Heuristic Approaches for Goal Recognition in Incomplete Domain Models

04/16/2018
by   Ramon Fraga Pereira, et al.
PUCRS
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Recent approaches to goal recognition have progressively relaxed the assumptions about the amount and correctness of domain knowledge and available observations, yielding accurate and efficient algorithms. These approaches, however, assume completeness and correctness of the domain theory against which their algorithms match observations: this is too strong for most real-world domains. In this paper, we develop goal recognition techniques that are capable of recognizing goals using incomplete (and possibly incorrect) domain theories. We show the efficiency and accuracy of our approaches empirically against a large dataset of goal and plan recognition problems with incomplete domains.

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1 Introduction

Goal recognition is the problem of identifying the correct goal intended by an observed agent, given a sequence of observations as evidence of its behavior in an environment and a domain model describing how the observed agent generates such behavior. Approaches to solve this problem vary on the amount of domain knowledge used in the behavior, or plan generation, model employed by the observed agent [Sukthankar et al.2014], as well as the level of observability and noise in the observations used as evidence [Sohrabi et al.2016]. Recent work has progressively relaxed the assumptions about the accuracy and amount of information available in observations required to recognize goals [E.-Martín et al.2015, Sohrabi et al.2016, Pereira and Meneguzzi2016, Pereira et al.2017a]. Regardless of the type of domain model formalism describing the observed agent’s behavior, all recent approaches assume that the planning domain models are correct and complete, restricting its application to realistic scenarios in which the domain modeler either has an incomplete or incorrect model of the behavior under observation.

Specifically, real-world domains have two potential sources of uncertainty: (1) ambiguity in how actions performed by agents are realized; and (2) ambiguity from how imperfect sensor data reports features of the environment. The former stems from an incomplete understanding of the action being modeled and requires a domain modeler to specify a number of alternative versions of the same action to cover the possibilities. For example, an action to turn on the gas burner in a cooker may or may not require the observed agent to press a spark button. The latter stems from imperfections in the way actions themselves may be interpreted from real-world noisy data, e.g.

, if one uses machine learning algorithms to classify objects to be used as features (

e.g., logical facts) of the observations [Granada et al.2017], certain features may not be reliably recognizable, so it is useful to model a domain with such feature as optional.

In this paper, we develop heuristic approaches to goal recognition to cope with incomplete planning domain models [Nguyen et al.2017] and provide five key contributions. First, we formalize goal recognition in incomplete domains (Section 2) combining the standard formalization of RamirezG_AAAI2010 RamirezG_IJCAI2009,RamirezG_AAAI2010 for plan recognition and that of Nguyen_AIJ_2017 Nguyen_AIJ_2017. Second, we adapt an algorithm from [Hoffmann et al.2004] to extract possible landmarks in incomplete domain models (Section 3). Third, we develop a notion of overlooked landmarks that we can extract online as we process (on the fly) observations that we can use to match candidate goals to the multitude of models induced by incomplete domains. Fourth, we develop two heuristics that account for the various types of landmark as evidence in the observations to efficiently recognize goals (Section 4). Finally, we build a new dataset for goal recognition in incomplete domains based on an existing one [Pereira et al.2017a, Pereira and Meneguzzi2017] by removing varying amounts of information from complete domain models and annotating them with possible preconditions and effects that account for uncertain and possibly wrong information (Section 5). We evaluate the approaches on this dataset and show that they are fast and accurate for recognizing goals in complex incomplete domain models at most percentages of incompleteness.

2 Problem Formulation

2.1 STRIPS Domain Models

We assume that the agents being observed reason using planning domains described in the STRIPS [Fikes and Nilsson1971] domain model , where: is a set of predicates with typed variables. Grounded predicates represent logical values according to some interpretation as facts, which are divided into two types: positive and negated facts, as well as constants for truth () and falsehood (); is a set of operators , where can be divided into positive effects (the add list) and negative effects (the delete list). An operator with all variables bound is called an action and allows state change. An action instantiated from an operator is applicable to a state iff and results in a new state . A planning problem within and a set of typed objects is defined as , where: is a set of facts (instantiated predicates from and ); is a set of instantiated actions from and ; is the initial state (); and is a partially specified goal state, which represents a desired state to be achieved. A plan for is a sequence of actions that modifies the initial state into a state in which the goal state holds by the successive execution of actions in a plan .

2.2 Incomplete STRIPS Domain Models

The agent reasoning about the observations and trying to infer a goal has information described using the formalism of incomplete domain models from Nguyen_AIJ_2017 Nguyen_AIJ_2017, defined as . Here, contains the definition of incomplete operators comprised of a six-tuple , where: and have the same semantics as in the STRIPS domain models; and possible preconditions that might be required as preconditions, as well as and that might be generated as possible effects respectively as add or delete effects. An incomplete domain has a completion set comprising all possible domain models derivable from the incomplete one. There are possible such models where , and a single (unknown) ground-truth model that actually drives the observed state. An incomplete planning problem derived from an incomplete domain and a set of typed objects is defined as , where: is the set of facts (instantiated predicates from ), is the set of incomplete instantiated actions from with objects from , is the initial state, and is the goal state.

Most approaches to planning in incomplete domains [Weber and Bryce2011, Nguyen and Kambhampati2014, Nguyen et al.2017] assume that plans succeed under the most optimistic conditions, namely that: possible preconditions do not need to be satisfied in a state; possible add effects are always assumed to occur in the resulting state; and delete effects are ignored in the resulting state. Formally, an incomplete action instantiated from an incomplete operator is applicable to a state iff and results in a new state such that . Thus, a valid plan that achieves a goal from in an incomplete planning problem is a sequence of actions that corresponds to an optimistic sequence of states. Example 1 from WeberBryce_ICAPS_2011 WeberBryce_ICAPS_2011 illustrates an abstract incomplete domain and a valid plan for it.

Example 1

Let be incomplete planning problem, where: ; , where:

  • ; and .

The sequence of actions is a valid plan to achieve goal state from the initial state . It corresponds to the optimistic state sequence: . The number of completions for this example is (2 possible preconditions and 3 possible effects, i.e., 1 possible add effect and 2 possible delete effects).

2.3 Goal Recognition in Incomplete Domains

Goal recognition is the task of recognizing and anticipating agents’ goals by observing their interactions in an environment. Whereas most planning-based goal recognition approaches assume complete domain model [Ramírez and Geffner2009, Ramírez and Geffner2010, Keren et al.2014, E.-Martín et al.2015, Sohrabi et al.2016, Pereira and Meneguzzi2016, Pereira et al.2017a], we assume that the observer has an incomplete domain model while the observed agent is planning and acting with a complete domain model. To account for such uncertainty, the model available to the observer contains possible preconditions and effects as defined in Section 2.2. Like most planning approaches in incomplete domains [Weber and Bryce2011, Nguyen and Kambhampati2014, Nguyen et al.2017], we reason about possible plans with incomplete actions (observations) by assuming that they succeed under the most optimistic conditions. We formalize goal recognition over incomplete domain models in Definition 1.

Definition 1 (Goal Recognition Problem)

A goal recognition problem with an incomplete domain model is a quintuple , where:

  • is an incomplete domain model (with possible preconditions and effects). is the set of typed objects in the environment, in which is the set of instantiated predicates from , and is the set of incomplete instantiated actions from with objects from ;

  • an initial state;

  • is the set of possible goals, which include a correct hidden goal (i.e., ); and

  • is an observation sequence of executed actions, with each observation . corresponds to the sequence of actions (i.e., a plan) to solve a problem in a complete domain in .

A solution for a goal recognition problem in incomplete domain models is the correct hidden goal that the observation sequence of a plan execution achieves. As most goal recognition approaches, observations consist of the action signatures of the underlying plan111Our approaches are not limited to using just actions as observations and can also deal with logical facts as observations., more specifically, we observe incomplete actions with possible precondition and effects, in which some of the preconditions might be required and some effects might change the environment. While a full (or complete) observation sequence contains all of the action signatures of the plan executed by the observed agent, an incomplete observation sequence contains only a sub-sequence of actions of a plan and thus misses some of the actions actually executed in the environment.

3 Landmark Extraction in Incomplete Domains

In planning, landmarks are facts (or actions) that must be achieved (or executed) at some point along all valid plans to achieve a goal from an initial state [Hoffmann et al.2004]. Landmarks are often used to build heuristics for planning algorithms [Richter et al.2008, Richter and Westphal2010]. Whereas landmark-based heuristics extract landmarks from complete and correct domain models in the planning literature, we extend the landmark extraction algorithm of Hoffmann et al. in Hoffmann2004_OrderedLandmarks to extract definite and possible landmarks in incomplete STRIPS domain models. This algorithm uses a Relaxed Planning Graph (RPG), which is a leveled graph that ignores the delete-list effects of all actions, thus containing no mutex relations [Hoffmann and Nebel2001]. Once the RPG is built, this algorithm extracts a set of landmark candidates by back-chaining from the RPG level in which all facts of the goal state are possible, and, for each fact in , it checks which facts must be true until the first level of the RPG. For example, if fact is a landmark and all actions that achieve share as precondition, then is a landmark candidate. To confirm that a landmark candidate is indeed a necessary condition, and thus a landmark, the algorithm builds a new RPG removing actions that achieve the landmark candidate and checks the solvability over this modified problem. If the modified problem is unsolvable, then the landmark candidate is a necessary landmark. This means that the actions that achieve the landmark candidate are necessary to solve the original planning problem. Deciding the solvability of a relaxed planning problem using an RPG structure can be done in polynomial time [Blum and Furst1997].

We adapt the extraction algorithm from Hoffmann2004_OrderedLandmarks Hoffmann2004_OrderedLandmarks222We chose this specific algorithm due to its simplicity and runtime efficiency. to extract landmarks from incomplete domain models by building an Optimistic Relaxed Planning Graph (ORPG) instead of the original RPG. An ORPG is leveled graph that deals with incomplete domain models by assuming the most optimistic conditions. Thus, besides ignoring the delete-effects of all actions, this graph also ignores possible preconditions and possible delete-effects, whereas we use all possible add effects. The ORPG allows us to extract definite and possible landmarks, formalized in Definitions 2 and 3.

Definition 2 (Definite Landmark)

A definite landmark is a fact (landmark) extracted from a known add effect of an achiever333An achiever is an action at the level before a candidate landmark in the RPG that can be used to achieve this candidate landmark. (action) in the ORPG.

Definition 3 (Possible Landmark)

A possible landmark is a fact (landmark) extracted from a possible add effect of an achiever (action) in the ORPG.

Figure 1: ORPG for Example 1. Green arrows represent preconditions, Orange arrows represent add effects, and Purple dashed arrows represent possible add effects. Light-Blue boxes represent the set of definite landmarks and Light-Yellow boxes represent the set of possible landmarks. Hexagons represent actions.

