Factores que afectan la actividad de los enemigos naturales

Ramirez and Geffner (2009) developed an approach for identifying the goal set G ∈ G where some optimal plan for G is compatible with the sequence of ob- served actions. In this work, a goal recognition problem is defined as a tuple T = h P , G , O i where P is a planning domain in the form of P = h F , A , I i where F is the set of predicates, A is the set of actions, and I is the set of initial conditions; G is a set of possible goal sets or hypotheses; and O is an observed action sequence O = (o1, ..., on)where onis an action in the sequence. The obser- vation sequence O may be incomplete, but is sequentially ordered. However, the time step in which the actions have been observed is unknown. The solution to this goal recognition problem is a set G∗, which consists of all goals subsets G ∈ G having an optimal plan π consistent with O.

In order to solve a goal recognition problem using planning, the goal recogni- tion problem T = h P , G , O i is transformed into a problem T0 = h P0, G0, O0i, where O0 is empty and for each of the observed actions oi ∈ O a new set of pred- icates Foand a new set of actions Ao are created such that Fo = {po|o ∈ O} and Ao = {ao|o ∈ O}. Each new action ao has the same preconditions and effects as its counterpart a except for the new predicate po, which is added to add(ao), and poi−1, which refers to the observed action that precedes action a ∈ O and is, there- fore, added to prec(ao). Hence, in the transformed goals recognition problem T0, the planning domain is P0 = h F0, A0, I0i where F0 _{= F ∪ F}

o, A0 = A ∪ Ao, and I0 = I, and the set of possible goal sets G0 = {G ∪ Fo| G ∈ G}. The extra goals Focan be only achieved by the new actions Ao, which because of the added pre- conditions can be only applied after all the corresponding preceding oi ∈ O are executed. Therefore, the transformed problem P0along with each G0 ∈ G0 can be executed in a planner to find a plan π that is consistent with O. The set solution G∗consists of each goal set G0 ∈ G0that gets a consistent plan.

In particular, this approach uses the optimal planner HSP∗_f(Haslum and Geff- ner, 2000) to exactly compute G∗. Although, it gives the optimal solution, it is computationally expensive. In addition, if there is a non-optimal plan consistent with the observations, this method fails. To remove both limitations, this work also considers approximate methods. The first method uses the suboptimal plan- ner FF (Hoffmann and Nebel, 2001) to approximate the set, while the second is based on the hsa heuristic, which approximates the solution by computing a re- laxed plan avoiding planning altogether.

The limitation of this work is the assumption that agents act optimally, which limits the theory to the space of optimal plans. Hence, suboptimal plans com- patible with the observations are not considered. To overcome this limitation, Ramirez and Geffner (2010) developed a more principled approach for estimat- ing the probability of each possible goal, based on how much the observed ac- tions contribute to achieving that goal. This theory is rooted in the space of all possible plans that might be used to achieve each goal, and defines likelihood of a goal given a sequence of observations as the cost difference between achieving the goal complying with the observations, and not complying with them. This cost is computed by means of two calls to a planner for each possible goal. This approach defines a goal recognition problem as a tuple T = hP, G, O, P ri where P is a planning domain and initial conditions, G is a set of possible goal set or hy- potheses, O is the observed action sequence O = (o1, ..., on), and P r is the prior probability distribution over the goal sets G ∈ G. The solution to a goal recog- nition problem is a probability distribution over each goal set G ∈ G giving the relative likelihood of each goal set. These posterior goal probabilities P (G|O) can be characterized using Bayes Rule as:

P r(G|O) = P r(O|G) P r(G) (2.9)

where P r(G) is the prior distribution over G ∈ G, and P r(O|G) is the likelihood of observing O when the goal is G. Ramirez goes on to characterize the likeli- hood P r(O|G) in terms of cost differences for achieving G under two conditions: complying with the observations O, and not complying with the observations O. More precisely, Ramirez characterizes the likelihood, P r(O|G), in terms of a Boltzman distribution:

P r(O|G) = α exp{−β ∆(G, O)}

1 + exp{−β ∆(G, O)} (2.10)

where α is a normalizing constant, β is a positive constant, and ∆(G, O) is the cost difference between achieving the goal with and without the observations:

∆(G, O) = Cost(G|O) − Cost(G|O) (2.11) Putting Equations 2.9 and 2.10 together yields:

P r(G|O) = α exp{−β ∆(G, O)}

1 + exp{−β ∆(G, O)}P r(G) (2.12) By computing ∆(G, O) for each possible goal, Equation 2.12 can be used to compute a probability distribution over those goals. The two costs necessary to

compute ∆(G, O) can be found by optimally solving the two planning problems G|O and G|O. Ram´ırez shows how the constraints O and O can be compiled into the goals, conditions, and effects of the planning problem so that a standard planner can be used to find plans for G|O and G|O.

To illustrate the Ram´ırez approach, consider the example shown in Figure 2.7, where an agent can move up, left, and right at cost 1. It has two possible goals, G1 and G2, and O = (o1) as the observed sequence. For goal G1, Cost(G1|O) = 4, and Cost(G1|O) = 4. (The costs are the same since o1 is on an optimal path to G1and there is more than one optimal path that reaches G1but does not include o1.) Hence, ∆(G1, O) = 0, and P r(G1|O) = α(0.5). In contrast, Cost(G2|O) = 3 and Cost(G2|O) = 3. This results in ∆(G2, O) = 0, and P r(G2|O) = α(0.5). This means that G1 and G2are the same likely to occur given O = (o1).

G1 G2

o1

Figure 2.7: A 2x3 plan network for goals G1and G2.

A major drawback to the Ramirez’s approach is the computational expense of finding two plans for every possible goal. Moreover, the constraints O and O make the planning problems more difficult to solve. As a result, even for rel- atively simple problems it can take a significant amount of time to find all the plans necessary for this computation. This makes the approach impractical to use for any sort of real-time goal recognition problem.