Hierarchical task network-based emergency task planning with incomplete information, concurrency and uncertain duration

Accepted Manuscript

Hierarchical task network-based emergency task planning with incomplete information, concurrency and uncertain duration Dian Liu , Hongwei Wang , Chao Qi , Peng Zhao , Jian Wang PII: DOI: Reference:

S0950-7051(16)30306-9 10.1016/j.knosys.2016.08.029 KNOSYS 3656

To appear in:

Knowledge-Based Systems

Received date: Revised date: Accepted date:

5 February 2016 23 August 2016 31 August 2016

Please cite this article as: Dian Liu , Hongwei Wang , Chao Qi , Peng Zhao , Jian Wang , Hierarchical task network-based emergency task planning with incomplete information, concurrency and uncertain duration, Knowledge-Based Systems (2016), doi: 10.1016/j.knosys.2016.08.029

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Graphical abstract

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Highlights

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 A paradigm for conditional temporal HTN planning with uncertain durations is proposed.  Rules for generating correct and valid temporal constraints are designed to achieve concurrency.  Eliminating redundancy of temporal constraints network promotes the time performance.

ACCEPTED MANUSCRIPT Hierarchical task network-based emergency task planning with incomplete information, concurrency and uncertain duration Dian Liub,c, Hongwei Wanga,b,c,*, Chao Qia,b,c, Peng Zhaob,c, Jian Wangb,c a School of Management, Huazhong University of Science and Technology, Wuhan 430074, China b Department of System Science and Engineering, School of Automation, Huazhong University of Science and Technology, Wuhan 430074, China c MOE Key Laboratory of Image Processing and Intelligent Control, Huazhong University of Science and Technology, Wuhan 430074, China

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Abstract: This paper focuses on the emergency task planning problem with the following characteristics: incomplete initial environment information, concurrent execution and uncertain execution durations. To address this problem, the conditional temporal hierarchical task network (HTN) planning with uncertain durations should be investigated. However, the research work on conditional HTN planning mainly concentrates on the planning domain without temporal features and pays no attention to the handling of concurrency and uncertain durations. This motivates us to propose a planning paradigm based on the conditional HTN planning. The planning paradigm incorporates a temporal reasoning technique suitable for conditional HTN planning to achieve concurrency and a temporal management approach to confirm the satisfiability of temporal constraints efficiently. For temporal reasoning, rules for generating temporal constraints are designed to avoid three types of flaws due to interactions between durative actions; a mechanism for detecting invalid temporal constraints is also presented. Regarding temporal management, Conditional Simple Temporal Network with Uncertainty (CSTNU) is utilized to represent the temporal constraints involving uncontrollable intervals and observations, and an approach for eliminating redundant temporal constraints is proposed to reduce the temporal constraint to be represented in a CSTNU to promote efficiency. Finally, an experimental study of an earthquake rescue domain demonstrates the viability of the proposed planning paradigms.

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1. Introduction

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Keywords Emergency task planning; HTN planning; incomplete information; temporal reasoning; temporal management

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In recent years, emergency response decision making has gained more and more attention from researchers and practitioners worldwide due to the frequent occurrences of natural and manmade disasters. As an essential part of emergency response decision making, emergency task planning is conducted to develop a feasible emergency action plan to accomplish a series of identified emergency response tasks under a highly volatile emergency environment. Due to the suddenness of disaster, the urgency of rescue, and the complexities of an emergency environment, emergency situation information cannot be completely collected immediately after disasters [1]. However, some initially absent information will be gathered gradually during execution and has an impact on the decisions regarding what to do next. Conversely, time urgency is always critical to emergency response and makes it imperative to concurrently execute emergency actions to complete emergency response as soon as possible. Furthermore, uncontrollable factors in an emergency environment render uncertain execution duration of emergency actions. In the real world, developing an emergency action plan is essentially a deliberation process to determine how to eliminate the gap between emergency goal tasks and executable tasks. Usually, *

Corresponding author. E-mail address: [email protected]

ACCEPTED MANUSCRIPT an emergency goal task can be identified by emergency decision makers based on emergency response preplans according to the current emergency situation [2]. The cognitive process involves refining emergency goal tasks into executable tasks according to emergency domain knowledge, which is similar to the underlying idea of Hierarchical Task Network (HTN) planning [3,4]. Specifically, HTN planning provides powerful expressivity to encode domain knowledge at different levels of abstraction and utilizes this domain knowledge to decompose non-primitive tasks into smaller subtasks until all the tasks are executable. Compared to classical planning paradigms [5], HTN planning is effective and scalable because it uses structured domain knowledge to guide the search process [6]. This is why HTN planning is adopted as the foundation of this paper to resolve the emergency task planning problem.

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In consideration of the above-mentioned characteristics of emergency response operations, emergency task planning necessitates a set of requirements for the HTN-based planning technique.

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(1) The desired planning technique should be able to deal with incomplete information regarding the initial planning state, as well as sensing actions whose outcomes are unpredictable and can only be observed at execution time. The process is also referred to as conditional planning [7]. Here, sensing actions, which provide new information without changing the environment state, are necessary to model the information gathering in the planning process. (2) The essence of concurrency is that the execution processes of durative actions are allowed to overlap. To avoid flaws due to interactions between durative actions, temporal constraints are needed to maintain a correct precedence relation between the start/end time points of actions. Therefore, the planning technique should have the ability to generate temporal constraints to enable concurrency.

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(3) Uncertain duration means that the amount of time required to execute an emergency action is uncontrollable-but-bounded and cannot be determined until the execution finishes. This type of uncertain duration will bring about temporal constraints with uncontrollable intervals; as a result, dynamic controllability [8], instead of consistency of temporal constraints, should be considered in the management of temporal constraints.

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To address these requirements simultaneously, the conditional temporal HTN planning problem with uncontrollable temporal constraints should be investigated; there are two important points to be emphasized. First, when generating temporal constraints via temporal reasoning to achieve concurrency, it is important to filter the invalid temporal constraints between actions from exclusive conditional branches in conditional planning. Second, observations, which are the products of conditional planning, should also be considered in the management of uncontrollable temporal constraints (i.e., temporal management) because they have impact on the propagation of the temporal constraints.

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However, the current conditional HTN planning [9–12] only focuses on instantaneous actions without temporal features and consequently has no ability to handle concurrency and uncertain durations. Although temporal reasoning techniques used in temporal HTN planning [13–15] are able to generate temporal constraints, they are limited to the deterministic domain, which has complete information regarding the planning state, and hence are not suitable for the conditional planning. Moreover, the Simple Temporal Network (STN) [16], which is most utilized in temporal planning to represent and propagate temporal constraints, is inadequate to confirm the dynamic controllability of temporal constraints because it focuses on the checking of consistency of temporal constraints with controllable intervals. In the current research of temporal constraints management, only the Conditional Simple Temporal Network with Uncertainty (CSTNU) [17] is capable of representing temporal constraints involving uncontrollable intervals and observations, whereas it has never been used in planning.

