Planning and Optimization October 27, 2016 — C3. Delete Relaxation: AND/OR Graphs Planning and Optimization C3.1 AND/OR Graphs C3. Delete Relaxation: AND/OR Graphs C3.2 Forced Nodes Malte Helmert and Gabriele R¨ oger C3.3 Most and Least Conservative Valuation Universit¨ at Basel October 27, 2016 C3.4 Summary M. Helmert, G. R¨ oger (Universit¨ at Basel) Planning and Optimization October 27, 2016 1 / 29 M. Helmert, G. R¨ oger (Universit¨ at Basel) Planning and Optimization October 27, 2016 2 / 29 C3. Delete Relaxation: AND/OR Graphs AND/OR Graphs Motivation ◮ Our next goal is to devise efficiently computable heuristics based on delete relaxation. C3.1 AND/OR Graphs ◮ The heuristics we will consider can all be understood in terms of computations on graphical structures called AND/OR graphs. ◮ In this chapter, we introduce AND/OR graphs and study some of their major properties. ◮ In the next chapter, we will relate AND/OR graphs to relaxed planning tasks. M. Helmert, G. R¨ oger (Universit¨ at Basel) Planning and Optimization October 27, 2016 3 / 29 M. Helmert, G. R¨ oger (Universit¨ at Basel) Planning and Optimization October 27, 2016 4 / 29
C3. Delete Relaxation: AND/OR Graphs AND/OR Graphs C3. Delete Relaxation: AND/OR Graphs AND/OR Graphs AND/OR Graphs AND/OR Graph Example Definition (AND/OR Graph) An AND/OR graph � N , A , type � is a directed graph � N , A � with a node label function type : N → {∧ , ∨} partitioning nodes into ◮ AND nodes ( type ( v ) = ∧ ) and ◮ OR nodes ( type ( v ) = ