Dialectical Theory for Multi-Agent Assumption-based Planning

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1 Dialectical Theory for Multi-Agent Assumtion-based Planning Damien Pellier, Humbert Fiorino Laboratoire Leibniz, 46 avenue Félix Viallet F Grenboble, France {Damien.Pellier,Humbert.Fiorino}.imag.fr Abstract. The urose of this aer is to introduce a dialectical theory for lan synthesis based on a multi-agent aroach. This aroach is a romising way to devise systems based on agent lanners in which the roduction of a global shared lan is obtained by conjecture/refutation cycles. Contrary to classical aroaches, our contribution relies on agents dialectical reasoning: in order to take into account the artial knowledge and the heterogeneous skills of the agents, we roose to consider the lanning roblem as a defeasible reasoning where agents exchange roosals and counter-roosals and are able to conjecture i.e., formulate lan stes based on hyothetical states of the world. The dialogue between agents is a joint investigation rocess allowing agents to rogressively rune objections, solve conjectures and elaborate solutions ste by ste. 1 Introduction The roblem of lan synthesis achieved by autonomous agents in order to solve comlex and collaborative tasks is still an oen challenge. Increasingly new alication areas can benefit from this research domain: for instance, cooerative robotics [1] or comosition of semantic web services [2] when considering actions as services and lans as comosition schemes. From our oint of view, multi-agent lanning can be likened to the rocess used in automatic theorem roving. In a sense, a lan can be considered to be a articular roof based on secific rules, called actions. In this aer, we draw our insiration from the roof theory described by Lakatos. According to [3], a correct roof does not exist in the absolute. At any time, an exerimentation or a test can refute a roof. If one single test leads to a refutation, the roof is reviewed and it is considered as mere conjecture which must be reaired in order to reject this refutation and consequently to become less questionable. The new roof can be subsequently tested and refuted anew. Therefore, the roof elaboration is an iterative non monotonous rocess of conjectures - refutations - reairs. The same is true of our aroach. The lan synthesis roblem is viewed as a dialectical and collaborative goal directed reasoning about actions. Each agent can refine, refute or reair the ongoing team lan. If the reair of a reviously refuted lan succeeds, it becomes more robust but it can still be refuted later. If the reair of the refuted lan fails, agents leave this art of the reasoning and exlore another ossibility: finally bad sub-lans are ruled out because there is no agent able to ush the investigation rocess further. As in an argumentation with oonents and roonents, the current lan

2 2 is considered as an accetable solution when the roosal/counter-roosal cycles end and there is no more objection. The originality of this aroach relies on the agent s caabilities to elaborate lans under artial knowledge and/or to roduce lans that artially contradict its knowledge. In other words, in order to reach a goal, such an agent is able to rovide a lan which could be executed if certain conditions were met. Unlike classical lanners, the lanning rocess does not fail if some conditions are not asserted in the knowledge base, but rather rooses an Assumtion-Based Plan or conjecture. Obviously, this conjecture must be reasonable: the goal cannot be considered achieved and the assumtions must be as few as ossible because they become new goals for the other agents. For instance, suose that a door is locked: if the agent seeks to get into the room behind the door and the key is not in the lock, the lanning rocedure fails even though the agent is able to fulfill 100% of its objectives behind the door. Another ossibility is to suose for the moment that the key is available and then lan how to oen the door etc. whereas finding the key might become a new goal to be delegated. To that end, we designed a lanner that relaxes some restrictions regarding the alicability of lanning oerators. Our aroach differs from former ones in two oints. First of all, unlike aroaches that emhasize the roblem of controlling and coordinating a osteriori local lans of indeendent agents by using negotiation [4], argumentation [5], or synchronization [6] etc., the dialectical theory for lan synthesis resented here focuses on generic mechanisms allowing agents to jointly elaborate a global shared lan and carry out collective actions. Secondly, by elaboration, we mean lan roduction and not instantiation of redefined global lan skeletons [7, 8]. This is achieved by comosing agents skills i.e., the actions they can execute for the benefit of the grou. Thus, the issues are: how can agents roduce lans as arts of the global roof with their artial and incomlete beliefs? what kind of refutations and reairs agents can roose to roduce robust lans? and how to define the conjecture - refutation rotocol so as to converge to an accetable solution lan? In this aer, we introduce a multi-agent assumtion-based lanning aroach. In section 2, we resent the rimary notions used in this aroach. Then, in section 3, we define the concet of roof board used by agents to collaboratively build a solution lan and finally, in section 4, the dialectical mechanisms for the conjecture-refutation rocess is resented. 2 Primary Notions We start by defining the language used to describe agents beliefs. This language is based on a first-order language L in which there is a finite number of redicates symbols and constants symbols but no function symbols. A state is a set of ground atoms of L. Since L has no functions symbols, the set S of all ossibles states s is guaranteed to be finite. An atom holds in s iff s. If g is a set of literals (i.e., atoms and negated atoms), we will say that s satisfied g (denoted s = g). Now, let us introduce, the definition of a lanning oerator used by agents. An lanning oerator defines a transition oeration from a state to another one.