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In this paper, based on a conditional HTN planning framework, a planning paradigm that aims to resolve the emergency task planning problem with incomplete information, concurrency and uncertain durations is proposed. Particularly, an extended temporal reasoning technique based on the start-end separation of durative actions is proposed to generate correct and valid temporal constraints to achieve concurrency. Meanwhile, a temporal management approach is presented to confirm the satisfiability of generated temporal constraints involving uncontrollable intervals and observations. In temporal reasoning, rules are designed to generate correct temporal constraints to avoid flaws that result from the interactions between actions; moreover, a mechanism is devised to filter invalid temporal constraints. Regarding temporal management, CSTNU is employed to represent temporal constraints involving uncontrollable intervals and observations and an approach for eliminating redundant temporal constraints is presented to reduce temporal constraints to be represented in CSTNU for the purpose of promoting the efficiency of confirming the satisfiability of those temporal constraints.

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The rest of this paper is organized as follows. Section 2 reviews the related works on conditional HTN planning, as well as temporal reasoning and temporal management techniques. Section 3 introduces a formalism of the planning problem to be investigated, as well as a new solution form. Section 4 presents the proposed planning algorithm. Section 5 exhibits an experimental study of a practical earthquake case to explain the viability of the proposed planning algorithm. Section 6 concludes this study with a discussion of possible future work.

2. Related works

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To our best knowledge, the planning problem proposed in this paper has never been discussed in the current literature. However, some relevant research focusing on one or two aspects of the planning problem has been investigated. Specifically, conditional HTN planning is related to the HTN planning problem with incomplete initial information, while temporal HTN planning is related to the HTN planning problem with the concurrency requirement. This section reviews the literature relevant to those studies. 2.1 Conditional HTN planning

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In the past decade, several conditional HTN planners have been developed to address the HTN planning problem with incomplete information regarding the initial environment and sensing action, which are typical characteristics of conditional planning [18]. Those planners can be divided into two categories according to the model used to represent the belief state and solution.

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One category of planners that use a stochastic model of the belief state and actions includes CSHOP [9] and PC-SHOP [10]. Both planners have been shown to be able to generate plans of a tree structure that can handle contingencies at execution time through the use of conditional operators that choose which branch to follow depending on what observations are made. The other category of planners, including CondSHOP2 [11] and FOCUS [12], uses logical formulas to represent the belief state and generates a policy that consists of a set of pairs of belief state and action. Each pair specifies which action to execute depending on the current belief state. Although the above planners are able to solve the conditional HTN planning problem, they assume that actions change the state of the world instantaneously and neglect the temporal properties of the planning domain, such as duration, timed conditions and effects. Clearly, those planners are insufficient to express durative actions with uncertain durations and handle concurrency. In this paper, some extensions to the existing conditional HTN planning are made to cope with the concurrency of durative actions with uncertain duration.

ACCEPTED MANUSCRIPT 2.2 Temporal HTN planning As an important topic in temporal planning, concurrency can cause a problem either because of interactions between actions in the domain or because of a deadline that forces the actions to be compressed in time [19,20]. To achieve concurrency, some temporal planning techniques have been proposed for the existing HTN planners.

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Castillo et al. [13] presented some enhancements to the temporal reasoning of HTN planner SIADEX [21] to cope with temporal knowledge representation such as temporal causal dependencies, deadlines, temporal landmarks and timed initial literals. In their work, durative actions are allowed to have delayed effects at arbitrary times between the start and the end, but they do not explicitly explain how to deal with various preconditions of PDDL2.1 level 3 [22], which is desired in this paper. Tang et al. [14] presented a concurrency control mechanism in the temporal HTN planner XePlanner to achieve concurrency of durative actions with instantaneous and invariant preconditions, as well as delayed effects at arbitrary time during execution. Nevertheless, the planner assigns rigid time to the start and end of durative actions during the planning process, sacrificing the flexibility of the action plan. Based on this work, Tang and Shen [15] proposed a novel temporal reasoning technique for an HTN planning-based emergency decision making model to generate a flexible concurrent action plan of durative actions with the same description as XePlanner. Furthermore, Goldman [23] provided a method for encoding PDDL 2 level 3 durative actions in an HTN formalism compatible with the HTN planner SHOP2 [24] and presented a search control for efficient planning with these actions. Although the abovementioned temporal HTN planning techniques can address concurrency to a different extent, those works are limited to the deterministic domain, in which incomplete initial information and sensing do not exist. The limitation results in those techniques being developed without consideration of the impacts of exclusive branches in the conditional planning on temporal reasoning.

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Apart from temporal reasoning for extracting temporal constraints, representation and propagation of temporal constraints are necessary for temporal planning to manage the temporal constraints. In the above-mentioned temporal HTN planning, STN is most employed to represent temporal constraints, while the path consistency algorithm (PC-algorithm) is utilized to propagate the time constraints and verify their satisfiability. However, the current PC-algorithms [25–28] need to narrow the temporal intervals, while the intervals of uncontrollable temporal constraints are not allowed to be squeezed. The algorithms [29–31] for verifying the dynamic controllability of the Simple Temporal Network with Uncertainty (STNU) [32], which augments a STN to include a set of contingent links that represent temporal constraints with uncontrollable intervals, can guarantee that the uncontrollable intervals not being squeezed; however, those algorithms do not consider the impacts of observations on the propagation of temporal constraints. Only CSTNU, which is a combination of STNU and the Conditional Simple Temporal Problem (CSTP) [33], which augments a STN to include observations, is able to represent temporal constraints involving uncontrollable intervals and observations. The propagation-based checking algorithm for dynamic controllability of a CSTNU has also been proposed in [34]. Nevertheless, the work regarding how to coordinate the CSTNU-based temporal management with the required temporal reasoning technique in the planning process has never been discussed in the literature. In this paper, an extended temporal reasoning technique is investigated to handle concurrency in the context of conditional HTN planning. The work regarding how to use CSTNU to manage the temporal constraints, which involves uncontrollable intervals and observations, is also addressed.

ACCEPTED MANUSCRIPT 3. Formalism of Conditional Temporal HTN planning problem with uncertain durations This section introduces a formalism to describe the conditional temporal HTN planning problem with uncertain durations based on the existing description of the conditional HTN planning problem. To describe information gathering during the execution and durative actions with uncertain durations, the formalism extends the representation of sensing actions and planning operators. Moreover, the solution of the planning problem differs from that of the traditional conditional planning problem and thus has a new structure, to be defined later.

