Christer Backstrom and Bernhard Nebel. Complexity results for SAS + planning. Computational Intelligence, 11(4):625--655, 1995.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....is a reasonable restriction from a plan execution point of view. 5 Related Work Other work in the AI planning literature concerned with computational complexity results mostly focuses on domain independent planning, analyzing different variants of the planning problem and special cases thereof [2, 4, 5]. This work mainly covers purely syntactical restrictions of general planning, such as limiting the number of operator preconditions or effects, but also discusses the complexity of STRIPS style planning in (arbitrary) fixed domains [5] There are very few articles in the planning literature ....

Christer Backstrom and Bernhard Nebel. Complexity results for SAS + planning. Computational Intelligence, 11(4):625--655, 1995.

....be made clear in chapter 6. 2.2 An Example: The Cappucino Brewer Below we will illustrate the SAS formalism using an example based on a simple idea of how an automatic cappucino brewer could work. This example is chosen for illustrative purpose only and is taken from a forthcoming paper by Backstrom and Nebel [ 1996 ] The brewer (see Figure 2.1) consists of a steam container, a coffee supply, a coffee filter and a milk steamer. In order to brew a cup of espresso there must be fresh coffee in the filter and steam in the steam container. If the milk in the milk steamer has been steamed to foam we might also ....

Christer Backstrom and Bernhard Nebel. Complexity results for SAS + planning. Computational Intelligence, 1996.

....leading to tractability. A line of research concerning the complexity of state variable planning was initiated by Backstrom and Klein in [ Backstrom and Klein, 1990 ] This paper was the first in a row of papers [ Backstrom and Klein, 1991a, Backstrom and Klein, 1991b, Backstrom, 1992b, Backstrom and Nebel, 1993 ] on the complexity of state variable planning in the SAS formalism. By starting with a simple, severely restricted problem and successively removing and replacing restrictions, several tractable subproblems could be identified. Naturally, the complexity situation is even worse for first order ....

....blocks world domain. It has, for example, been proven that the blocks world minimal plan generation problem is NP complete, whereas nonminimal blocks world planning can be carried out in polynomial time [ Gupta and Nau, 1991 ] A similar result holds for the SAS US plan generation problem [ Backstrom and Nebel, 1993 ] The most obvious alternative definition of tractability is to settle for tractability in the average case. One problem with this approach is to define models for realistic average cases. Although attempts to define average case planning problems has been made [ Bylander, 1993 ] there seems ....

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Christer Backstrom and Bernhard Nebel. Complexity results for SAS + planning. In Ruzena Bajcsy, editor, Proceedings of the 13th International Joint Conference on Artificial Intelligence (IJCAI-93), pages 1430--1435, Chamb'ery, France, August--September 1993. Morgan Kaufmann.

....can be tested in polynomial time [51] We continued pursuing a bottom up approach; starting with the SASPUBS problem, we repeatedly removed restrictions such that the resulting problem is still tractable. This led to proving tractability for the SASPUS [60] SAS PUS [51, 52] and SAS US [63, 64] problems and we developed sound and complete, polynomial time planning algorithms for each of these problems. In one of these steps we removed restrictions on the basic formalism (from SAS to SAS ) rather than removing any of the P, U, B and S restrictions. This first phase of our research ....

....restrictions on the basic formalism (from SAS to SAS ) rather than removing any of the P, U, B and S restrictions. This first phase of our research was concluded by an analysis 12 of the complexity for all combinations of the SAS and SAS formalisms and of the P, U, B and S restrictions [63, 64]. In this analysis we studied both the existence problem (finding out whether a solution exists or not) and the generation problem (generating a solution or answer that none exists) and we considered both bounded and unbounded solutions. The result was that under the restrictions considered, SAS ....

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Christer Backstrom and Bernhard Nebel. Complexity results for SAS + planning. Computational Intelligence, 1993.