3 3 Definition 1 (Planning Oerator). A lanning oerator is a trile o = name(o), recond(o), effects(o) whose elements are as follows: name(o), the name of the oerator,n(x 1,..., x k ) where n is a symbol and x 1,...,x k define oerator s arameters. recond(o) and effects(o), the reconditions and effects of o, resectively defining the literals that must be held in the state where the oerator is alied and the literals that must be added (denoted effect(o) + ) or removed (denoted effect(o) ), to comute the transition oeration. Although we use the same oerator reresentation as in classical lanning, the oerator semantic in our aroach is different. In classical lanning, an oerator is alicable to a state s if o is ground and s is a state such recond(o) s. Our aroach relaxes this constraint: all oerators are alicable to a state s. Hence, we must distinguish facts that hold in s and facts that do not hold. The second are called assumtions. An assumtion defines a literal recond(o) such do not hold in s. We use assum(o) to denote the set of assumtions needed to aly an oerator o in a articular state s. The state resulting of the alication of o to s i is the state: s i+1 = ((s i assum(a)) effects (a)) effects + (a) For instance, consider the initial belief state of an agent s 0 = {at(cont,loc1)} and a simle oerator that can be erformed by this agent to move a container from a location to another one: name(o) = move(c,l1,l2); recond(o) = {connected(l1,l2), at(c,l1)} and effect(o) = { at(c,l1), at(c,l2)}. In this examle, the agent has no information about the connection between the locations loc1 and loc2. In order to aly the move oerator, the agent must assume the assumtion connected(loc1,loc2). The state resulting of the alication of the move oerator is the state: s 1 = {connected(loc1, loc2), at(cont,loc2)}. Before going further and introducing our multi-agent lanning model, we must clarify one oint. We say that an assumtion is a recondition of an oerator o that do not hold in the state s where the oerator is alied. Thus, there are two cases: i) if a recondition is not contained in s, the fact must be added to the agent s belief and simly considered as a hyothetical fact; ii) if a recondition does not hold because its negation is contained in s, the agent must first remove the negation before adding the recondition. We call this kind of assumtion a fact negation. Assumtions are imortant oortunities for imroving collaborative synergy between agents. They can be refined by the other agents in order to roduce the suosed facts (e.g., by connecting the two locations loc1 and loc2). They are viewed as subgoals that must be fulfilled by other agents. Definition 2 (Agent). An agent is a trile ag = name(ag), oerators(ag), beliefs(ag), where: name(ag),the name of the agent; oerators(ag), a set of oerators, i.e., the skills of the agent; beliefs(ag), a set of literals, i.e., the initial beliefs of the agent. In classical lanning, a lanning domain is defined by a set of oerators. In our aroach, oerators are included in agents descrition. Thus, we define a multi-agent lanning domain as a set of agents.