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To formally describe a conditional temporal HTN planning problem, the belief state is needed to represent the incomplete initial information of the world, sensing actions are necessary to model the possible outcomes of information gathering, a task network is required to describe the identified emergency goal tasks to be solved, durative operators are required to depict the process of durative actions with uncertain durations, and methods are necessary to specify the ―prescriptions‖ to decompose non-primitive tasks into smaller tasks. Compared to the formalism of the existing conditional HTN planning problem, the representation of sensing actions and planning operators should be extended. According to the analysis, a conditional temporal HTN planning problem is defined as a 3-tuple based on the formalism presented in conditional HTN planner FOCUS. Definition 1. A conditional temporal HTN planning problem with uncertain durations is a tuple , where:  

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is the initial belief state, which inherits the representation of the belief state in FOCUS; is the initial task network, denoted as a pair , of which is a set of tasks to be solved and is a set of constraints between tasks in ;  is the planning domain knowledge, denoted as a 3-tuple < >, of which is a finite set of sensing actions, is a finite set of durative operators, and is a finite set of methods. The expression of the method has the same form as that of HTN planner SHOP2 [24]. Each method has the form , where is a task that can be solved by this method, each is a precondition expression and each is a partially

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ordered set of subtasks that should accomplish the task. The representation of the sensing action and durative operator will be introduced in detail below.

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A sensing action only provide new information without changing the environment and is required to specify all the possible outcomes of an attempt to gather new information. In FOCUS, a sensing action is defined to observe the current truth value of a literal ; its effect is binary, being either or . In the practical domain, however, there may be more than two possible outcomes when gathering new information. For example, in the earthquake rescue domain, there may be three possible outcomes (i.e., good, damaged, and badly damaged) when checking an unknown road condition. Considering this need, a sensing action is defined as below. Definition 2. A sensing action is a tuple , where head is a task called the head of , pre is a precondition expression, and is a set of mutually exclusive observations that denotes all the possible outcomes after the execution of . The real outcome of a sensing action can only be determined at the execution time, but during the planning process, the agent must consider the belief states after the execution of a sensing action. We use to denote the belief states following the execution of in belief state : +. ( ) * |

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A planning operator is supposed to describe the preconditions and effects of a class of actual actions that change the environment without yielding new information. In this paper, actual actions in the emergency response domain are considered to have preconditions that should hold at the start, at the end or throughout its execution and effects that occur when the actions start or finish. However, the definition of the planning operator in the existing conditional HTN planning is similar to that of SHOP2, neglecting the feature of durative. To depict the duration, an enhanced durative operator is defined based on the definition of durative action in PDDL 2.1 level 3 [22]. The presented definition also takes into account uncertain duration and dynamic duration, which means that the duration changes with the evaluations of the parameters involved in the action. For example, the duration of a transportation action changes with the distance between the origin point and destination, and the distance can only be determined in the planning process.

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Definition 3. An enhanced durative operator is defined as a tuple , where head is a primitive task that can be completed by the action; , and denote the condition that must hold at the start of, at the end of and throughout the action, respectively, called the at-start precondition, at-end precondition and overall precondition; and denote the negative and positive effects that occur immediately after starts; and are negative and positive effects when is finished; denotes the duration of the action, denoted by an interval , -. The upper and lower bound of the duration are the functions of parameters in head or preconditions and can be computed when the operator is instantiated in the planning process. For uncertain duration, holds, which means that the action may take an arbitrary amount of time within the interval. When holds, the duration is controllable.

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For the defined planning problem, the solution should not only specify what actions are executed to complete the initially identified tasks but also record the relationships among the time points that denote the start or end of durative actions or the occurrence of observations. However, the solution generated by the existing conditional HTN planner is either a policy consisting of beliefstate-action pairs or a conditional plan of a tree structure, without recording the time relationships between actions. To meet the above-mentioned need, a solution is defined here as the combination of a conditional plan and a conditional simple temporal network with uncertainty, which records the temporal relationships between actions or between actions and observations.

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Definition 4. A solution is a tuple , where is a conditional plan represented by a set of tuples that describe actions and observations contained in the plan, is a CSTNU whose nodes denote the start or end of durative actions or the occurrence time of observations and whose edges are relative time relationships between those nodes.

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In a conditional plan, tuples corresponding to durative actions, sensing actions and observations have different forms. A durative action tuple has the form , where is an instantiated primitive task that can be completed by this action, and are the index of start and end time of this action, respectively, and is a conjunction of observation literals attached to the branch to which this action belongs. A sensing action tuple has the form , where is the head of this sensing action, is the index of the time when this sensing action occurs 1 , and is identical to that of the previous definition. An observation tuple has the form , where is an observation literal depicting the observed outcome, is the index of the occurrence time

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ACCEPTED MANUSCRIPT of , is the index of the sensing action that generates this observation, and identical to that of the previous definition.

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The structure of CSTNU here inherits the definition introduced in [17] and is defined as a tuple ( ), each of which has a correspondence with elements of tuples contained in the conditional plan. is a set of real-valued time-points, each of which corresponds to the time index in three types of tuples. is a set of propositional letters, each of which represents an observation literal of an observation tuple. is a function that assigns a label to each time point in ; the label can be derived from the element of an action or observation tuple. is a bijection that associates a unique observation time-point with each propositional letter, representing the one-toone correspondence between and of an observation tuple. is a set of contingent links, each of which connects the start time point with the end time point of a durative action with uncertain duration. is a set of labeled simple temporal constraints, each of which have the form ( ), where , is a real number, and is a label that can be empty or a conjunction of propositional letters.

4. CTPU-HTN planning algorithm

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To get such a solution, emergency domain knowledge, which can be extracted from the emergence response preplans, standard operation rules, expert knowledge and experiences, etc., is employed to guide the emergency task planning process. The extracted domain knowledge is then represented based on the formalism of the planning domain knowledge in HTN planning, as mentioned in Definition 1. The process of using the domain knowledge to generate a plan is illustrated in the next section.

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To solve the defined planning problem in Section 3, a planning algorithm is proposed in this section. First, the overall framework of the algorithm is given based on the analysis of critical issues in the solving process, followed by the detailed planning process. Then, two highlights are introduced in detail to explain the means to cope with concurrency and uncertain durations. 4.1 Overall framework and main planning process

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To solve the conditional temporal HTN planning problem with uncertain durations, it is necessary to expand the initial task network continually until the task network only contains primitive tasks. During the expansion, primitive tasks are refined as actions to be added to the conditional plan by applying sensing actions or enhanced durative operators; the following issues are considered.

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(1) A sensing action has several possible outcomes that can only be determined at execution time; the planning process should be able to generate a branch plan for each possible outcome, ensuring that the executors know what to do no matter which outcome occurs. (2) An enhanced durative operator has separate effects at the start and at the end; it is unsuitable to apply the start and end effects simultaneously to the planning state, which would result in failure if the end effect deletes literals achieved by the start effect and the deleted literals provide support for applying another durative operator. For this reason, the planning process should separately apply the start and end effects and have the ability to determine when to apply the effects to avoid the planning being lost. (3) To achieve concurrency, the durative operators should be allowed to overlap, which may lead to flaws because of incorrect start or end times for a durative operator. To avoid these flaws, the planning process should be able to generate temporal constraints in the process of applying a durative operator to maintain the correct time precedence relation between the start/end time points of durative operators.

ACCEPTED MANUSCRIPT (4) Whether the concurrency is feasible depends on the satisfiability of the temporal constraints involving uncontrollable intervals due to uncertain durations and observations representing the outcomes of the sensing action. Therefore, the planning process should also have the ability to confirm the satisfiability of the generated temporal constraints. Based on the above analysis, an algorithm called CTPU-HTN is proposed to realize the desired planning process; the overall framework of CTPU-HTN is shown in Fig. 1. Conditional HTN planning

Planning process

Temporal reasoning

Apply sensing action

Avoid flaws

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Apply durative operator Apply start snap action Apply end snap action

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Depth-first AND-search

Temporal management

Eliminate redundant temporal constraints

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Fig. 1. Overall framework of CTPU-HTN

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The proposed algorithm is built on the conditional HTN planning framework presented in planner FOCUS, which essentially performs AND-OR search over the space of belief states, where each AND-branch corresponds to a possible observation of the sensing action and each OR-branch corresponds to the choice of alternative methods for decomposing non-primitive tasks. The difference is that CTPU-HTN carries out a depth-first AND-search instead of choosing the ANDbranch arbitrarily. In this way, CTPU-HTN is able to generate a conditional plan branching based on the observations of the sensing actions. To address the separate effects of durative actions, a straightforward selection rule is proposed to correctly apply a durative operator in the process of expanding the task network. Considering the requirement of concurrency, an extended temporal reasoning technique is presented and an accelerated temporal management approach is also developed to confirm whether the concurrency is available under the existence of uncertain durations. The details are listed below.

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(1) To correctly apply the start and end effects of a durative operator, the idea of start-end separation inspired by the research in [23] is employed here to split a durative action into start and end snap actions as in [35]. Then, the planner should refine a primitive task twice, the first time to apply the start snap action and the second time to apply the end snap action. To determine when to apply the end snap action after the paired start snap action has been applied, a straightforward selection rule is employed based on the fact that applying a snap action may disable a non-primitive task to be decomposed anymore, while decomposing a non-primitive task may require the effects of a snap action to support its precondition. The main idea of the selection rule is to give priority to achieving a new task in the current top tasks (denoted by ), which are those regarding which no task in the current task network is constrained to precede, and to refine an old task for the second time only when no new task in the top tasks is applicable in the current planning state. A new task here may be a non-primitive task or a primitive task that has not yet been refined, reserved in the subset of top tasks. An old task is a primitive task that has been started but not yet finished, reserved in the complementary set of in top tasks, denoted as . (2) To avoid possible flaws, rules based on the interactions between durative actions are designed to generate temporal constraints to stipulate precedence relations between the start/end time points of durative actions. In the conditional planning framework, however, the designed rules

ACCEPTED MANUSCRIPT may give rise to invalid temporal constraints between actions generated under inconsistent branches. To avoid this situation, a detection mechanism is presented to identify the invalid constraints based on the definition of consistent branch labels. The whole process is referred to as temporal reasoning for AND/OR search; the details will be introduced in Section 4.2.

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(3) CSTNU is used to represent the temporal constraints involving uncontrollable intervals and observations; the satisfiability of the constraints can be determined by checking the dynamic controllability of the CSTNU. Finding that some of the generated temporal constraints represent redundant precedence relations, an approach for eliminating the redundant temporal constraints is proposed to decrease the amount of temporal constraints to be represented in the CSTNU, promoting, as a result, the efficiency of verifying the dynamic controllability to some extent. The whole process is called temporal management, which will be detailed in Section 4.3.

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Based on the above framework, the detailed planning algorithm is depicted in Fig. 2. The inputs to the planning algorithm are the initial belief state , the initial task network , the planning domain knowledge > and the empty branch label. The planning process involves recursively choosing a task to achieve from the top tasks according to the selection rule. Here, a primitive task that can be refined by applying a sensing action is called a sensing task, while a primitive task that can be refined by applying a durative operator is called an actuation task. When the chosen task is a non-primitive task, a method is used to decompose the non-primitive task; its subtasks then replace the non-primitive task in the task network, as well as the top tasks (line 2733, Fig. 2). When the chosen task is a sensing task, each observation of the applicable sensing action generates an AND-branch and is added to the branch label to identify this branch. The task will be removed from the task network after AND-search succeeds (line 3-11, Fig. 2). When the chosen task is an actuation task, it is refined for the first time by applying the suitable start snap action and simply moved to the subset of the top tasks (line 13-19, Fig. 2). When no task in is applicable in the current planning state, it arbitrarily chooses a task from to refine for a second time and removes the task from the task network after this refinement (line 19-25, Fig. 2).

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Note that during the process of applying start and end snap actions, temporal reasoning and temporal management are needed to determine whether concurrency is available if adding the applied operator to the current plan.

