Son Thanh To

On the effectiveness of CNF and DNF representations in contingent planning

This paper investigates the effectiveness of two state representations, CNF and DNF, in contingent planning. To this end, we developed a new contingent planner, called CNFct, using the AND/OR forward search algorithm PrAO [To et al., 2011] and an extension of the CNF representation of [To et al., 2010] for conformant planning to handle nondeterministic and sensing actions for contingent planning. The study uses CNFct and DNFct [To et al., 2011] and proposes a new heuristic function for both planners. The experiments demonstrate that both CNFct and DNFct offer very competitive performance in a large range of benchmarks but neither of the two representations is a clear winner over the other. The paper identifies properties of the representation schemes that can affect their performance on different problems.

A generic approach to planning in the presence of incomplete information: Theory and implementation

Abstract This paper proposes a generic approach to planning in the presence of incomplete information . The approach builds on an abstract notion of a belief state representation, along with an associated set of basic operations. These operations facilitate the development of a sound and complete transition function, for reasoning about effects of actions in the presence of incomplete information, and a set of abstract algorithms for planning. The paper demonstrates how the abstract definitions and algorithms can be instantiated in three concrete representations— minimal-DNF , minimal-CNF , and prime implicates —resulting in three highly competitive conformant planners: Dnf , Cnf , and PIP . The paper relates the notion of a representation to that of ordered binary decision diagrams, a well-known belief state representation employed by many conformant planners, and several target compilation languages that have been presented in the literature. The paper also includes an experimental evaluation of the planners Dnf , Cnf , and PIP and proposes a new set of conformant planning benchmarks that are challenging for state-of-the-art conformant planners.

On the impact of belief state representation in planning under uncertainty

Planning Under Uncertainty Planning under uncertainty is one of the most general and hardest problems considered in the area [Rintanen 2004]. Uncertainty can take the form of incomplete information, wrong information, multiple action outcomes, and varying action durations. My doctoral thesis concentrates on planning with incomplete knowledge and multiple action outcomes, specifically conformant planning and contingent planning. Conformant planning [Goldman and Boddy 1996] is a problem of finding sequences of actions for the agent to act, with no observations, to achieve the goal from any possible initial state of the world in presence of incomplete initial information. Conformant planning is useful when observations are expensive, dangerous, and/or impossible. Conformant planning has attracted the attention of several researchers. A number of efficient and sophisticated conformant planners have been developed CFF [Brafman and Hoffmann, 2004], POND [Bryce et al., 2006], t0 [Palacios and Geffner, 2007], and CPA [Tran et al., 2009]. Contingent planning is a more general problem of conformant planning that allows non-deterministic actions and observation of some properties of the world, in addition to incomplete information. The contingent plan allows the agent to act, at the execution time, conditionally depending on the observed values; and guarantees to achieve the goal no matter what the actual initial world the agent starts from and which actual action outcomes occur. Contingent planning is more practical and harder, as opposed to conformant planning. Many work have been done on contingent planning resulting in various contingent planner. Among the most competitive contingent planners are contingent-FF [Hoffmann and Brafman, 2005], POND, and CLG [Albore et al., 2009]. Previous Approaches To deal with incomplete information about the world, the notion of belief state has been introduced—defined as a set of possible states. This notion is convenient for capturing the semantic of incomplete information and uncertain action effects and for defining a transition function between belief states. The use of this representation in the implementation of a planner, however, is inefficient and impractical due to its exponential size. To address this, numerous research work have been done with proposals of different approaches. Significant progress can be observed by the introduction of variety of planning systems that can solve problems of different size at different level of hardness, usually using the approach that encodes the planning problem as a search problem in the belief state space. However, the scalability of these planners, though are among the best planning systems in the literature, is still modest, mostly due to disadvantages of the methods they use to represent belief states. For example, the representation using binary decision diagrams (BDDs) [Bryant, 1992], used in POND, is usually very large and its size is sensitive to the order of the variables. Moreover, computing successor belief states in BDDs representation requires generation of intermediate formulae of exponential size. In contrast, the method used in CFF and contingent-FF, that encodes belief states implicitly through the action sequences leading to them from the initial belief state, needs a little space but incurs an excessive amount of repeated computation. Furthermore, checking whether a proposition holds after the execution of an action sequence is exponentially harder, compared with the case of the execution of one action, in general. t0 and CLG transform the problem into a search problem in the state space, whose literals represent the beliefs over the original problem. This approach is fairly efficient as t0 and CLG outperforms the others in a set of benchmarks. However, the number of literals in the translated problem can be exponential in the number of unknown literals in the original problem, making the state space extremely large and preventing the planners to scale up. Finally, the method that approximates belief states used in CPA is efficient in several problems. Yet the size of the approximated formulae, if satisfies the complete condition, explodes in many other problems. Motivation and Approaches of this Work This work provides a systematic methodology for dealing with planning under uncertainty, focusing on the representation of belief states that can be used in a forward search paradigm in the belief space for solutions. A good representation should be compact so that a planner implementing it can perform and scale up well as the larger the formulae, the more the computation and the more the memory consumption (i.e., the slower the system and the less the scalability). On the other hand, it should also have properties that allow for definition of an efficient transition function for computing successor belief states, e.g., checking satisfaction in a DNF formula is easy. Defining a direct complete transition