....can be tested in polynomial time [51] We continued pursuing a bottom up approach; starting with the SASPUBS problem, we repeatedly removed restrictions such that the resulting problem is still tractable. This led to proving tractability for the SASPUS [60] SAS PUS [51, 52] and SAS US [63, 64] problems and we developed sound and complete, polynomial time planning algorithms for each of these problems. In one of these steps we removed restrictions on the basic formalism (from SAS to SAS ) rather than removing any of the P, U, B and S restrictions. This first phase of our research ....

....restrictions on the basic formalism (from SAS to SAS ) rather than removing any of the P, U, B and S restrictions. This first phase of our research was concluded by an analysis 12 of the complexity for all combinations of the SAS and SAS formalisms and of the P, U, B and S restrictions [63, 64]. In this analysis we studied both the existence problem (finding out whether a solution exists or not) and the generation problem (generating a solution or answer that none exists) and we considered both bounded and unbounded solutions. The result was that under the restrictions considered, SAS ....

Christer Backstrom and Bernhard Nebel. Complexity results for SAS + planning. In Ruzena Bajcsy, editor, Proceedings of the 13th International Joint Conference on Artificial Intelligence (IJCAI-93), Chamb'ery, France, August--September 1993. Morgan Kaufmann. 27

.... v i v j (4) Thus, the upper bound for MinPlanSize, presented in Equation 2 can be exponential in the size of the problem description. Now we show an example, for which such an exponential upper bound can be achieved. This particular example was used in different context by Backstrom and Nebel in [2]. Consider a propositional planning problem with jVj = n, and Parents(v i ) fv1 ; v i 1 g for 1 i n. The operator set consist of 2n operators fA1 ; A 0 1 ; An ; A 0 n g where pre(A i ) post(A 0 i ) 0 post(A 0 i ) pre(A i ) 1 prv(A i ) j] prv(A 0 i ) j] ....

Christer Backstrom and Bernhard Nebel, `Complexity results for SAS + planning', Computational Intelligence, 11(4), 625--655, (1995).

.... and Highpoint that always produces hierarchies allowing a hierarchical planner to generate plans of length within a constant factor of the shortest length (actually, not even within a logarithmic factor in the size of the instance) We have chosen in this paper to concentrate on state ab 3 See Backstrom and Nebel [1993] or [Garey and Johnson, 1979, pp. 11 12] for a discussion of this topic. 127 straction as defined and used by Knoblock [ 1994 ] i.e. using a total order hierarchical planner. We are currently investigating the consequences of using a partial order hierarchical planner like AbTweak [ Yang and ....

Christer Backstrom and Bernhard Nebel. Complexity results for SAS + planning. In Ruzena Bajcsy, editor, Proceedings of the 13th International Joint Conference on Artificial Intelligence (IJCAI-93), pages 1430--1435, Chamb'ery, France, August--September 1993. Morgan Kaufmann. 128

....methods than heuristics. Some tractability results have been reported in the literature for restrictions on the propositional Strips formalism [ Bylander, 1992a, Bylander, 1994, Erol et al. 1992 ] and for restrictions on the related state variable formalism SAS [ Backstrom and Klein, 1991, Backstrom and Nebel, 1993 ] These results are all based on essentially syntactic restrictions on the set of operators. Syntactic restrictions are very appealing to study, since, typically, they are easy to define and not very costly to test. However, to gain any deeper insight into what makes planning problems hard and ....

....graph, many interesting and useful properties of this graph can be indirectly exploited. In particular, we identified three structural restrictions which together make planning tractable and properly generalize the tractable problems we have previously defined using syntactical restrictions [ Backstrom and Nebel, 1993 ] We also presented a polynomial time, sound and complete algorithm for generating optimal plans under the new structural restrictions. Despite being structural, our restrictions can be tested in polynomial time. Further, this approach would not be very useful for a planning formalism based on ....