4 4 Definition 3 (Multi-Agent Planning Domain). A multi-agent lanning domain D is defined as a set of agents. Finally, we need to define the notion of multi-agent lanning roblem. A multiagent lanning roblem must define the goals that must be reached and the set of agents that must solve it. Definition 4 (Multi-Agent Planning Problem). A multi-agent lanning roblem is a coule P = AG, g, where: AG defines a set of agents names; g is a set of literals that must be reached by the agents defined in AG. Consider a simle domain containing four agents: a farmer, a miller, a baker and a conveyor. The farmer sows wheat, which must be harvested. The miller grinds the farmer s wheat to roduce flour. The baker makes bread with miller s flour and finally the conveyor is in charge of moving the goods needed by the other agents. An instance of a multi-agent lanning roblem can define with AG = {famer, miller, baker, convoyor} and g = {has-goods(baker, bread, 2)}. 3 Conjectures Sace Search The lan synthesis relies on dialectical exchanges between agents as exected in a debate. Agents interact collaboratively in the dialogue so as to construct a lan without assumtion, fulfilling the assigned goals. In order to build such a lan and organize the dialog between agents, we need a structure, called roof board. This structure has two main functions: it must be able to reresent the sace search as in classical lanning and it must be able to secify the dialectical rules used by agents to interact. 3.1 Conjectures and Plans First, let us refine the notion of conjecture used in our aroach. We have informally introduced a conjecture as a lan that can be executed if certain conditions were met. In classical lanning, a lan is a set of ground oerators organized into some structure, e.g., a sequence. However, a sequence of oerators is a articular lan that reflects the intrinsic constraints of the oerators. It seems to be to much restrictive for a multi-agent aroach of collaborative lanning, e.g., it is no ossible to define concurrent actions. Therefore, to find out what is needed in a conjecture, consider an informal lanning ste (shown figure 1) on the simle examle reviously introduced with the farmer, the miller, the baker and the conveyor. baker 1 : I can make 2 breads to solve the goal, but I need 2 flour containers available in loc1. conveyor 1 : I can transort the flours containers at loc1, but I don t know where I must load the goods. miller 1 : I roose to sell you the flours containers. I needed to be ayed 4 euros for that and find someone to transort flour containers from loc2 to loc1. Moreover, I need a wheat container available in loc2 to grind the flour.

5 5 baker 2 : Thank you for your hel, miller, but I have not enough money. miller 2 : Ok, give me only 2 euros. baker 2 : Good deal, I ay you. conveyor 2 : Thus, I understand that I must load the flour in loc2. farmer 1 : I roose to sell you a wheat container. I need to be ayed 1 euros for that and find someone to transort the container from loc3 to loc2. miller 3 :... connected(loc2,loc1) at(loc2) move(conveyor,loc2,loc1) at(loc1) at(loc2) at(loc1) loaded(flour) at(loc2) available(flour,loc2) unload(conveyor,flour,loc1) load(cenveyor,flour,loc2) available(flour,loc1) loaded(flour) loaded(flour) available(flour,loc2) available(flour,loc1) has(baker,flour,2) has(baker,cash,2) make(baker,bread,2) ay(baker,miller,flour,2) a0 has(baker,bread,2) a n has(baker,bread,2) has(baker,flour,0) available(flour,loc1) cash(miller,baker,flour,2) has(miller,flour,2) sell(miller,baker,flour,2) cash(miller,baker,flour,2) has(baker,cash,0) connected(loc2,loc1) at(loc2) available(flour,loc2) has(baker,cash,2) has(baker,flour,2) has(miller,flour,0) available(flour,loc1) available(wheat,loc2) has(miller,wheat,1) grind(miller,wheat,1) has(miller,flour,2) has(miller,wheat,0) Fig. 1. Examle of conjecture: each boxes is an oerator with reconditions above and effects below. Solid arrows are ordering constraints, dashed arrows are causal links and binding constraints are imlicit or shown directly in the oerator arameters. This reresentation is based on [9]. Oerators. Initially, baker 1 rooses to add the oerator make-bread to reach the goal g = { has(baker,bread,2)}. This oerator make two assumtions: available(flour, loc1) and has(baker,flour,2). These assumtions must be refined. Thus, conveyor 1 and miller 1 roose recursively to add others oerators or sub-conjecture to reach these two new goals. Ordering Constraints. Consider the sub-conjecture added by conveyor 1 ; it achieves its urose only if it is constrained to come before the make-bread oerator. But should this conjecture come before or after the miller conjecture? Both otions are ossible. We use the least commitment rincile of not adding constraints unless it is strictly needed. If no constraint are secified the conjecture between conveyor 1 and miller 1, these two conjectures will be able to run concurrently. Causal links. Because there is no exlicit notion of current state (distributed on the agents), an ordering constraint does not say, for instance, that the flour stays available at loc1 until make-bread oerator is erformed. Hence, we need to encode exlicitly in the conjecture the reason why the conveyor 1 sub-conjecture was added: to satisfy the assumtion available(flour, loc1). The relation between the baker s conjecture and the conveyor s one with resect to available(flour, loc1), is called a causal link.