Figure 1 shows the ORPG for Example 1. The set of definite landmarks is (Light-Blue in Figure 1), and the set of possible landmarks is (Light-Yellow in Figure 1). The classical landmark extraction algorithm from Hoffmann2004_OrderedLandmarks (without the most optimistic conditions), returns as landmarks ignoring as a fact landmark because it does not assume the most optimistic condition that possible add effects always occur. Therefore, the action was not considered as a possible achiever action. Although we use this modification to recognize goals in incomplete domain models, these landmarks can easily be used to build heuristics for planning in incomplete domains.

4 Heuristic Goal Recognition Approaches

noinline,color=red!80noinline,color=red!80todo: noinline,color=red!80FRM: Are overlooked landmarks from the set of definite landmarks? Or can possible preconditions/effects be overlooked landmarks?noinline,color=orange!80noinline,color=orange!80todo: noinline,color=orange!80RAMON: It depends on two factors: if the fact is extracted from known add effects, then it is a definite landmark. However, if it is extracted from possible add effects, then it is an overlooked landmark. We just check the observed facts on-the-fly.

Key to our goal recognition approaches are observing the evidence of achieved landmarks during observations to recognize which goal is more consistent with the observations. To do so, our approaches combine the concepts of definite and possible with that of overlooked landmarks. An overlooked landmark is an actual landmark, i.e., a necessary fact for all valid plans towards a goal, that was not detected by approximate landmark extraction algorithms. Since we are dealing with incomplete domain models, and it is possible that they have few (or no) definite and/or possible landmarks, we extract overlooked landmarks from the evidence in the observations as we process them in order to enhance the set of landmarks useable by our heuristic. To find such landmarks we build a new ORPG removing observed actions that achieve a potentially overlooked fact landmark and check the solvability of this modified problem. If the modified problem is indeed unsolvable, then this fact is an overlooked landmark.

4.1 Goal Completion Heuristic

We now combine our notions of landmarks to develop a goal recognition heuristic for recognizing goals in incomplete domain models. Our heuristic estimates the correct goal in the set of candidate goals by calculating the ratio between achieved

definite (), possible (), and overlooked () landmarks and the amount of definite (), possible (), and overlooked () landmarks. The estimate computed using Equation 1 represents the percentage of achieved landmarks for a candidate goal from observations.

(1)

4.2 Uniqueness Heuristic

Most goal recognition problems contain multiple candidate goals that share common fact landmarks, generating ambiguity that jeopardizes the goal completion heuristic. Clearly, landmarks that are common to multiple candidate goals are less useful for recognizing a goal than landmarks that exist for only a single goal. As a consequence, computing how unique (and thus informative) each landmark is can help disambiguate similar goals for a set of candidate goals. Our second goal recognition heuristic is based on this intuition, which we develop through the concept of landmark uniqueness, which is the inverse frequency of a landmark among the landmarks found in a set of candidate goals. Intuitively, a landmark that occurs only for a single goal within a set of candidate goals has the maximum uniqueness value of 1. Equation 2 formalizes the computation of the landmark uniqueness value for a landmark and a set of landmarks for all candidate goals .

(2)

Using the concept of landmark uniqueness value, we estimate which candidate goal is the intended one by summing the uniqueness values of the landmarks achieved in the observations. Unlike our previous heuristic, which estimates progress towards goal completion by analyzing just the set of achieved landmarks, the landmark-based uniqueness heuristic estimates the goal completion of a candidate goal by calculating the ratio between the sum of the uniqueness value of the achieved landmarks of and the sum of the uniqueness value of all landmarks of a goal . Our new uniqueness heuristic also uses the concepts of definite, possible, and overlooked landmarks. We store the set of definite and possible landmarks of a goal separately into and , and the set of overlooked landmarks into . Thus, the uniqueness heuristic effectively weighs the completion value of a goal by the informational value of a landmark so that unique landmarks have the highest weight. To estimate goal completion using the landmark uniqueness value, we calculate the uniqueness value for every extracted (definite, possible, and overlooked) landmark in the set of landmarks of the candidate goals using Equation 2. Since we use three types of landmarks and they are stored in three different sets, we compute the landmark uniqueness value separately for them, storing the landmark uniqueness value of definite landmarks into , the landmark uniqueness value of possible landmarks into , and the landmark uniqueness value of overlooked landmarks into . Our uniqueness heuristic is denoted as and formally defined in Equation 3.

(3)

5 Experiments and Evaluation

We now describe the experiments carried out to evaluate our goal recognition and discuss the results over our new dataset.

5.1 Dataset and Setup

For experiments, we used openly available goal and plan recognition datasets [Pereira and Meneguzzi2017], which contain thousands of recognition problems. These datasets contain large and non-trivial planning problems (with optimal and sub-optimal plans as observations) for 15 planning domains, including domains and problems from datasets that were developed by Ramírez and Geffner RamirezG_IJCAI2009,RamirezG_AAAI2010. All planning domains in these datasets are encoded using the STRIPS fragment of PDDL. Each goal/plan recognition problem in these datasets contains a complete domain definition, an initial state, a set of candidate goals, a correct hidden goal in the set of candidate goals, and an observation sequence. An observation sequence contains actions that represent an optimal or sub-optimal plan that achieves a correct hidden goal, and this observation sequence can be full or partial. A full observation sequence represents the whole plan that achieves the hidden goal, i.e., 100% of the actions having been observed. A partial observation sequence represents a plan for the hidden goal, varying in 10%, 30%, 50%, or 70% of its actions having been observed. To evaluate our goal recognition approaches in incomplete domain models, we modify the domain models of these datasets by adding annotated possible preconditions and effects. Thus, the only modification to the original datasets is the generation of new, incomplete, domain models for each problem, varying the percentage of incompleteness in these domains.

