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Abstracting Abstraction in Search II: Complexity Analysis Proceedings of the Fifth Annual Symposium on Combinatorial Search Abstracting Abstraction in Search II: Complexity Analysis Christer Backstr¨ om¨ and Peter Jonsson Department of Computer Science, Linkoping¨ University SE-581 83 Linkoping,¨ Sweden [email protected] [email protected] Abstract In our earlier publication (Backstr¨ om¨ and Jonsson 2012) we have demonstrated the usefulness of this framework in Modelling abstraction as a function from the original state various ways. We have shown that a certain combination of space to an abstract state space is a common approach in com- our transformation properties exactly captures the DPP con- binatorial search. Sometimes this is too restricted, though, cept by Zilles and Holte (2010). A related concept is that and we have previously proposed a framework using a more flexible concept of transformations between labelled graphs. of path/plan refinement without backtracking to the abstract We also proposed a number of properties to describe and clas- level. Partial solutions to capturing this concept have been sify such transformations. This framework enabled the mod- presented in the literature, for instance, the ordered mono- elling of a number of different abstraction methods in a way tonicity criterion by Knoblock, Tenenberg, and Yang (1991), that facilitated comparative analyses. It is of particular inter- the downward refinement property (DRP) by Bacchus and est that these properties can be used to capture the concept of Yang (1994), and the simulation-based approach by Bundy refinement without backtracking between levels; how to do et al. (1996). However, how to define general conditions this has been an open question for at least twenty years. In that capture this concept has remained an open question in this paper, we continue our previous research by analysing the literature until our previous paper. There we proved that the complexity of testing the various transformation proper- different combinations of our transformation properties can ties for both explicit and implicit graph representations. be used to capture this concept, and we can even distinguish several different degrees of the concept. Furthermore, we re- 1 Introduction formulated five different abstraction methods in planning as transformations. This enabled us to analyse and compare the Abstraction in combinatorial search is often modelled as a methods in new ways by determining their inherent transfor- function f from the vertices of one graph to the vertices mation properties. of another graph, the latter graph being the abstraction of The reader is strongly recommended to also read our pre- the first. It is also common that the function is a homomor- vious paper (Backstr¨ om¨ and Jonsson 2012) in connection phism. This approach is very natural and has proven useful with this one, since that paper focused on presenting the in many cases, cf. Holte et al. (1996), Helmert, Haslum, framework with motivating examples on how to use it, as and Hoffmann (2007) and Zilles and Holte (2010). It is not well as proving various theorems about it. This paper must always sufficient, however; one such case is abstraction in be viewed as a continuation of that work, focusing only on planning. Motivated by the desire to model and compare the computational complexity of deciding which properties various abstraction methods for planning, we have earlier a transformation has; certain aspects of the framework might (Backstr¨ om¨ and Jonsson 2012) defined a more flexible ab- seem unmotivated if reading this paper alone. Sections 2 to 5 straction framework based on transformations between la- presents the framework used in our previous paper, but omits belled graphs. A transformation is a pair hf; Ri, where f all proofs and some of the discussions. All major definitions is a transformation function and R is a label relation. The and results about the framework itself remain. The new re- function f maps vertices in the original graph to sets of ver- sults are the complexity analyses in Sections 6 to 8. The tices in the abstract graph and the relation R specifies how paper ends with a discussion of these results in Section 9. labels in the two graphs are related, which implicitly speci- fies how subsets of arcs in the graphs are related. While the 2 STGs and STG Transformations transformation