Plan Recognition as Planning Miquel Ram´ırez Hector Geffner Universitat Pompeu Fabra ICREA & Universitat Pompeu Fabra 08003 Barcelona, SPAIN 08003 Barcelona, SPAIN [email protected] [email protected] Abstract and consider the plan recognition problem over a domain the- ory and a possible set G of goals. The plan recognition task In this work we aim to narrow the gap between plan is then formulated as the problem of identifying the goals recognition and planning by exploiting the power G ∈Gsuch that some optimal plan for G is compatible and generality of recent planning algorithms for ∗ with the observations. Such goals are grouped into the set recognizing the set G of goals G that explain a se- G∗. The reason for focusing on the optimal plans for G is that quence of observations given a domain theory. Af- they represent the possible behaviors of a perfectly rational ter providing a crisp definition of this set, we show agent pursuing the goal G. By a suitable transformation, we by means of a suitable problem transformation that G G∗ ∗ show that a goal is in if there is an action sequence that a goal G belongs to G if there is an action sequence achieves optimally both the goal G and the goal G extended π that is an optimal plan for both the goal G and the with extra goals Go representing the observations. Exploiting goal G extended with extra goals representing the this result, we show how the set of goals G∗ can be com- observations. Exploiting this result, we show how ∗ puted exactly and approximately by minor modifications of the set G can be computed exactly and approxi- existing optimal and suboptimal planning algorithms, and ex- mately by minor modifications of existing optimal isting polynomial heuristics, that are then tested experimen- and suboptimal planning algorithms, and existing tally over several domains. The suboptimal algorithms and polynomial heuristics. Experiments over several the heuristics appear to provide good approximations of the domains show that the suboptimal planning algo- optimal goal set G∗, while scaling up as well as state-of-the- rithms and the polynomial heuristics provide good ∗ art planning algorithms and heuristics. approximations of the optimal goal set G while The paper is organized as follows. We provide first the ba- scaling up as well as state-of-the-art planning al- sic planning terminology, define the plan recognition problem gorithms and heuristics. over a domain theory, and introduce the transformation that allows us to compile away the observations. We then focus on exact and approximate computational methods, present the 1 Introduction result of the experiments, and discuss limitations and gener- Plan recognition is an ubiquitous task in a number of ar- alizations of the proposed framework. eas, including natural language, multi-agent systems, and as- sisted cognition [Schmidt et al., 1978; Cohen et al., 1981; 2 Planning Background Pentney et al., 2006]. Plan recognition is planning in re- A Strips planning problem is a tuple P = F, I, A, G where verse: while in planning, we seek the actions to achieve F is the set of fluents, I ⊆ F and G ⊆ F are the initial a goal, in plan recognition, we seek the goals that explain and goal situations, and A is a set of actions a with precon- the observed actions. Both are abductive tasks where some dition, add, and delete lists Pre(a), Add(a), and Del(a) re- hypothesis needs to be adopted to account for given data: spectively, all of which are subsets of F . plans to account for goals, and goals to account for par- For each action a ∈ A, we assume that there is a non- tially observed plans. Work in plan recognition, however, negative cost c(a) so that the cost of a sequential plan π = has proceeded independently of the work in planning, us- a1,...,an is c(π)= c(ai). A plan π is optimal if it has ing mostly handcrafted libraries or algorithms that are not minimum cost. For unit costs, i.e., c(a)=1for all a ∈ A, related to planning [Kautz and Allen, 1986; Vilain, 1990; plan cost is plan length, and the optimal plans are the shortest Charniak and Goldman, 1993; Lesh and Etzioni, 1995; Gold- ones. Unless stated otherwise, action costs are assumed to be man et al., 1999; Avrahami-Zilberbrand and Kaminka, 2005]. 