We vary the percentage of incompleteness of a domain from 20% to 80%. For example, consider that a complete domain has, for all its actions, a total of 10 preconditions, 10 add effects, and 10 delete effects. A derived model with 20% of incompleteness needs to have 2 possible preconditions (8 known preconditions), 2 possible add effects (8 known add effects), and 2 possible delete effects (8 known delete effects), and so on for other percentages of incompleteness. Like [Nguyen and Kambhampati2014, Nguyen et al.2017], we generated the incomplete domain models by following three steps involving randomly selected preconditions/effects: (1) move a percentage of known preconditions and effects into lists of possible preconditions and effects; (2) add possible preconditions from delete effects that are not preconditions of a corresponding operator; and (3) add into possible lists (of preconditions, add effects, or delete effects) predicates whose parameters fit into the operator signatures and are not precondition or effects of the operator. These three steps yield three different incomplete domain models from a complete domain model for each percentage of domain incompleteness with different possible lists of preconditions and effects. We ran all experiments using a single core of a 12 core Intel(R) Xeon(R) CPU E5-2620 v3 @ 2.40GHz with 16GB of RAM, and a JavaVM with a 2GB memory and a 2-minute time limit.

5.2 Evaluation Metrics

We evaluated our approaches using three metrics: recognition time in seconds (Time); Accuracy (Acc %)444This metric is analogous to the Quality (Q) metric , used for most planning-based goal recognition approaches [Ramírez and Geffner2009, Ramírez and Geffner2010, E.-Martín et al.2015, Sohrabi et al.2016]., representing the rate at which the the algorithm returns the correct goal among the most likely goals in ; and Spread in (S) as the average number of returned goals. As each percentage of domain incompleteness has three different incomplete domain models, the percentage columns (20%, 40%, 60%, and 80%) in Table 1 report averages for Time, Acc %, and S by taking into account the results of the three incomplete domain models. The first column of Table 1 shows the total number of goal recognition problems for each domain name, and each row expresses averages for the number of candidate goals ; the percentage of the plan that was actually observed (% Obs); and the average number of observations per problem .

We adapt the Receiver Operating Characteristic (ROC) curve metric to show the trade-off between true positive and false positive results. An ROC curve is often used to compare the true positive predictions as well as the false positive predictions of the experimented approaches. Here, each prediction result of our goal recognition approaches represents one point in the space, and thus, instead of a curve, our graphs show the spread of our results over ROC space. In the ROC space, the diagonal line represents a random guess to recognize a goal from observations. This diagonal line divides the ROC space in such a way that points above the diagonal represent good classification results (better than random guess), whereas points below the line represent poor results (worse than random guess). The best possible (perfect) prediction for recognizing goals are points in the upper left corner (0,100).

5.3 Results

Incompleteness of (%) 20% 40% 60% 80%
# Obs (%) Time Acc % S Time Acc % S Time Acc % S Time Acc % S Time Acc % S Time Acc % S Time Acc % S Time Acc % S

Blocks

(12192)

20.3 10 1.82 0.049 33.7% 1.45 0.044 18.4% 1.66 0.034 29.3% 1.74 0.031 28.7% 2.57 0.039 41.1% 2.46 0.035 32.5% 1.88 0.041 45.0% 2.89 0.037 32.2% 2.78
30 4.93 0.050 44.9% 1.33 0.046 41.5% 1.07 0.051 50.9% 1.67 0.050 50.9% 1.06 0.054 60.8% 2.04 0.052 50.9% 1.05 0.058 56.4% 2.11 0.055 51.1% 1.05
50 7.66 0.053 52.6% 1.31 0.052 52.0% 1.06 0.060 62.1% 1.54 0.059 61.4% 1.02 0.061 69.8% 1.69 0.061 61.1% 1.02 0.067 65.0% 1.71 0.065 59.5% 1.04
70 11.1 0.059 72.4% 1.33 0.058 69.5% 1.04 0.065 76.6% 1.44 0.063 76.6% 1.02 0.069 85.8% 1.67 0.066 75.3% 1.02 0.071 82.2% 1.76 0.068 73.8% 1.04
100 14.5 0.062 94.6% 1.32 0.057 83.7% 1.09 0.067 96.4% 1.38 0.066 96.0% 1.08 0.078 99.6% 1.61 0.070 94.2% 1.06 0.081 99.3% 1.66 0.077 93.5% 1.11

Campus

(900)

2.0 10 1.0 0.004 75.6% 1.22 0.003 44.4% 1.0 0.003 84.4% 1.24 0.003 51.1% 1.11 0.005 57.8% 1.29 0.004 60.0% 1.40 0.005 80.0% 1.49 0.005 75.6% 1.58
30 2.0 0.006 77.8% 1.11 0.005 55.6% 1.0 0.006 80.0% 1.22 0.005 68.9% 1.09 0.008 62.2% 1.24 0.007 73.3% 1.29 0.007 84.4% 1.47 0.007 86.7% 1.42
50 3.0 0.007 91.1% 1.11 0.006 73.3% 1.02 0.008 91.1% 1.13 0.007 82.2% 1.04 0.009 64.4% 1.16 0.008 77.8% 1.24 0.009 77.8% 1.47 0.009 73.3% 1.31
70 4.47 0.006 97.8% 1.02 0.005 82.2% 1.0 0.007 95.6% 1.09 0.007 93.3% 1.0 0.011 77.8% 1.09 0.010 84.4% 1.02 0.011 82.2% 1.29 0.010 68.9% 1.22
100 5.4 0.009 100.0% 1.0 0.008 82.2% 1.0 0.009 97.8% 1.09 0.009 95.6% 1.04 0.010 86.7% 1.13 0.009 84.4% 1.0 0.012 86.7% 1.24 0.010 75.6% 1.18

depots

(4368)

