducts which are never used are examples of producible resources. The battery on a planetary rover is an example of a continuous resource. Within the capacity of the battery any amount of energy can be drawn at a time. In contrast disk space on a hard drive is consumed in discrete chunks (bytes). This is an example of a discrete resource. A printer is an example of a single-capacity resource since it prints only one job at a time. On the other hand, a passenger aircraft contains numerous seats representing a multicapacity resource. A fuel container typically has a fixed volume, and therefore a fixed capacity. Alternatively, a battery whose capacity degrades over time is an example of a variable capacity resource. Seats on an airplane are also examples of a deterministic resource because the state of the resource is known precisely. The energy of a battery is an example of a stochastic resource because of the inherent uncertainty in the amount of the resource available. The fuel tank in a car is an example of an exclusive resource because refueling is not allowed while the engine is running. Data bandwidth is an example of a shared resource because multiple activities can use the bandwidth simultaneously. A cargo bay has specific restrictions for both weight and volume. Loading a cargo bay consumes both weight and volume at different rates. Weight and volume are two distinct dimensions of the same resource so this is an example of a multi-dimensional resource (Smith & Becker 1997). Keeping these examples in mind, we proceed to define properties that precisely categorize resources. Resource Properties In this section we present a set of properties that can be used to describe a quantitative resource i.e. a resource with capacity and availability described in terms of numeric quantities. We assume all measures of quantity are represented by a generic unit rather than actual units of measure. We also assume that all conversion operations are defined and occur outside of the resource definitions and operations. Furthermore, we assume that all quantities are greater than zero. We adopt the following notation for domain definitions. Let be to the domain of real numbers; let be the domain of natural numbers;1 let be the domain of whole numbers; let be the domain of time. Notice that we explicitly defer commitment to whether time is represented by natural numbers or real numbers, leaving the choice to implementation. Furthermore, let be the universe of reWe assume that includes the number zero. sources and let be the universe of transactions.2 Let