concept can capture many abstraction meth- ods, it is a more general concept which is not specifically We first introduce our framework for studying abstractions. tailored to abstraction. In order to classify, analyse and com- Although the definitions may appear somewhat complex and pare transformations we also introduced a number of trans- difficult to understand at first sight, there is a reason: we formation properties. want to prove results, not merely devote ourselves to dis- cussions. We begin by defining some general notation and Copyright c 2012, Association for the Advancement of Artificial concepts, then we introduce state transition graphs and our Intelligence (www.aaai.org). All rights reserved. transformation concept. 10 If X is a set, then jXj denotes the cardinality of X.A f1(00) = f1(01) f1(10) = f1(11) partition of a set X is a set P of non-empty subsets of X c such that (1) [p2P p = X and (2) for all p; q 2 P , if p 6= q, G2: 0 1 then p \ q = ?. Let f : X ! Y be a function, then Rng(f) = ff(x) j x 2 Xg is the range of f. Often, a func- tion f will be from X to 2Y (for some sets X and Y ). In this case, Rng(f) ⊆ 2Y , that is, the value of f is a subset of a b a Y , not an element in Y . For such functions we also define G1: 00 01 10 11 f(Z) = [x2Z f(x) for all Z ⊆ X. Definition 1. A state transition graph (STG) over a set L of labels is a tuple G = hS; Ei where S is a set of vertices and f3(01) f3(10) E ⊆ S×S×L is a set of labelled arcs. The set of labels in G f3(00) f3(11) is implicitly defined as L(G) = L(E) = f` j hs; t; `i 2 Eg. 6 5 A sequence s0; `1; s1; `2; : : : ; `k; sk of states in S and labels in L(E) is a (state) path in G if either (1) k = 0 or (2) 7 4 hsi−1; si; `ii 2 E for 1 ≤ i ≤ k. The set S is called a state space and its members states. G3: d More than one arc in the same direction between two states is allowed, as long as the arcs have different labels. The in- 0 3 tention of the labels is to provide a means to identify a subset 1 2 of arcs by assigning a particular label to these arcs. This is useful, for instance, in planning where a single action may induce many arcs in an STG. If all arcs have the same la- bel, which is allowed, then the STG collapses to an ordinary directed graph. Arcs may be written as hs; ti and paths as a b a 1: 00 01 10 11 s0; : : : ; sk if the exact labels are not relevant. G Definition 2. Let G1 = hS1;E1i and G2 = hS2;E2i be two STGs. A total function f : S ! 2S2 is a transfor- 1 Figure 1: The functions f1 and f3 in Example 3. mation function from G1 to G2 if Rng(f) is a partition of S2.A label relation from G1 to G2 is a binary relation R ⊆ L(G1) × L(G2). An (STG) transformation from G1 to considerably, but it does not automatically carry over also to G2 is a pair τ = hf; Ri where f is a transformation function all types of properties of transformations; some care must be from G1 to G2 and R is a label relation from G1 to G2. exercised in such cases, as we will see examples of later. The transformation function f specifies how the transfor- Definition 4. Let G1 = hS1;E1i and G2 = hS2;E2i be mation maps states from one STG to the other while the label two STGs, let f be a transformation function from G1 to relation R provides additional information about how sets of G2 and let R be a label relation from G1 to G2. Then, the S1 arcs are related between the two STGs. Note that f is for- reverse transformation function f : S2 ! 2 is defined as mally a function from S to 2S2 , that is, it has a subset of 1 f(t) = fs 2 S1 j t 2 f(s)g and the reverse label relation S2 as value. We use a function rather than a relation since R ⊆ L( ) × L( ) is defined as R(` ; ` ) iff R(` ; ` ). it makes the theory clearer and simpler and is more in line G2 G1 2 1 1 2 with previous work in the area. Consider the functions f1 and f3 from Example 3 once again. We see that f1(0) = f00; 01g and f1(1) = f10; 11g, Example 3. Consider two STGs: : 00 !a 01 !b 10 G1 while f3(0) = f3(7) = f00g, f3(1) = f3(6) = f01g, !a 11 and : 0 !c 1. Also define f : ! G2 1 G1 G2 f3(2) = f3(5) = f10g and f3(3) = f3(4) = f11g. such that f (xy) = fxg. We see immediately that f 1 1 Lemma 5. Let f be a transformation function from an STG is a transformation function from G1 to G2. Define f2 : G1 = hS1;E1i to an STG G2 = hS2;E2i. Then: G1 ! G2 such that f2(xy) = fx; yg; this function is 1) for all s 2 S , s 2 S , s 2 f(s ) iff s 2 f(s ).
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