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Christer Backstrom and Bernhard Nebel. Complexity results for SAS + planning. In Ruzena Bajcsy, editor, Proceedings of the 13th International Joint Conference on Artificial Intelligence (IJCAI-93), pages 1430--1435, Chamb'ery, France, August--September 1993. Morgan Kaufmann.

....third formalism is the ground (i.e. propositional) variant of TWEAK [11] and it is, thus, closely related to the first two formalisms. We refer to this formalism as Ground TWEAK (GT) The fourth, and final, formalism in our analysis is the extended simplified action structures (SAS ) formalism [2, 3, 6, 17], which derives from the SAS formalism [4, 5] and the original action structures formalism [23] These four formalism seem to form a sequence of successively more and more expressive power, in the order presented above. The PSN formalism adds to the CPS formalism the capability of expressing ....

....technically necessary, this distinction between the precondition and the prevail condition has shown to have conceptual advantages in some cases. For instance, it has been possible to identify certain restrictions that result in computationally tractable subcases of the SAS planning problem [2, 3, 4, 5, 6, 17]. Distinguishing the changed and unchanged parts of the preconditions has also made it easier to define criteria for possible parallel execution of operators [2, 4, 5] 6 variables having some defined value in either of these conditions (a variable cannot be defined in both these conditions) has ....

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Christer Backstrom and Bernhard Nebel. Complexity results for SAS + planning. In Bajcsy [7].

....complete even in a very weak sense. The paper is concluded with a brief discussion of the results. 2 Basic Formalism We base our work in this paper on the propositional Strips formalism with negative goals [ Bylander, 1994 ] which is equivalent to most other variants of propositional Strips [ Backstrom, 1995 ] Definition 2.1 An instance of the PSN planning problem is a quadruple Pi = hP ; O; I; Gi where ffl P is a finite set of atoms; ffl O is a finite set of operators where o 2 O has the form Pre ) P ost where Pre is a satisfiable conjunction of positive and negative atoms in P , ....

....power than the general PSN planning problem which is Pspace complete 4 . Yet, there do 4 Under the plausible assumption that NP6=Pspace 9 not exist P T;S A universal plans for this class of planning problems. Note that this is not caused by the existence of exponentially size minimal plans [ Backstrom and Nebel, 1995 ] since all minimal plans in this class are polynomially bounded. Finally, we would like to compare Theorem 4.5 with a negative result by Selman [ 1994 ] Theorem 4.6 Unless NP P poly, there exists a blocks world planning goal for which there is no P T;S A universal plan for generating the ....

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Christer Backstrom and Bernhard Nebel. Complexity results for SAS + planning. Computational Intelligence, 11(4):625--655, 1995.

....to guarantee efficiency and correctness. Further, we believe that if having such a deep understanding of the problem structure, it is better to use other methods than heuristics. Some tractability results for planning has been reported in the literature lately [ Backstrom and Klein, 1991, Backstrom and Nebel, 1993, Bylander, 1994, Erol et al. 1992 ] However, apart from being very restricted, they are all based on essentially syntactic restrictions on the set of operators. Syntactic restrictions are very appealing to study, since they are typically easy to define and not very costly to test. However, to ....

....the state transition graph must be done with great care, however. This graph is typically of size exponential in the size of the planning problem instance, making it extremely costly to test arbitrary properties. In this paper, we take an intermediate approach. We adopt the statevariable model of Backstrom and Nebel [ 1993 ] and define restrictions not on the whole state transition graph, but on the domain transition graph for each state variable in isolation. This is less costly since each such graph is only of polynomial size. Although not being a substitute for restrictions on the whole state transition graph, ....

[Article contains additional citation context not shown here]

Christer Backstrom and Bernhard Nebel. Complexity results for SAS + planning. In Ruzena Bajcsy, editor, Proceedings of the 13th International Joint Conference on Artificial Intelligence (IJCAI-93), pages 1430--1435, Chamb'ery, France, August--September 1993. Morgan Kaufmann.

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