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Algorithm 1 CTPU-HTN ( ) Legend: = current belief state, = current task network, >, = branch label

= top tasks of

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Procedure CTPU-HTN ( ) 1 if then return 2 choose a task by using the selection rule 3 if is a sensing task and is a sensing action whose head unifies with then 4 for each observation 5 6 7 remove from and 8 CTPU-HTN ( ) 9 if then return ) to 10 append ( 11 else if is an actuation task then 12 if and is a durative action whose head unifies with then 13 apply-start-action( ) 14 if then return 15 delete from and move it to 16 CTPU-HTN ( ) 17 if then return 18 ( ) 19 else 20 is the durative action that has been started to refine but not yet finished 21 apply-end-action( ) 22 if then return 23 remove from and , add the immediate successor of in to

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24 CTPU-HTN ( ) 25 if then return 26 else 27 * | is a ground method such that unifies with and satisfies } 28 if then return 29 for each 30 replace with subtasks of (denoted by ) in , remove from and add the top tasks of to 31 CTPU-HTN ( ) 32 if then return 33 return Fig. 2. CTPU-HTN algorithm 4.2. Temporal reasoning for AND/OR search

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When applying a snap action, the interactions between literals in preconditions and effects of durative actions may bring about three types of flaws if the effects occur at an inappropriate time. To avoid the possible flaws, rules are designed to generate temporal constraints to maintain correct precedence relations between actions. Considering that rules based on literal interactions may cause invalid temporal constraints between actions belonging to inconsistent branches, a detection mechanism is developed to identify and discard invalid temporal constraints before they are generated. 4.2.1 Rules for generating temporal constraints to avoid flaws During the application of snap actions, their effects are used to update the current planning state. The positive effects will add literals that may provide support to the preconditions of another snap action; the negative effects will delete literals that may threaten the supporting relationship between positive effects and preconditions. Meanwhile, CTPU-HTN performs forward chaining

ACCEPTED MANUSCRIPT search, which implies the commitment that effects that have added/deleted a literal to/from the current planning state should come before effects that are going to delete/add the same literal. If the effects do not occur in the right order, this will result in three types of flaws, as explained below. (1) Unsatisfied precondition. The positive effect of snap action achieves a literal in the current planning state; is contained in the precondition of snap action . If comes before , then its precondition cannot be satisfied and becomes an unsatisfied precondition.

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(2) Threat to supporting relationship. The negative effect of snap action deletes the literal and there is a supporting relationship between and in terms of literal . If happens between and , the supporting relationship will be broken. (3) Commitment violation. The positive effect of snap action adds a literal to the planning state, while the negative effect of snap action deletes the same literal from the planning state. If comes before , it may result in early applied actions becoming inapplicable because the underlying planning state has changed due to the promotion of .

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To avoid these three types of flaws, rules are designed to generate temporal constraints that are required to maintain the correct precedence relation between snap actions. (1) Rule for generating temporal constraints to avoid unsatisfied precondition. For each literal in the precondition of snap action , another snap action that most recently ( ) achieves is bound to come before , resulting in the constraint ( ) ( ). For each literal in the overall precondition of a durative action , another snap action that most recently achieves should come before the start of , resulting in the )

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constraint (

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(2) Rule for generating temporal constraints to avoid threat to supporting relationship. For each literal in the negative effect of snap action , any other snap action requires that ( ) should come before , resulting in the constraint ( ) . When literal is in the ) ( ) overall precondition of a durative action , the resulting constraint is ( .

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(3) Rule for generating temporal constraints to avoid commitment violation. For each literal in the positive effect of snap action , the snap action that most recently adds/deletes is promoted to come before . For each literal in the negative effect of snap action , the snap action that most recently deletes/adds is promoted to come before . 4.2.2 Mechanism for detecting invalid temporal constraints

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In the conditional planning framework, actions may come from various AND-branches; constraints between two actions that come from an inconsistent branch make no sense because those two actions would not happen in the same scenario. Thus, the conditional HTN planning framework is likely to generate invalid temporal constraints because interactive literals may be contained in two actions belonging to inconsistent branches. To avoid this, a detection mechanism is proposed here to identify those invalid constraints based on the branch labels of AND-branches. Whether a temporal constraint is valid can be determined according to the following rules.

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The invariant precondition holds throughout an open interval according to the semantics of PDDL2.1.3; hence, the lower bound is 0 instead of ( ).

ACCEPTED MANUSCRIPT (1) For action belonging to the AND-branch labeled and action belonging to the AND-branch labeled , the constraint between and is valid and reserved when is consistent with ; otherwise, the constraint is invalid and discarded. (2) Specifically, literal in the positive effect of snap action that belongs to the AND-branch labeled is most recently added into the state by observation of a sensing action belonging to an AND-branch labeled ; then, the constraint between and is valid if and only if both and are consistent with .

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The above detection involves the concept of consistent branch labels. A branch label is a conjunction of observation literals used to record the observed facts under the current ANDbranch. The definition of consistent branch labels is given below. Definition 5. Two branch labels are called consistent if and only if the union of observation literals of those two branch labels is consistent. Branch labels and are consistent if one of the following conditions is true. or

is ; *

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+ and there does not exist any literals

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4.2.3 Temporal reasoning technique

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Combining the generation of correct temporal constraints and the detection of invalid temporal constraints, a temporal reasoning algorithm is proposed and typically embodied in the procedure of applying a start snap action, as shown in Fig. 3. The process of applying an end snap action is similar, except for the need to consider the overall preconditions, and is not detailed here.

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To make it easier to understand the procedure of applying a start snap action, some data structures used in the procedure are explained below. These records are also useful for constructing the CSTNU in the management of temporal constraints. () ( ) gives the time point of the snap action by which literal was most recently deleted/added, respectively  ( ) is a set of pairs ( ); if is a literal in the overall precondition, then gives the end time point of a durative action and ; if is a literal in the at-start/at-end precondition, then gives the start or end time point of a durative action and .  is a set of nodes that represents snap actions or observations; each node has the form ( ), where gives the time point of a snap action or observation and ( ) specifies the node that satisfies ; gives the index of the observation literal if the node represents an observation; otherwise, it is ; is a branch label.  is a set of temporal constraints between time points; each constraint has the form ( ) ( ( , -)), which denotes the constraint ( ) .T is an option used to indicate that the interval , - is uncontrollable-but-bounded.

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

Legend: is a global variable for indexing the time points of snap actions, Procedure apply-start-action( ) 1 if does not satisfy then return ) 2 if ( does not satisfy then return 3 ) and ( 4 add ( ) to ( ) 5 add ( ) to

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( ) exists and ( ( )) is consistent with then ( ) ( )) add ( to add ( ) to ( ) for for ( ) ( ) do ) if ( is consistent with then ) ( ) add ( to if ( ) exists and ( ( )) is consistent with then ) add ( ( ( )) to if ( ) exists and ( ( )) is consistent with then ) add ( ( ( )) to () for if ( ) exists and ( ) and ( ( )) is consistent with then ) add ( ( ( )) to if both ( ( )) and ( ( )) are consistent with then ) add ( ( ( )) to () for if ( ) exists and ( ) and ( ( )) is consistent with then ) add ( ( ( )) to add ( ) to ( ) if ( ) is not dynamically controllable then return ( ) return Fig. 3. Procedure of applying start snap action

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6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30

4.3 Eliminate the redundant temporal constraints

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In section 4.2, temporal constraints are generated to correct the application of snap actions. Whether the application is allowed in the current planning state depends on the satisfiability of the generated temporal constraint (line 29, Fig. 3). Because those temporal constraints involve uncontrollable intervals and observations, CSTNU is utilized to represent them. To confirm their satisfiability, a unique propagation-based algorithm [34] in the current literature is employed to check the dynamic controllability of the CSTNU. However, the checking algorithm is timeconsuming and will reduce the overall efficiency of the planning algorithm. Based on the findings that two types of redundant temporal constraints may be generated in the temporal reasoning, an approach for eliminating redundant temporal constraints is proposed to deduce the amount of temporal constraints in the CSTNU, aiming to promote the efficiency to some extent.