6 6 Binding Constraints. Oerators are added in a conjecture with systematic variable renaming. For instance, we must ensure that the conveyor conjecture concerns the same container flour and the same location loc1 as those in oerator make-bread. Definition 5 (Conjecture). A conjecture is a tule χ = A,, B, C, where: A = {a 1,...,a k } is a set of artially instantiated oerators. is a set of ordering constraints on A of the form (a i a j ). B is a set of binding constraints on A of the form x = y, x y or x D x, where D x is the domain of x. C is a set of causal links of the form (a i aj ), such that a i and a j are oerators in A, the constraint a i a j is in, assumtion is an effect of a i and a recondition of a j and finally the binding constraints between a i and a j about are in B. The roof board is a conjecture sace defining a directed grah whose vertices are conjectures and whose edges corresond to the transition oeration roosed by the agent. An outgoing edge from a vertex χ is a transition oeration that transforms χ into a successor χ. A transition oeration can be: a refinement (i.e., adding oerators to rove an assumtion), a refutation (i.e., highlighting inconsistencies in the conjecture) and a reair of a reviously highlighted inconsistency. Therefore, multi-agent assumtion-based lanning is a search in the roof board from a initial conjecture to a node recognized as a solution lan. Note that due to no exlicit current state reresentation, goals and initial state must be defined as articular conjectures. Since reconditions are ossibly assumtions, the roositions corresonding to the goals are reresented as reconditions of a dummy oerator a n. Similarly, the initial state is reresented as the effects of a dummy action a 0. The effects of a 0 define the union of the agents beliefs. We make the assumtion that the agents beliefs are consistent. 3.2 Solution Plan Let us now secify what is a solution lan to a lanning roblem P = AG, g. A solution lan is a conjecture that has articular roerties. First, a conjecture is a solution lan if the conjecture makes no assumtion. But according to the conjecture definition, it is not enough. A solution conjecture must define a consistent set of ordering constraints, binding constraints and causal links. These roerties allow us to define three kinds of refutations. Proosition 1 (Solution Plan 1 ). A conjecture χ = A,, B, C is a solution lan to a lanning roblem P = AG, g, if χ has no assumtion and χ can not be refuted. Definition 6 (Ordering Refutation). An action a k of a conjecture χ refutes an ordering constraint a i a j iff a k a i and a j a k. Definition 7 (Binding Refutation). An action a k of a conjecture χ refutes an binding constraint iff one of the following condition holds: 1. if there is an oerator a k that contains a variable x such that x D x and x is not consistent with B. 1 can be roved inductively on the number of oerators in A.

7 7 2. if there is an oerator a k that contains two variables x and y such that x = y is not consistent with B. 3. if there is an oerator a k that contains two variables x and y such that such that x y is not consistent with B. Definition 8 (Causal Refutation). An action a k of a conjecture χ refutes a causal link a i aj, iff: a k has an effect q and q is not consistent with, i.e., and q are unifiable. ordering constraints (a i a k ) and (a k a j ) are consistent with. binding constraints resulting of the unification of and q are consistent with B. 4 Dialectical Mechanisms In order to tackle the dialectical mechanisms to collaboratively build a solution lan, let us remember the definition of the roof board. The roof board defines a conjectures sace where edges reresent transition oerations: refine, refute or reair. A conjecture is a solution lan if it does not contain assumtion and if no agent is able to refute it. This definition gives us two tis to secify the dialectical mechanism. Indeed, the first condition can be reached by refining or reairing. On the contrary, the second condition needs a deliberation rocess to guarantee that no agent can refute the conjecture. Therefore, we distinguish two layers: i) an informational layer that defines the rules to exchange refinements, refutations and reairs about the current conjecture. Each new conjecture suggested by an agent roduces new goals to be achieved by the other agents; ii) a contextualization layer in which agents can decide to sto interacting when they believe a solution was found or not reachable. Moreover agents can decide to change the dialogue context by forwarding or backtracking into the roof board if the current conjecture has been refuted or none of the agents can refine its assumtions. 