A generic approach to planning in the presence of incomplete information: Theory and implementation (Extended Abstract)

This paper proposes a generic approach to planning in the presence of incomplete information. The approach builds on an abstract notion of a belief state representation, along with an associated set of basic operations. These operations facilitate the development of a sound and complete transition function, for reasoning about effects of actions in the presence of incomplete information, and a set of abstract algorithms for planning. The paper demonstrates how the abstract definitions and algorithms can be instantiated in three concrete representations: minimal-DNF, minimalCNF, and prime implicates, resulting in three highly competitive conformant planners: DNF, CNF, and PIP. The paper includes an experimental evaluation of the planners DNF, CNF, and PIP and proposes a new set of conformant planning benchmarks that are challenging for state-of-the-art conformant planners. 1 Motivation and Related Work Planning in the presence of incomplete information [Goldman and Boddy 1996; Smith and Weld, 1998] is the problem of generating a plan that can achieve a given goal regardless of what the actual truth value of unknown information about the initial world is. This paper focuses on conformant planning—a special class of planning problems in the presence of incomplete information— that deals with incomplete information about the initial state and with deterministic and non-sensing actions. One of the challenges in conformant planning is the problem of reasoning about action effects in the presence of incomplete information. Specifically, consider an action a with a set of conditional effects of the form (a : ψ → η), where ψ and η are a set of literals. Each ∗This paper is based on the journal article [To, Son, and Pontelli, 2015]. effect (a : ψ → η) denotes that the set of literals η will be true in the resulting state after the execution of a in the current state if the e-condition ψ is true in the current state. Due to incomplete information, the current state and, hence, the truth value of ψ in it can be unknown. As a consequence, the fact that η must be true in the successor state can be unknown, making the computation of the successor state particularly challenging. This agrees with the results in [Baral et al., 2000; Haslum and Jonsson, 1999], that conformant planning for domains with conditional effects is at a higher complexity level than that without conditional effects. Since its introduction, conformant planning has attracted the attention of several researchers—leading to the development of several state-of-the-art conformant planners. It is important to observe that most efficient conformant planners are best-first search and progression-based planners, whose development starts with the selection of a representation language for incomplete (partially known) states and the definition of a transition (progression) function that, given a state S and an action, computes the next state S′ of the world, where S and S′ are generally incomplete and encoded in the representation. To deal with incomplete information about the world, the notion of a belief state—defined as the set of possible states—has been introduced and is widely used in planning in presence of incomplete information [Bonet and Geffner, 2000; Smith and Weld, 1998]. An advantage of this notion lies in its simplicity in representing and reasoning about the effects of actions in presence of incomplete information. Indeed, any formalism for Reasoning about Action and Change (RAC), that assumes completeness of the state, can be straightforwardly generalized to deal with incomplete states, by dealing with each individual possible state in the belief state separately, and by collecting the set of resulting states as the successor belief state. More precisely, let s be a state and a be an action executable in s (s satisfies the precondition of a). The effect of executing a in s is the set of literals that become true in the successor state, defined as e(a, s) = {` | ∃(a : ψ → η). ψ ⊆ s ∧ ` ∈ η} (1) Proceedings of the Twenty-Sixth International Joint Conference on Artificial Intelligence (IJCAI-17) 5076 The transition function Φ(a, s) that returns the result of the execution of a in s is defined by Φ(a, s) = s \ e(a, s) ∪ e(a, s) (2) where e(a, s) denotes the set of negations of literals in e(a, s). Then the function Φ̂(a, S) = {Φ(a, s) | s ∈ S} characterizes the transition between belief states and can be used for reasoning with incomplete information. The downside of the direct use of belief states lies in its size, which is exponential in the number of unknown fluents. This presents two challenges for the planners employing this representation. First, it can quickly increase the memory usage for maintaining the set of generated belief states to avoid repeated search. This can often lead to the ‘Out of memory’ situation before the planner finds a solution. Second, it directly influences the