Christer Bäckström

COMPLEXITY RESULTS FOR SAS+ PLANNING

We have previously reported a number of tractable planning problems defined in the SAS+ formalism. This article complements these results by providing a complete map over the complexity of SAS+ planning under all combinations of the previously considered restrictions. We analyze the complexity of both finding a minimal plan and finding any plan. In contrast to other complexity surveys of planning, we study not only the complexity of the decision problems but also the complexity of the generation problems. We prove that the SAS+‐PUS problem is the maximal tractable problem under the restrictions we have considered if we want to generate minimal plans. If we are satisfied with any plan, then we can generalize further to the SAS+‐US problem, which we prove to be the maximal tractable problem in this case.

Computational aspects of reordering plans

This article studies the problem of modifying the action ordering of a plan in order to optimise the plan according to various criteria. One of these criteria is to make a plan less constrained and the other is to minimize its parallel execution time. Three candidate definitions are proposed for the first of these criteria, constituting a sequence of increasing optimality guarantees. Two of these are based on deordering plans, which means that ordering relations may only be removed, not added, while the third one uses reordering, where arbitrary modifications to the ordering are allowed. It is shown that only the weakest one of the three criteria is tractable to achieve, the other two being NP-hard and even difficult to approximate. Similarly, optimising the parallel execution time of a plan is studied both for deordering and reordering of plans. In the general case, both of these computations are NP-hard. However, it is shown that optimal deorderings can be computed in polynomial time for a class of planning languages based on the notions of producers, consumers and threats, which includes most of the commonly used planning languages. Computing optimal reorderings can potentially lead to even faster parallel executions, but this problem remains NP-hard and difficult to approximate even under quite severe restrictions.This article studies the problem of modifying the action ordering of a plan in order to optimise the plan according to various criteria. One of these criteria is to make a plan less constrained and the other is to minimize its parallel execution time. Three candidate definitions are proposed for the first of these criteria, constituting a sequence of increasing optimality guarantees. Two of these are based on deordering plans, which means that ordering relations may only be removed, not added, while the third one uses reordering, where arbitrary modifications to the ordering are allowed. It is shown that only the weakest one of the three criteria is tractable to achieve, the other two being NP-hard and even difficult to approximate. Similarly, optimising the parallel execution time of a plan is studied both for deordering and reordering of plans. In the general case, both of these computations are NP-hard. However, it is shown that optimal deorderings can be computed in polynomial time for a class of planning languages based on the notions of producers, consumers and threats, which includes most of the commonly used planning languages. Computing optimal reorderings can potentially lead to even faster parallel executions, but this problem remains NP-hard and difficult to approximate even under quite severe restrictions.

A unifying approach to temporal constraint reasoning

Abstract We present a formalism, Disjunctive Linear Relations (DLRs), for reasoning about temporal constraints. DLRs subsume most of the formalisms for temporal constraint reasoning proposed in the literature and is therefore computationally expensive. We also present a restricted type of DLRs, Horn DLRs, which have a polynomial-time satisfiability problem. We prove that most approaches to tractable temporal constraint reasoning can be encoded as Horn DLRs, including the ORD-Horn algebra by Nebel and Burckert and the simple temporal constraints by Dechter et al. Thus, DLRs is a suitable unifying formalism for reasoning about temporal constraints.

State-variable planning under structural restrictions: algorithms and complexity

Abstract Computationally tractable planning problems reported in the literature so far have almost exclusively been defined by syntactical restrictions. To better exploit the inherent structure in problems, it is probably necessary to study also structural restrictions on the underlying state-transition graph. The exponential size of this graph, though, makes such restrictions costly to test. Hence, we propose an intermediate approach, using a state-variable model for planning and defining restrictions on the separate state-transition graphs for each state variable. We identify such restrictions which can tractably be tested and we present a planning algorithm which is correct and runs in polynomial time under these restrictions. The algorithm has been implemented and it outperforms Graphplan on a number of test instances. In addition, we present an exhaustive map of the complexity results for planning under all combinations of four previously studied syntactical restrictions and our five new structural restrictions. This complexity map considers both the optimal and non-optimal plan generation problem.

Planning in polynomial time: the SAS-PUBS class

This article describes a polynomial‐time, O(n3), planning algorithm for a limited class of planning problems. Compared to previous work on complexity of algorithms for knowledge‐based or logic‐based planning, our algorithm achieves computational tractability, but at the expense of only applying to a significantly more limited class of problems. Our algorithm is proven correct, and it always returns a parallel minimal plan if there is a plan at all.

On the computational complexity of temporal projection, planning, and plan validation

Abstract One kind of temporal reasoning is temporal projection—the computation of the consequences of a set of events. This problem is related to a number of other temporal reasoning tasks such as plan validation and planning. We show that one particular, simple case of temporal projection on partially ordered events turns out to be harder than previously conjectured, while planning is easy under the same restrictions. Additionally, we show that plan validation is tractable for an even larger class of plans—the unconditional plans—for which temporal projection is NP-hard, thus indicating that temporal projection may not be a necessary ingredient in planning and plan validation. Analyzing the partial decision procedure for the temporal projection problem that has been proposed by other authors, we notice that it fails to be complete for unconditional plans, a case where we have shown plan validation tractable.

Expressive equivalence of planning formalisms

Abstract A concept of expressive equivalence for planning formalisms based on polynomial transformations is defined. It is argued that this definition is reasonable and useful both from a theoretical and from a practical perspective; if two languages are equivalent, then theoretical results carry over and, more practically, we can model an application problem in one language and then easily use a planner for the other language. In order to cope with the problem of exponentially sized solutions for planning problems an even stronger concept of expressive equivalence is introduced, using the novel ESP reduction . Four different formalisms for propositional planning are then analyzed, namely two variants of STRIPS, ground TWEAK and the SAS + formalism. Although these may seem to exhibit different degrees of expressive power, it is proven that they are, in fact, expressively equivalent under ESP reduction. This means that neither negative goals, partial initial states nor multi-valued state variables increase the expressiveness of “standard” propositional STRIPS.

Tractable plan existence does not imply tractable plan generation

We present a class, 3S, of planning instances such that the plan existence problem is tractable while plan generation is provably intractable for instances of this class. The class is defined by simple structural restrictions, all of them testable in polynomial‐time. Furthermore, we show that plan generation can be carried out in time bounded by a polynomial in the size of the input and the size of the generated solution. For this class, we propose a provably sound and complete incremental planner, i.e., a planner that can usually output an executable prefix of the final plan before it has generated the whole plan.

Computational complexity of relating time points with interval

Abstract Several algebras have been proposed for reasoning about qualitative constraints over the time line. One of these algebras is Vilains point–interval algebra, which can relate time points with time intervals. Apart from being a stand-alone qualitative algebra, it is also used as a subalgebra in Meiris approach to temporal reasoning, which combines reasoning about metric and qualitative temporal constraints over both time points and time intervals. While the satisfiability problem for the full point–interval algebra is known to be NP-complete, not much is known about its 4 294 967 296 subclasses. This article completely determines the computational complexity of these subclasses and it identifies all of the maximal tractable subalgebras—five in total.

Efficient planning for a miniature assembly line

This paper presents a provably correct and efficient, polynomial time, planning tool and its application to a miniature assembly line for toy cars. Although somewhat limited, this process has many similarities with real industrial processes. One of our previous polynomial-time planning algorithms has been extended and adapted to work for a larger class of planning problems, including this particular process. The plans produced by the planner are then translated into GRAFCET charts, which are compiled into code for a programmable logic controller. Although capable of producing ordinary assembly plans, the system is mainly intended for producing plans in error recovery situations.