1. In this work we aim to narrow the gap between plan recog- In the plan recognition setting over a domain theory, ac- nition and planning by exploiting the power and generality tion costs c(a) carry implicit information about the probabil- of recent classical planning algorithms. For this, we move ity that the agent will use action a to solve a problem P ,as away from the plan recognition problem over a plan library the agent should rationally avoid plans with higher cost than 1778 needed. The most likely plans, assuming full rationality, are Definition 3. The exact solution to a theory T = P, G,O is ∗ the optimal plans, and hence, we will focus initially on them. given by the optimal goal set GT which comprises the goals ∗ Algorithms for computing optimal plans cast the planning G ∈Gsuch that for some π ∈ ΠP (G), π satisfies O. problem as a problem of heuristic search with admissible heuristics derived automatically from the problem encoding [Haslum and Geffner, 2000] . An heuristic h(s) is admis- A B C sible if h(s) ≤ h∗(s) for all the states, where h∗(s) is the optimal cost from s. Optimal planners use such heuristics D HI in the context of admissible search algorithms like A* and EFG IDA*. Computing optimal plans, however, is much harder than computing plans that are good but not necessarily opti- J K mal, and indeed suboptimal planners such as FF and FD, that use more focused search algorithms and more informative heuristics, scale up much better [Hoffmann and Nebel, 2001; Figure 1: Plan Recognition in a Navigation Domain Helmert, 2006]. We will thus use optimal plans for providing a crisp definition of the plan recognition task, and develop ef- Figure 1 shows a simple plan recognition problem. Room ficient methods that borrow from suboptimal planning meth- A (marked with a circle) is the initial position of the agent, ods and heuristics for producing good approximations that while Rooms C, I and K (marked with a square) are its pos- scale up. sible destinations. Arrows between Rooms A and B, and F and G, are the observed agent movements in that order. In the resulting theory T , the only possible goals that have optimal 3 Plan Recognition over a Domain Theory plans compatible with the observation sequence are I and K. The plan recognition problem given a plan library for a set G In the terminology above, the set of possible goals G is given of possible goals G can be understood, at an abstract level, as by the atoms at(C), at(I), and at(K), while the optimal goal ∗ the problem of finding a goal G with a plan π in the library, set GT comprises at(I) and at(K), leaving out the possible written π ∈ ΠL(G), such that π satisfies the observations. goal at(C). We define the plan recognition problem over a domain theory Before proceeding with the methods for computing the op- ∗ in a similar way just changing the set ΠL(G) of plans for G timal goal set GT exactly or approximately, let us first com- ∗ in the library by the set ΠP (G) of optimal plans for G given ment on some of the limitations and strengths of this model of the domain P . We will use P = F, I, O to represent plan- plan recognition. The model can be easily extended to handle ning domains so that a planning problem P [G] is obtained by observations on fluents and not only on actions. For this, the concatenating a planning domain with a goal G, which is any observation of a fluent p can be encoded as the observation of set of fluents. We can then define a plan recognition problem a ’dummy’ action NOOP(p) with precondition and effect p. or theory as follows: Similarly, it is not difficult to account for actions that must be Definition 1. A plan recognition problem or theory is a among the observations when they have been executed. For triplet T = P, G,O where P = F, I, A is a planning this, it suffices to remove such actions from the domain theory domain, G is the set of possible goals G, G ⊆ F , and when they haven’t been observed. Notice that the same tricks do not work in library-based approaches that are less flexible. O = o1,...,om is an observation sequence with each oi be- ing an action in A. Likewise, the assumption that the observations are ordered in a linear sequence is not critical, and can be replaced with We also need to make precise what it means for an ac- no much trouble by a partial order. Library-based recogni- tion sequence to satisfy an observation sequence made up of tion can also be accommodated by a suitable compilation of actions. E.g., the action sequence π = {a, b, c, d, e, a} sat- (acyclic) libraries into Strips [Lekavy´ and Navrat,´ 2007].A isfies the observation sequences O1 = {b, d, a} and O2 = critical aspects of the model is that it does not weight the hy- {a, c, a}, but not O3 = {b, d, c}. This can be formalized with potheses G in G but it just filters them.