4.1 Informational Layer The characterization of the solution lan brings elements needed for the secification of the seech acts used in the informational layer. The main rincile of the multi-agent assumtion based lanning is to let the agents choose a transition oeration to aly to the roof board until χ contains no more assumtions and until χ cannot be refuted. The basic stes of agent s dialectical mechanisms are the following: Select a conjecture χ on which to aly a transition oeration. Select a transition oeration to aly to χ. Find ways to resolve the transition oeration. Select a resolver for the transition oeration. Assert the resolver, i.e., refine, refute or reair. For each transition oeration that can be alied, we introduce a seech act: i) a seech act refine is erformed by an agent to exress the refinement of a conjecture. A refinement can be secified by adding a set of oerators, a set of ordering constraints, a set of binding constraints and finally a set of causal links (e.g., miller 1 in examle 1); ii) a

8 8 seech act refute is erformed by an agent to exress the refutation of a conjecture. A refutation highlights that an action roduces a set of ordering inconsistencies or a set of binding inconsistencies or finally a set of causal inconsistencies. The comutation of the inconsistencies are based on the formal definition of the three kinds of refutation reviously resented (e.g., baker 2 in examle 1); iii)a seech act reair is erformed by an agent to exress that a conjecture can be reaired by adding and removing resectively a set of oerators, a set of ordering constraints, a set of binding constraints and finally a set of causal links (e.g., miller 2 in examle 1). Note that all informational seech acts can be erformed only if they were not already roosed by other agents. This condition guarantees that the roof board defines a loo free directed grah. In order to find ways to resolve a transition oeration agents use the following mechanisms: Refinement. If a conjecture χ contains an oerator a j that makes an assumtion (see figure 2): i) If a causal link (a i aj ) can be established such that a i is already in the conjecture, the refinement will contain the causal link (a i aj ), the ordering constraint (a i a j ) and the binding constraints to unify with the effects of a i ; ii) Otherwise, agents must comute a sub conjecture χ to rove. The refinement will contain all the elements of χ, a causal link (a i aj ) to secify which oerator a i of χ reaches the assumtion done by a j and a ordering constraint (a i a j ). Note that we have already shown in [10] how an agent can roduce such conjecture. Case 1 Case 2 ai ai ai aj aj aj aj Before After Before After Fig. 2. The left figure shows a refinement when an oerator already reached an assumtion and right figure shows a refinement by adding a new conjecture. Reair 2. If there is a causal refutation on (a i aj ) by an action a k that has an effect q, and q is unifiable with, then the resolvers are any of the following: i) add an ordering constraint such that a k occurs before the causal link; ii) add an ordering constraint such that a k occurs after the causal link; iii) add a binding constraint that makes q and non-unifiable. Refutation. The causal refutation can be comuted by testing all triles of actions of a conjecture χ. The ordering refutation can be comuted by testing that the ordering constraint reresent a loo free grah. Finally, the binding refutation of tye 1 and 2 (see definition 7) can be comuted in linear time, whereas the tye 3 raises a general NP-comlete Constraint Satisfaction Problem (CSP). 2 Reairs of binding refutation and ordering refutation are not discussed here.

9 9 4.2 Contextualization Layer The informational layer defines the basic mechanisms to build a solution lan. Is that enough? Not quite. The dialectical mechanism must guarantee the soundness and the comleteness of the collaborative lan synthesis rocess. Now let us consider the roof board as a search in an AND/OR tree. The assumtions and the refutations corresond to AND branches because all of them must be resolved in order to find a solution. For each assumtion and refutation the ossible resolvers (i.e., refinement and reair) corresond to OR branches because only one of them is needed in order to find a solution. In order to guarantee the comleteness, agents must coordinate their exloration. Therefore, we consider that agents can aly a transition oeration only on a secific conjecture in the roof board, called current conjecture. This conjecture defines the dialog context. The seech acts define in the contextualization layer allow agents to change the dialog context. We introduce four contextualization seech acts: i) a seech act ro.solve is erformed by an agent when it believes that a solution lan χ is reached. When the seech act ro.solve is roosed each agent checks if it can refute χ. If χ cannot be refuted each agent acknowledges the solve roosition. Otherwise, they refute χ and the dialectical rocess is extended; ii) a seech act ro.failure is erformed by an agent when it believes that no solution lan exists. Like the revious seech act, when seech act ro.failure is erformed, each agent checks if there is a conjecture in the roof board on which