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In the process of temporal reasoning, most generated temporal constraints describe the precedence relationships between time points; this is prone to produce two types of redundant constraints, which are classified according to whether the precedence relations represented by the temporal constraints are implied by one or two other temporal constraints. Namely, two cases will give rise to redundant temporal constraints. (1) If two temporal constraints involve identical time points but their time intervals overlap, then one of them is redundant or both of them are redundant with respect to a new temporal constraint, which involves identical time points but has a time interval that is the intersection of the original two intervals.

ACCEPTED MANUSCRIPT (2) The precedence relation described by a temporal constraint is implied by another two temporal constraints and hence is redundant. For example, there are two temporal constraints: and ; the temporal constraint is implied by those two constraints. To avoid adding redundant temporal constraints, different mechanisms are utilized to eliminate each type of redundant temporal constraint. For the first type of redundant temporal constraints, there are three mutually exclusive ways to eliminate the redundancy according to whether the temporal constraint to be added is a contingent link. Assuming that the constraint to be added is ( denoted by , -) and the other constraint that has already been added is denoted by , -), then: ( - is uncontrollable; then, is a contingent constraint, which means that the interval , is not allowed to be added because in a CSTNU, a contingent link dominates a requirement link, whose time interval is allowed to be squeezed; (2) rather than is a contingent constraint; then, is deleted and is added; (3) Neither nor is a contingent constraint; then, a new constraint with identical time points - is added, while and an interval that is the intersection of , - and , is deleted.

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For the second type of redundant temporal constraints, it is required to estimate whether the constraint is implied by another two already added constraints or implies an already added constraint together with another already added constraint. This relies on the calculation of time points that are reachable from a time point according to the transitive law of precedence relationships between time points, as well as time points that have access to a time point. For two time points and , four time-point sets need to be calculated, respectively denoted as , , , , where / is a set of time points that are reachable from / ; / is a set of time points that have access to / . Based on the four time points set, the second type of redundant temporal constraints can be eliminated according to the following two rules.

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(1) If there is a time point that is not only in but also in , which means that there exist two constraints ( ) and ( ), then the constraint is implied by those two constraints and cannot be added.

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(2) If there is a constraint ( ( )) such that and , which means that there exist two constraints ( ) and ( ), then those two constraints, together with the constraint , imply the constraint and is deleted (line 16-18, Fig. 4).

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In the above calculation, the transitive law of precedence relationships between time points is used, which is defined below.

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Definition 6. (Reachability and Accessibility) (1) Time point is reachable from time point if and only if there exists a constraint ( ) or a constraint ( ) in which is reachable from . (2) Time point has access to time point if and only if there exists a constraint ( ) or a constraint ( ) in which has access to . According to the above analysis, the algorithm for eliminating the redundant temporal constraints is depicted in Fig. 4. ( , -) is a new temporal constraint to be added; is the current temporal constraints list Procedure refine-constraints ( ) , -) in then 1 if there is a temporal constraint ( Legend:

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2 if is a contingent constraint then 3 else if is a contingent constraint then 4 delete , 5 else 6 delete and from , ( 7 ( ) ( )-) 8 else 9 if ( ( )) then 10 { | is a time-point that is reachable from by the transitive law of precedence} 11 { | is a time-point that has access to by the transitive law of precedence} 12 { | is a time-point that is reachable from by the transitive law of precedence} 13 { | is a time-point that has access to by the transitive law of precedence} 14 if then 15 is implied by another two constraints and cannot be added ( )) in and 16 else if there is a constraint ( then 17 is implied by together with the constraint between and , as well as the constraint between and 18 delete and add to 19 else 20 add to Fig. 4. Algorithm for eliminating redundant temporal constraints

5. Experimental Study

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In this section, an experimental study is conducted to demonstrate that the proposed algorithm CTPU-HTN works and to testify that the proposed elimination approach promotes the efficiency of the algorithm. Firstly, a practical earthquake case is introduced to underpin the design of planning problems that embody the characteristic of emergency task planning considered in this paper. To the best of our knowledge, there are no planners that can resolve the planning problem considering incomplete initial information, concurrency and uncertain durations as this paper does. We therefore make a comparison, with respect to time performance, between CTPU-HTN and a SHOP2-based solving approach, which generates a plan for each complete planning state consistent with initially known environment facts. All the experiments are run on a Pentium(R) Dual-Core CPU T4300 @2.1 GHz with 2 GB RAM in Ubuntu 14.04 LTS. 5.1 A practical earthquake case

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A magnitude-6.5 earthquake struck Ludian county in Zhaotong, Yunnan, China on 3 August 2014. As soon as this earthquake happened, the Chinese government quickly set up an emergency command group to respond to the disaster. Because many victims were trapped in collapsed buildings, professional rescue teams were required to enter the severely affected area with rescue equipment as soon as possible. Conversely, people settled in shelters needed life support materials to sustain life; thus, disaster relief materials needed to be delivered to the shelters. The completion of these two emergency goal tasks relied on the road network; road avalanches and landslides could obstruct the road network, gravely impeding the rescue operations. In practice, road damage assessment relies on on-site investigation or remote sensing technology [36]. Because the communication facilities in the badly stricken area were destroyed and golden rescue time was limited, the emergency command group could not gather all the information regarding the conditions of roads. As a result, the emergency command group confronted a situation where the initial information was incomplete when planning an emergency action plan to accomplish those two emergency goal tasks. The incomplete initial information includes road topology, necessary relief materials (RMs), mobilized professional rescue teams (PRTs), and transportation vehicles. Fig. 5 gives a schematic view of the road topology in the heavily stricken

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area, in which D1 is the epicenter of this earthquake, D2, D3 and D4 are another three badly stricken townships and countryside, and D5 is the unique entrance to the stricken area. The setting of that basic initial information is shown in Table 1. In this setting, because resources are not the focus of this paper, neither the amount of relief material nor the capacity of trucks or aircraft is given, while the approximate speed of trucks or aircraft is provided by the experts to estimate the transport time. Moreover, all the PRTs, RMs, and transportation vehicles are initially located at D5.