time complexity in computing the successor belief states. To address this problem, different approaches have been proposed in state-of-the-art planners, resulting in significant improvement in their performance, both planning time and scalability on the problem size. In the planner KACMBP [Cimatti, Roveri, and Bertoli, 2004], belief states are represented as OBDDs [Bryant, 1992] with the use of model checking techniques for expanding the search space. The OBDD formula was also used in the planner POND [Bryce et al., 2006] to represent literals and actions in the planning graph for computing heuristics. This BDD-based approach does not require any extra BDD manipulation operations and it can be applied in both directions: progression and regression. However, the transition function defined for OBDDs may require a huge memory consumption in the manipulation of the OBDDs. This is a potential reason why planners employing this formulation, like POND or KACMBP, do not scale well as shown in the experimental evaluation [To, Son, and Pontelli, 2015]. Hoffmann and Brafman (2006) proposed implicit encoding of belief states with the sequence of actions that leads to the belief state from the initial state. To determine whether an unknown proposition holds in a belief state, the resulting conformant planner CFF has to reason about a CNF formula that captures the semantic of the initial belief state and the entire action sequence from it to the current belief state. This representation requires very little memory. However, it incurs excessive repeated computations and checking whether two belief states under this representation are equivalent is challenging and expensive. Furthermore, checking whether a proposition holds after the execution of even one single action in the presence of incomplete information is co-NP complete. We believe those are the main reasons why CFF has difficulties in finding a solution for even small instances of harder problems, where there are unknown propositions in the e-conditions [Palacios and Geffner, 2009; Tran et al., 2009; To, Pontelli, and Son, 2009]. The approach employed in the planners T0 [Palacios and Geffner, 2007; Palacios and Geffner, 2009] and T1 [Albore, Ramirez, and Geffner, 2011] compiles incomplete information away by translating a conformant problem into a classical problem and then using the efficient classical planner FF [Hoffmann and Nebel, 2001] to solve the resulting problem. This approach demonstrates great performance improvements, e.g., T0 and T1 can solve many hard problems of large size. However, the complete translation is exponential in the conformant width of the problem, related to the number of relevant unknown literals, and the number of literals in the resulting problem can be exponential in that of the original problem. To reduce the complexity of the translation, T0 sacrifices the completeness of the translation in certain problems. This explains why T0 is unable to return a solution in several problems, that have solutions, or the translation fails for large instances of several domains, as shown in our experiments. GC[LAMA] [Nguyen et al., 2012] is another conformant planner that uses a classical planner. This approach is based on the observation that every solution for a conformant problem is also a solution for the classical problem obtained by replacing the initial belief state with a state in it. A tentative plan α for such a subproblem then can be corrected, by inserting an action sequence to produce precondition for the action the plan α violates at that place. If α cannot be fixed to become a plan for the conformant problem, GC[LAMA] continues with another plan of the same or different subproblem. GC[LAMA] is efficient on problems where the action effects are monotonic, i.e., useful actions create useful literals without destroying other useful literals, and the belief state is not large. Otherwise, it is not as efficient, as reported in the experimental evaluation. Instead of using a complete transition function in the search, the planner CPA [Son et al., 2005] approximates a belief state with the intersection of the states in it [Son and Baral, 2001]. The approximation is polynomial, yet it is incomplete and so are planners employing it. Later, a complete condition for the approximation was identified and corresponding techniques were developed in CPA to make the planner complete [Son and Tu, 2006; Tu et al., 2011]. The computation of successor belief states in this approach is very simple, i.e., it can be performed in the same manner as that for belief states. However, these techniques require the system to deal with sets of approximated states, which—in the worst case—are the same as the actual belief states being represented. Hence, the approximated formula explodes in many problems, as observed in the experiments, preventing the planner to effectively scale. The authors of CPA [Tran et al., 2009] developed preprocessing techniques which help reduce the size of the initial disjunctive formula in ce