8.9 10 3.13 0.079 30.2% 1.38 0.079 37.7% 1.35 0.071 37.3% 1.87 0.063 48.0% 2.01 0.075 39.3% 2.01 0.075 44.0% 2.02 0.067 52.8% 2.68 0.060 53.6% 2.81
30 8.61 0.079 40.5% 1.33 0.071 49.2% 1.08 0.067 57.5% 1.62 0.063 55.6% 1.10 0.083 48.0% 1.47 0.075 52.0% 1.10 0.075 67.1% 2.37 0.071 57.9% 1.40
50 14.04 0.079 61.5% 1.32 0.075 67.9% 1.04 0.075 75.8% 1.38 0.071 72.2% 1.03 0.087 63.9% 1.35 0.087 69.0% 1.05 0.095 86.5% 1.85 0.087 79.4% 1.15
70 19.71 0.079 78.6% 1.28 0.079 81.3% 1.04 0.079 90.5% 1.28 0.075 85.3% 1.02 0.099 79.8% 1.33 0.095 80.2% 1.02 0.107 92.1% 1.71 0.103 88.1% 1.06
100 27.43 0.083 89.3% 1.11 0.083 91.7% 1.02 0.083 100.0% 1.08 0.083 96.4% 1.01 0.107 90.5% 1.17 0.107 89.3% 1.01 0.119 98.8% 1.46 0.119 96.4% 1.12

Driver

(4368)

7.1 10 2.61 0.282 35.3% 1.45 0.280 36.9% 1.79 0.310 36.1% 1.37 0.303 40.5% 1.64 0.346 36.1% 2.01 0.330 43.7% 1.90 0.378 53.6% 3.13 0.355 46.8% 2.17
30 6.96 0.307 45.2% 1.90 0.299 45.6% 1.15 0.325 47.2% 1.86 0.311 45.2% 1.11 0.363 43.7% 1.92 0.358 46.8% 1.12 0.406 60.7% 2.01 0.401 57.5% 1.23
50 11.18 0.329 56.0% 1.32 0.312 62.7% 1.10 0.344 63.5% 1.73 0.329 69.4% 1.08 0.392 61.1% 1.84 0.377 63.9% 1.08 0.424 79.0% 1.95 0.395 77.8% 1.14
70 15.64 0.331 69.0% 1.26 0.319 79.4% 1.13 0.339 75.4% 1.62 0.326 82.5% 1.11 0.409 72.2% 1.76 0.400 77.4% 1.05 0.452 90.1% 1.89 0.434 88.5% 1.07
100 21.71 0.338 78.6% 1.21 0.325 90.5% 1.12 0.341 82.1% 1.46 0.330 86.9% 1.07 0.428 79.8% 1.65 0.412 88.1% 1.05 0.464 95.2% 1.82 0.423 92.9% 1.05

DWR

(4368)

7.3 10 5.71 0.221 19.0% 1.31 0.208 31.7% 1.20 0.365 44.0% 2.42 0.326 47.2% 2.34 0.413 48.8% 3.65 0.401 36.9% 1.67 0.468 63.9% 4.15 0.425 56.3% 3.15
30 16.0 0.237 37.3% 1.26 0.220 50.8% 1.08 0.392 66.3% 1.88 0.371 54.8% 1.33 0.449 72.6% 2.97 0.424 53.6% 1.19 0.603 77.0% 3.34 0.548 59.9% 1.46
50 26.21 0.274 50.0% 1.27 0.251 62.3% 1.06 0.348 75.0% 1.58 0.326 65.5% 1.12 0.480 80.2% 1.65 0.456 63.5% 1.10 0.718 84.1% 2.58 0.653 62.7% 1.15
70 36.86 0.279 67.1% 1.20 0.264 80.2% 1.02 0.484 91.7% 1.36 0.431 80.2% 1.04 0.561 91.7% 1.31 0.508 79.8% 1.04 0.802 92.9% 2.25 0.772 80.6% 1.04
100 51.89 0.286 85.7% 1.10 0.270 92.9% 1.02 0.507 98.8% 1.23 0.489 95.2% 1.01 0.624 100.0% 1.04 0.575 95.2% 1.04 0.917 100.0% 2.12 0.861 96.4% 1.04

Ferry

(4368)