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Fig. 5. A schematic view of partial road topology after the M(s) 6.5 Ludian earthquake Table 1. Basic initial information

RMs: RM1, RM2, RM3, RM4

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PRTs: PRT1, PRT2, PRT3, PRT4 Trucks: truck1, truck2, truck3, truck4 Aircrafts: aircraft1, aircraft2

Road network topology Road road1 road2 road3 road4

Origin D5 D1 D5 D5

Destination D1 D2 D3 D4

Distance(km) 29 26.2 48.2 23.8

To solve this emergency task planning problem, domain knowledge, especially knowledge regarding how to respond to different road conditions, is also required. According to grades of subgrade seismic damage [37], road damage can be divided into three grades: Grade-I, Grade-II, Grade-III. Under different grades of road damage, the roads confront different traffic function loss and hence different transportation choices are chosen to transport relief materials and professional rescue teams. The details are shown in Table 2. Table 2. Traffic function loss and transport choices under different grades of road damage Grades of Traffic function loss Transport choices road damage

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Grade-II

Grade-III

Vehicles can travel along the road with slow velocity Vehicles cannot travel along the road unless taking certain time to repair the road; people can walk along the road Vehicles cannot travel along the road unless taking a long time to repair the road; people cannot walk along the road unless simply repairing the road

5.2 Comparison of CTPU-HTN and SHOP2

Transport relief materials and professional rescue teams by truck Transport relief materials and professional rescue teams by truck after damaged road is repaired; Transport relief materials by aircraft; professional rescue teams enter stricken area on foot after damaged road is simply repaired

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Because no conditional planner is able to solve the above-mentioned planning problem, we consider making a comparison with respect to time performance between CTPU-HTN and HTN planner SHOP2 to verify the feasibility of the proposed paradigm. To solve the considered planning problem, SHOP2 needs to construct a plan for each deterministic planning problem with uncertain durations, which is a simplified version of the original planning problem, by replacing the incomplete initial planning state with a complete initial planning state consistent with initially known environment facts. To enable SHOP2 to handle identical concurrency, the designed temporal reasoning mechanism in subsection 4.2.1 is incorporated into SHOP2. Due to uncertain durations, the resulting temporal constraints network is of a STNU. For a fair comparison, CSTNU-editor3, which is a Java implementation of the CSTNU DC-checking algorithm, proposed in [34] to verify the dynamic controllability of CSTNU, is also utilized to verify the dynamic controllability of STNU by regarding a STNU as a CSTNU without observation points. Moreover, the domain knowledge input for the two approaches is equivalent.

5.2.1 Problem setting

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To make the comparison, a set of planning problems is devised that embodies the relevant characteristics. Before giving the comparison results, an illustration of the planning process for a problem is made to demonstrate the feasibility of CTPU-HTN. Finally, the results are discussed.

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In the above-mentioned earthquake case, the primary unknown information is the condition of roads; there are four roads that connect the entrance of the stricken area with the stricken townships and countryside. Therefore, 15 planning problems are designed to enumerate all possible combinations of unknown road conditions, each of which differs from one another regarding which roads are set to be unknown. To incorporate uncertain duration, the speeds of trucks and aircraft are set as the bounded interval; thus, the duration of transportation via truck or aircraft is uncontrollable. To embody concurrency, overlapping road repair with the loading of relief materials when the relief materials need to be transported through a damaged road is allowed. This consideration is represented in the HTN methods. The detailed settings of those planning problems are shown in Table 3. All the problems have identical emergency goal tasks, as represented in the last row of Table 3. By default, the other roads omitted in Table 3 are known to be in good condition (i.e., Grade-I). Table 3. Initial tasks and road conditions of designed planning problems Problem Uncertain road condition Problem Uncertain road condition problem1 (is-good road1 :unknown) problem2 (is-good road2 :unknown) problem3 (is-good road3 :unknown) problem4 (is-good road4 :unknown) 3

http://profs.scienze.univr.it/~posenato/software/cstnu/index.html

ACCEPTED MANUSCRIPT (is-good road1 :unknown) (is-good road1 :unknown) problem6 (is-good road2 :unknown) (is-good road3 :unknown) (is-good road1 :unknown) (is-good road2 :unknown) problem7 problem8 (is-good road4 :unknown) (is-good road3 :unknown) (is-good road2 :unknown) (is-good road3 :unknown) problem9 problem10 (is-good road4 :unknown) (is-good road4 :unknown) (is-good road1 :unknown) (is-good road1 :unknown) problem11 (is-good road2 :unknown) problem12 (is-good road2 :unknown) (is-good road3 :unknown) (is-good road4 :unknown) (is-good road1 :unknown) (is-good road2 :unknown) problem13 (is-good road3 :unknown) problem14 (is-good road3 :unknown) (is-good road4 :unknown) (is-good road4 :unknown) (is-good road1 :unknown) (is-good road2 :unknown) problem15 (is-good road3 :unknown) (is-good road4 :unknown) Speed of truck: 18~20 km/h; Speed of aircraft: 50~70 km/h Emergency goal tasks: (send PRTs), (deliver RMs)

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problem5

5.2.2 Illustration of the feasibility of CTPU-HTN

Before giving the experimental results, we take problem1 as an example to illustrate the planning process based on the proposed algorithm, as well as the feasibility of the generated plan.

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To solve problem1, the implemented planner recursively decomposes two emergency goal tasks into smaller tasks until achieving executable primitive tasks. When decomposing a non-primitive task requires an unknown road condition, a sensing action is applied to gather the required information. For every possible outcome, the planner generates a searching branch to continue the task decomposition. As the decomposition and application proceed, a hierarchical task decomposition network forms. Due to the space limitation, not a whole but a partial task decomposition network of problem1 is shown in Figure 6.

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As shown in Figure 6, because the condition of road1 is unknown, the non-primitive task (sendfrom-to D5 PRT1 D1) is decomposed into a sensing action (!check-road road1) and a smaller nonprimitive task (cond-transportP D5 PRT1 D1); the former precedes the latter. After the sensing action is applied, the planner would generate three branches corresponding to three damage grades of road1 and continue to decompose the non-primitive task (cond-transportP D5 PRT1 D1) according to the domain knowledge described in Table 2. In contrast, the non-primitive task (deliver-from-to D5 RM3 D3) is refined into a smaller non-primitive task that can be further decomposed into four durative actions because the condition of road3 is set to be good by default. The other non-primitive tasks can also be gradually decomposed into primitive tasks in a similar way. When all non-primitive tasks are decomposed into primitive tasks, a plan is generated. Because it is inconvenient to demonstrate the complete plan of problem1 here, a partial plan for task (send-from-to D5 PRT1 D1) and task (deliver-from-to D5 RM3 D3) is displayed in Figure 7 to illustrate the feasibility of CTPU-HTN.

