Assessing the Complexity of Plan Recognition

This paper presents a discussion of the theoretical complexity of plan recognition on the basis of an analysis of the number of explanations that any complete plan recognition algorithm must consider given various properties of the plan library. On the basis of these results it points out properties of plan libraries that make them computationally expensive. Introduction Plan recognition is a well studied problem in the Artificial Intelligence literature. Following others, we distinguish between plan recognition/task tracking and goal identification. By plan recognition we mean the process of identifying not only the top level goal an agent is pursuing but also the plan that is being followed and the actions that have been done in furtherance of the plan. The algorithms that have been used to address plan recognition range from, graph covering (Kautz 1986), to Bayes nets (Bui 2003, Horvitz 1998), to Probabilistic State Dependent Grammars (Pynadath 2000). While a significant amount of information is known about the complexity of these algorithms what has previously been lacking in the literature is a discussion of how hard the actual problem for an individual plan library is. Without a discussion of the complexity of the actual problem, we may find ourselves using very powerful algorithms to solve problems that are amenable to simpler algorithms. The rest of this paper is organized as follows, we will discuss HTN plans as a representation for plan libraries and define an explanation for a given set of observations. Then, making sure to divorce ourselves from any particular plan recognition algorithm, we will discuss the complexity of a plan library in terms of the number of possible explanations licensed by a given set of observations and the features of the domain that control this. This will leave us in a position to make predictions about the difficulty any complete algorithm for plan recognition will have with a particular domain. Copyright © 2004, American Association for Artificial Intelligence (www.aaai.org). All rights reserved. Modeling Plans and Explanations Much of the past work in plan recognition has at least tacitly been based on simple hierarchical task networks (HTN) (Kutluhan, Hendler, and Nau 1994a, 1994b) as the representation for plans. Figure 1 is an HTN plan library represented as partially ordered and/or trees. In most plan recognition systems this kind of plan library is given to the system to define the set of plans it is expected to recognize. In the figure, interior “and nodes” representing plan decomposition (all the children must be performed for the parent to be achieved) are represented by an undirected arc across the lines connecting the parent node to its children. Interior “or nodes”, which represent choice points in the plan (only one of the children must be performed to achieve the parent) do not have this arc. Finally, basic actions that are directly observable by a plan recognition system are shown as leaf nodes of the trees. Directed arcs represent ordering constraints between plan nodes. For example, in Figure 1, action zt must be executed before ips and ps. In this paper, we will be considering the complexity of plan recognition limited to plans that can be represented in this formalism. We define an explanation of a set of observations as a minimal forest of instances of plan trees with expansions chosen for “or” nodes sufficient to allow an assignment of each observation to a specific basic action in the plan. For example, Figure 2 shows one of many possible Brag Theft Dos Scan Get-Ctrl Get-Data Dos-Attack zt ips ps Get-ctrl-local Get-ctrl-rem ote syl bnd pod sni adl r2l u2r cons r2r cons si er cr er rf ab u2rattack