7.6 10 2.93 0.071 73.8% 1.59 0.068 51.2% 1.35 0.083 62.7% 1.71 0.081 48.4% 1.22 0.091 67.9% 2.02 0.087 61.9% 1.54 0.098 80.6% 3.10 0.094 64.3% 1.94
30 7.68 0.075 94.0% 1.43 0.073 69.0% 1.08 0.102 87.7% 1.62 0.095 71.8% 1.08 0.110 85.7% 1.78 0.104 77.0% 1.10 0.113 88.9% 1.97 0.108 73.8% 1.15
50 12.36 0.087 98.8% 1.32 0.084 73.4% 1.02 0.103 96.0% 1.44 0.099 76.2% 1.06 0.126 95.6% 1.63 0.110 79.8% 1.04 0.125 92.5% 1.85 0.112 76.2% 1.07
70 17.36 0.095 100.0% 1.24 0.088 82.1% 1.02 0.117 100.0% 1.37 0.112 86.5% 1.02 0.138 100.0% 1.58 0.125 93.3% 1.02 0.153 100.0% 1.11 0.144 90.9% 1.02
100 24.21 0.112 100.0% 1.0 0.106 88.1% 1.02 0.129 100.0% 1.0 0.118 88.1% 1.04 0.141 100.0% 1.0 0.133 100.0% 1.0 0.153 100.0% 1.02 0.140 97.6% 1.02

Intrusion

(5580)

16.7 10 1.92 0.022 46.7% 2.07 0.020 18.1% 3.1 0.034 34.6% 1.92 0.032 16.2% 2.08 0.043 25.7% 2.39 0.040 40.0% 6.43 0.049 16.5% 1.82 0.044 67.0% 10.69
30 4.48 0.031 86.3% 1.31 0.030 41.9% 1.57 0.039 65.7% 1.45 0.037 50.5% 1.03 0.046 34.9% 2.68 0.044 48.9% 2.28 0.050 23.8% 1.97 0.049 61.6% 6.93
50 6.7 0.038 93.7% 1.15 0.033 55.9% 1.52 0.041 79.4% 1.24 0.040 62.2% 1.04 0.048 49.2% 2.81 0.045 58.1% 1.78 0.053 31.4% 2.18 0.052 63.8% 5.7
70 9.55 0.049 95.2% 1.12 0.047 60.6% 1.52 0.052 85.7% 1.17 0.049 72.4% 1.01 0.059 61.0% 2.79 0.056 63.8% 1.76 0.066 44.1% 2.27 0.064 69.8% 4.37
100 13.07 0.053 100.0% 1.03 0.052 62.2% 1.51 0.060 87.4% 1.13 0.059 79.3% 1.02 0.062 87.4% 3.27 0.058 70.4% 1.76 0.074 58.5% 2.21 0.067 83.0% 3.14

IPC-Grid

(8076)

8.7 10 2.8 0.492 28.8% 1.08 0.484 28.8% 2.16 0.924 30.4% 3.24 0.622 30.4% 3.24 7.083 45.1% 3.24 6.801 49.5% 4.16 13.532 99.1% 7.58 12.831 96.9% 7.62
30 7.49 0.496 45.1% 1.19 0.488 44.4% 1.19 0.946 44.1% 2.76 0.714 42.5% 2.76 8.499 46.8% 2.76 6.955 61.9% 2.82 14.294 99.3% 6.01 13.674 97.2% 5.79
50 12.02 0.498 66.7% 1.11 0.495 57.5% 1.11 0.968 66.8% 2.06 0.75 66.0% 2.06 9.547 58.6% 2.06 7.231 69.5% 2.15 15.063 99.8% 3.99 14.119 98.3% 3.84
70 17.16 0.503 73.4% 1.10 0.501 63.0% 1.07 1.232 75.5% 1.83 0.891 72.5% 1.83 10.259 67.8% 1.83 8.414 75.6% 1.83 15.588 99.3% 3.22 14.72 98.7% 3.06
100 21.84 0.533 82.5% 1.05 0.516 67.8% 1.07 1.317 85.7% 1.06 0.945 81.0% 1.06 10.571 81.4% 1.46 9.765 80.3% 1.06 16.208 100.0% 2.26 15.051 100.0% 1.01

Kitchen

(900)

3.0 10 1.33 0.002 51.1% 1.47 0.002 57.8% 2.11 0.003 15.6% 1.33 0.002 33.3% 1.64 0.003 26.7% 1.07 0.003 53.3% 1.56 0.005 57.8% 1.33 0.004 57.8% 1.33
30 3.33 0.004 40.0% 1.18 0.003 68.9% 1.44 0.006 26.7% 1.29 0.005 64.4% 1.58 0.005 33.3% 1.07 0.005 60.0% 1.29 0.007 48.9% 1.33 0.006 48.9% 1.33
50 4.0 0.006 53.3% 1.07 0.005 60.0% 1.29 0.008 48.9% 1.36 0.007 62.2% 1.4 0.008 37.8% 1.04 0.009 51.1% 1.36 0.009 48.9% 1.33 0.008 48.9% 1.33
70 5.0 0.008 71.1% 1.21 0.007 62.2% 1.13 0.008 53.3% 1.33 0.008 53.3% 1.38 0.008 46.7% 1.09 0.008 42.2% 1.29 0.009 44.4% 1.33 0.009 46.7% 1.44
100 7.47 0.008 73.3% 1.21 0.007 77.8% 1.11 0.008 48.9% 1.33 0.008 62.2% 1.31 0.010 42.2% 1.09 0.009 44.4% 1.29 0.010 48.9% 1.33 0.010 57.8% 1.78

Logistics

(8076)