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(send-from-to D5 PRT2 D2) (send-from-to D5 PRT1 D1)

(is-good road1 :unknown)

(truck-transportP D5 PRT1 D1)

(!geton-truck PRT1 truck1 D5)

(!drive truck1 D5 D1) (!getoff-truck PRT1 truck1 D1)

(!repair-road road1)

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(truck-transportP D5 PRT1 D1)

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Fig. 6. Partial task decomposition network of problem1 Grade-I

(!geton-truck PRT1 truck1 D5) (!repair-road road1)

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(!geton-truck PRT1 truck1 D5)

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Grade-III (!drive truck3 D3 D5)

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(!load-truck truck3 RM3 D5)

(!drive truck3 D5 D3)

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Fig. 7. A partial plan for (send-from-to D5 PRT1 D1) and (deliver-from-to D5 RM3 D3)

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As shown in Figure 7, the partial plan contains two concurrent execution branches that, respectively, accomplish task (send-from-to D5 PRT1 D1) and task (deliver-from-to D5 RM3 D3). The execution branch for the former task is composed of three OR-branches, each of which responds to one possibility of the condition of road1. Namely, whichever observation of the condition of road1 occurs, the partial plan specifies a set of actions to respond to. Additionally, in the OR-branch corresponding to Grade-II, the action (!repair-road road1) can overlap with the action (!geton-truck PRT1 truck1 D5), while the action (!drive truck1 D5 D1) is only allowed to start after road repair is completed. Clearly, all ‗!drive‘ actions have uncertain duration because of the previous speed setting. All these things suggest that the proposed algorithm is able to generate a feasible plan to respond to incomplete initial information, concurrency and uncertain durations. 5.2.3 Comparison results

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The time performances of the two approaches when solving those problems are shown in Table 4. To demonstrate the influence of the elimination of redundant temporal constraints on the efficiency, the planning time of CTPU-HTN with or without elimination is also listed in the table. Table 4. Time performance of CTPU-HTN and SHOP2

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problem1 problem2 problem3 problem4 problem5 problem6 problem7 problem8 problem9 problem10

CTPU-HTN Planning time (s) With elimination Without elimination 4.407 4.4404 4.402 4.4394 4.384 4.4892 4.477 4.5298 10.747 11.140 9.645 9.813 10.932 11.213 10.766 11.182 10.812 11.156 10.906 11.105

SHOP2 Planning time (s) 21.953 21.983 21.574 21.6 66.352 67.798 65.727 66.544 65.412 64.026

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41.094 40.788 40.779 33.398 310.288

43.603 43.509 43.589 34.599 422.090

195.427 195.437 193.304 194.021 580.174

Table4.cvs here

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The results in Table 4 show that CTPU-HTN performs better than SHOP2 in terms of planning time for all fifteen problems. Although our method also takes into account all possibilities of initially unknown information, it only does this when the unknown information is relevant to the decomposition of a non-primitive task. In contrast, SHOP2 transforms the incomplete initial state into several complete initial states and generates a plan for each complete state. The amount of complete states is equal to the amount of observation combinations of that unknown information. Consequently, SHOP2 takes more time to solve the original planning problem plausibly. Although SHOP2 generates several plans, only the plan corresponding to the complete initial state that exactly describes the real world is viable. However, the emergency decision makers still do not have complete information when choosing a plan to execute; thus, the chosen plan can accomplish the emergency goal tasks opportunistically. When encountering a situation inconsistent with the initial state, the plan cannot continue unless some modifications are made or a new plan is generated. By comparison, as demonstrated in subsection 5.2, the plan generated by the proposed method can start execution from the incomplete initial state and choose a branch to continue along after gathering information when the execution encounters an unknown condition.

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6. Conclusions

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For all problems, the planning time with elimination is less than the planning time without elimination. This result demonstrates that elimination of redundant temporal constraints can indeed promote the planning efficiency. The more uncertain information the problem has, the more obvious the promotion is. The reason may be that more redundant temporal constraints are generated when uncertain information increases and that the time spent on dynamic controllability verification reduces sharply after those redundant constraints are eliminated.

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This study presents a planning paradigm CTPU-HTN to tackle the emergency task planning problem with incomplete initial information, uncertain durations and concurrent execution, in which temporal reasoning and temporal management are two crucial but difficult issues to address. Our planning paradigm can generate correct and valid temporal constraints through designed rules and a detection mechanism. Those rules are designed to avoid flaws because of interactions between overlapping durative actions, while the detection mechanism is devoted to filtering invalid temporal constraints. CSTNU is adopted to represent and propagate the generated temporal constraints that involve uncontrollable intervals and observations. Because redundant temporal constraints may be generated during the temporal reasoning, an approach of eliminating redundant constraints is presented to reduce the scale of CSTNU and consequently promote the efficiency of temporal management. The experimental study demonstrates that the proposed planning paradigm is able to solve the considered emergency task planning problem and that the proposed elimination approach can improve the efficiency of the planning paradigm to some extent. The performance of the planning paradigm is greatly influenced by the efficiency of the adopted DC-checking algorithm. With a more efficient DC-checking algorithm, the proposed planning algorithm can be practical in the real-world domain from the view point of time performance and solution feasibility.

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In this paper, we consider the fact that the complexity of an emergency environment and the urgency of rescues will lead to incomplete initial information and assume that the possible outcomes of sensing can be modeled in the planning process. However, in a highly volatile emergency environment, unpredictable contingences often happen arbitrarily during the emergency response process. For example, landslides, aftershocks and debris flows often occur unexpectedly during the emergency response process of an earthquake disaster. These contingences change the emergency environment and may break the execution preconditions of the emergency action plan. Even worse, it is impossible to enumerate all possibilities of these contingences; thus, these contingences cannot be taken into account in the planning process like the initially unknown information. Therefore, our future work will focus on the research of HTNbased on-line planning for dealing with these unpredictable contingences during the execution. It will aim to generate a repair plan for restoring the environment state to a point where the original plan can continue to execute. If no such point exists, a new plan will be produced through replanning.

Acknowledgments

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This work was supported by the National Key Research and Development Program of the Ministry of Science and Technology of China (Grant No. 2016YFC0802509), the National Science Fund for Distinguished Young Scholars of China (Grant No. 71125001), and the National Natural Science Foundation of China (Grant No. 71390524, 71371079).

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