they can aly a transition oeration. In this case, each agent acknowledges the failure roosition. Otherwise, the dialectical rocess continues; iii) a seech act ro.backward is erformed by an agent when it believes that no resolver can be roosed to go further in the current conjecture exloration; iv) a seech act ro.foreward is erformed by an agent when it believes that agent have no more resolvers to aly at the current conjecture. Note that all contextualization seech acts define a joint commitment between agents. For instance, all agents must agree on the lan solution before stoing the dialectical lan synthesis rocess. The comutation of the next current conjecture when the seech acts ro.backward and ro.foreward are roosed by agents is based on A* heuristics. Recall that A* uses a heuristic estimate f(χ) of the overall solution cost consisting of, in the one hand g(χ) = cost of the current conjecture χ and in the other hand h(χ) = estimate of the additional cost of the best comlete solution that extends χ. We roose to think f(χ) as a measure of conjecture comlexity, i.e., good conjecture are simle conjectures. What is significant to comute f(χ)? [11] indicates that the most romising heuristic measure for conjecture selection is the number of actions contained in the conjecture and the number of assumtions done. Therefore, we define g(χ) as the number of action of χ, i.e., the comlexity of the conjecture and h(χ), the number of assumtions done, since each remaining assumtion must be established by some sub-conjecture. Note that this heuristic can be used locally by the agent to choose the best resolver to submit to the other agents. 5 Conclusion The dialectical lan synthesis theory model resented in this aer relies on lan roduction and revision by conjecture/refutation cycles: for a given goal, agents try collab-

10 10 oratively to roduce a valid roof, i.e., a lan. In order to demonstrate the goal assigned to the system, agents interact by using a conventional dialogue aroach that can be slit in two layers: informational layer, which defines the conventions to refine, refute or reair conjectures and contextualization layer, which defines the conventions to allow agents to change the dialogue state. The dialogue rules are described according to the roof board. The roof board reresents the ublic art of the communication storing the different exchanges between agents. The advantage of the dialectical lan synthesis is to merge in the collaborative lan generation, the comosition and the coordination stes. It also includes the notion of uncertainty in the agents reasoning and allows the agents to make conjectures and to comose their heterogeneous cometences. Moreover, we aly conjecture/refutation to structure the multi-agent reasoning as a collaborative investigation rocess. However, former works on synchronization, coordination and conflict resolution are integrated through the notions of refutation/reairs. From our oint of view, this aroach is suitable for alications in which agents share a common goal and in which the slitting of the lanning and the coordination stes (when agents have indeendent goals, they locally generate lans and then solve their conflicts) becomes difficult due to the agents strong interdeendence. References 1. Alami, R., Fleury, S., Herrb, M., Ingrand, F., Robert, F.: Multi robot cooeration in the martha roject. IEEE Robotics and Automation Magazine 5 (1997) Wu, D., Parsia, B., Sirin E, Hendler, J., Nau, D.: Automating daml-s web services comosition using sho2. In: Proceedings of International Semantic Web Conference. (2003) 3. Lakatos, I.: Proofs and Refutations: The Logic of Mathematical Discovery. Cambridge University Press, Cambridge, England (1976) 4. Zlotkin, G., Rosenschein, J.: Negotiation and conflict resolution in non-cooerative domains. In: Proceedings of the American National Conference on Artificial Intelligence, Boston, Massachusetts (1990) Tambe, M., Jung, H.: The benefits of arguing in a team. Artificial Intelligence Magazine 20 (1999) Clement, B., Barrett, A.: Continual coordination through shared activities. In: Proceedings of the International Conference on Autonomous Agent and Muti-Agent Systems. (2003) Grosz, B., Kraus, S.: Collaborative lans for comlex grou action. Artificial Intelligence 86 (1996) D Inverno, M., Luck, M., Georgeff, M., Kinny, D., Wooldridge, M.: The dmars architecture: A secification of the distributed multi-agent reasoning system. Autonomous Agents and Multi-Agent Systems 9 (2004) Ghallab, M., Nau, D., Traverso, P.: Automated Planning Theory and Practice. Morgan Kaufmann Publishers (2004) 10. Pellier, D., Fiorino, H.: Assumtion-based lanning. In: In Proceedings of the International Conference on Advances in Intelligence Systems Theory and Alications, Luxemburg (2004) 11. Gerevini, A., Schubert, L.: Accelerating artial-order lanners: Some techniques for effective search control and runing. Journal of Artificial Intelligence Research 5 (1996)