10.5 10 2.82 0.205 42.5% 2.01 0.200 41.6% 1.98 0.273 52.3% 2.72 0.266 53.8% 2.42 0.233 59.0% 4.07 0.229 60.3% 3.85 0.312 80.8% 5.51 0.295 77.8% 5.43
30 8.01 0.227 66.0% 1.53 0.227 60.3% 1.17 0.314 72.5% 1.68 0.303 62.5% 1.12 0.301 79.3% 2.85 0.299 72.1% 2.12 0.436 88.9% 3.08 0.394 77.1% 2.36
50 13.07 0.246 81.5% 1.31 0.221 70.4% 1.09 0.349 81.0% 1.53 0.325 73.0% 1.07 0.355 85.6% 2.21 0.340 81.0% 1.68 0.514 92.6% 2.23 0.466 82.6% 1.82
70 18.33 0.271 91.7% 1.14 0.265 75.2% 1.03 0.362 93.2% 1.19 0.354 85.0% 1.02 0.401 93.5% 1.45 0.377 87.4% 1.13 0.546 97.6% 1.44 0.508 89.5% 1.19
100 24.41 0.312 97.3% 1.04 0.240 78.7% 1.02 0.383 99.5% 1.08 0.377 93.4% 1.03 0.419 97.3% 1.09 0.444 100.0% 1.0 0.567 100.0% 1.02 0.451 99.5% 1.01

Miconic

(4368)

6.0 10 3.96 0.282 56.3% 1.58 0.278 48.0% 1.25 0.361 84.5% 2.21 0.353 57.9% 2.19 0.306 64.7% 2.93 0.294 54.8% 1.84 0.302 68.7% 3.12 0.290 59.1% 2.2
30 11.14 0.286 81.3% 1.29 0.272 77.4% 1.13 0.339 95.2% 1.67 0.325 77.0% 1.24 0.348 81.7% 1.82 0.336 73.0% 1.17 0.353 83.3% 2.07 0.340 75.0% 1.34
50 18.07 0.294 94.4% 1.19 0.281 91.3% 1.02 0.341 97.2% 1.31 0.340 89.7% 1.06 0.355 88.9% 1.46 0.348 87.3% 1.05 0.366 87.7% 1.79 0.357 83.7% 1.21
70 25.32 0.315 98.4% 1.07 0.284 96.8% 1.01 0.347 99.6% 1.13 0.343 97.2% 1.02 0.362 92.5% 1.51 0.349 94.4% 1.04 0.370 94.8% 1.26 0.361 90.9% 1.03
100 35.57 0.324 100.0% 1.02 0.311 100.0% 1.02 0.351 100.0% 1.02 0.349 100.0% 1.02 0.373 94.0% 1.07 0.355 97.6% 1.02 0.384 100.0% 1.26 0.380 100.0% 1.02

Rovers

(4368)

6.0 10 3.0 0.579 51.6% 1.81 0.567 45.2% 1.31 0.659 41.7% 1.88 0.648 52.0% 1.32 0.738 41.3% 1.94 0.726 51.6% 1.60 0.675 68.7% 3.19 0.661 61.5% 2.80
30 7.93 0.583 64.7% 1.49 0.571 65.5% 1.11 0.676 57.9% 1.49 0.663 61.5% 1.07 0.746 61.5% 1.78 0.709 57.9% 1.17 0.753 81.0% 2.45 0.721 63.5% 1.63
50 12.75 0.591 76.2% 1.19 0.575 79.4% 1.04 0.687 79.4% 1.28 0.675 82.5% 1.02 0.768 78.2% 1.49 0.753 78.6% 1.04 0.782 92.5% 1.96 0.755 76.6% 1.05
70 17.96 0.595 88.9% 1.13 0.579 90.9% 1.01 0.702 86.9% 1.14 0.695 90.9% 1.02 0.806 90.5% 1.26 0.799 90.5% 1.01 0.794 99.6% 1.74 0.783 92.1% 1.02
100 24.93 0.602 94.0% 1.07 0.583 95.2% 1.0 0.714 92.9% 1.04 0.701 94.0% 1.02 0.821 97.6% 1.12 0.815 98.8% 1.01 0.845 100.0% 1.28 0.843 98.8% 1.0

Satellite

(4368)

6.4 10 2.07 0.105 50.0% 1.88 0.100 41.7% 1.89 0.108 50.4% 2.06 0.102 47.6% 2.09 0.112 58.3% 2.02 0.107 48.8% 1.93 0.124 65.1% 2.46 0.115 56.0% 2.45
30 5.43 0.119 59.9% 1.52 0.112 53.2% 1.21 0.124 63.1% 1.63 0.115 65.1% 1.26 0.128 71.8% 1.66 0.119 65.1% 1.23 0.131 75.8% 1.77 0.127 69.8% 1.28
50 8.71 0.123 65.1% 1.25 0.118 65.1% 1.12 0.141 75.0% 1.37 0.130 75.0% 1.09 0.152 81.0% 1.46 0.144 76.6% 1.11 0.155 85.3% 1.52 0.147 79.8% 1.08
70 12.29 0.143 72.6% 1.14 0.137 71.8% 1.05 0.155 85.7% 1.29 0.151 89.7% 1.06 0.164 86.1% 1.19 0.159 85.3% 1.06 0.173 90.9% 1.26 0.165 90.9% 1.05
100 16.89 0.158 82.1% 1.08 0.146 85.7% 1.0 0.169 95.2% 1.06 0.167 97.6% 1.01 0.173 94.0% 1.08 0.170 98.8% 1.04 0.198 100.0% 1.10 0.172 100.0% 1.02

Sokoban

(4368)

7.1 10 3.18 1.339 26.6% 1.33 0.861 34.1% 1.41 Timeout - - Timeout - - Timeout - - Timeout - - Timeout - - Timeout - -
30 8.82 1.243 29.0% 1.23 0.897 40.9% 1.05 Timeout - - Timeout - - Timeout - - Timeout - - Timeout - - Timeout - -
50 14.07 1.191 32.5% 1.22 0.913 42.9% 1.03 Timeout - - Timeout - - Timeout - - Timeout - - Timeout - - Timeout - -
70 19.86 1.173 35.3% 1.19 0.964 59.1% 1.02 Timeout - - Timeout - - Timeout - - Timeout - - Timeout - - Timeout - -
100 27.71 1.145 45.2% 1.14 0.993 76.2% 1.01 Timeout - - Timeout - - Timeout - - Timeout - - Timeout - - Timeout - -

Zeno

(4368)

6.9 10 2.61 0.909 37.3% 1.38 0.837 49.6% 1.19 1.452 38.1% 1.43 1.409 42.5% 1.21 1.341 56.7% 2.12 1.258 44.8% 1.36 1.694 79.4% 3.02 1.655 56.3% 2.29
30 6.75 0.909 50.8% 1.43 0.849 57.1% 1.04 1.476 52.0% 1.41 1.425 58.7% 1.06 1.365 71.0% 1.73 1.294 57.1% 1.05 2.014 88.1% 2.1 1.869 66.3% 1.37
50 10.82 0.937 63.1% 1.29 0.857 79.8% 1.04 1.512 66.3% 1.33 1.472 72.6% 1.03 1.413 83.3% 1.43 1.349 75.8% 1.01 2.274 93.7% 1.4 2.111 78.6% 1.01
70 15.21 0.948 79.0% 1.19 0.869 90.5% 1.02 1.563 77.4% 1.22 1.488 85.7% 1.01 1.448 91.7% 1.19 1.401 86.5% 1.01 2.452 96.8% 1.12 2.353 88.1% 1.02
100 21.14 0.952 91.7% 1.07 0.881 97.6% 1.01 1.571 92.9% 1.08 1.512 96.4% 1.02 1.512 96.4% 1.04 1.447 97.6% 1.02 2.759 100.0% 1.04 2.548 96.4% 1.02
Table 1: Experimental results of our approaches for recognizing goals in incomplete STRIPS domain models.
Figure 2: ROC space for all four percentage of domain incompleteness.

Table 1 shows the experimental results of our goal recognition approaches in incomplete domain models. Our approaches yield high accuracy at low recognition time for most planning domains apart from IPC-Grid and Sokoban, which took substantial recognition time. Sokoban exceeds the time limit of 2 minutes for most goal recognition problems because this dataset contains large problems with a huge number of objects, leading to an even larger number of instantiated predicates and actions. For example, as domain incompleteness increases (i.e., the ratio of possible to definite preconditions and effects), the number of possible actions (moving between cells and pushing boxes) in a grid with 9x9 cells and 5 boxes increases substantially because as there are very few definite preconditions for several possible preconditions. The average number of possible complete domain models is huge for several domains, showing that the task of goal recognition in incomplete domains models is quite difficult and complex.

Figure 2 shows four ROC space graphs corresponding to recognition performance over the four percentages of domain incompleteness we used in our experiments. We aggregate multiple recognition problems for all domains and plot these results in ROC space varying the percentage of domain incompleteness. Although the true positive rate is high for most recognition problems at most percentages of domain incompleteness, as the percentage of domain incompleteness increases, the false positive rate also increases, leading to several problems being recognized with a performance close to the random guess line. This happens because the number of extracted landmarks decreases significantly as the number of definite preconditions and effects diminishes, and consequently, all candidate goals have few (if any) landmarks. For example, in several cases in which domain incompleteness is 60% and 80%, the set of landmarks is quite similar, leading our approaches to return more than one candidate goal (increasing the Spread in ) as the correct one. Thus, there is more uncertainty in the result of the recognition process as incompleteness increases.

6 Conclusions and Future Work

We have developed a novel goal recognition approaches that deal with incomplete domain models that represent possible preconditions and effects besides traditional models where such information is assumed to be known. The main contributions of this paper include the formalization of goal recognition in incomplete domains, two heuristic approaches for such goal recognition, novel notions of landmarks for incomplete domains, and a dataset to evaluate the performance of such approaches. Our novel notions of landmarks include that of possible landmarks for incomplete domains as well as overlooked landmarks that allow us to compensate fast but non-exhaustive landmark extraction algorithms, the latter of which can also be employed to improve existing goal and plan recognition approaches [Pereira and Meneguzzi2016, Pereira et al.2017b, Pereira et al.2017a]. Experiments over thousands of goal recognition problems in 15 planning domain models show that our approaches are fast and accurate when dealing with incomplete domains at all variations of observability and domain incompleteness.

Although our results are significant, this is just the first work to solve the problem of goal recognition in incomplete domains and a number of refinements can be investigated as future work. Since our approaches only explore the set of possible add effects to build an ORPG, we can further improve our approaches by exploring the set of possible preconditions for extracting landmarks. Second, we intend to use a propagated RPG to reason about impossible incomplete domain models, much like in [Weber and Bryce2011], to build a planning heuristic. Third, we aim to use a bayesian framework to compute probabilistic estimations of which possible complete domain is most consistent with the observations. Fourth, we can use recent work on the refinement of incomplete domain models based on plan traces [Zhuo et al.2013] as part of a complete methodology to infer domains with incomplete information based on plan traces. Finally, and most importantly, being able to recognize goals in incomplete domains allows us to employ learning techniques to automated domain modeling [Jiménez et al.2012] and cope with its possible inaccuracies.

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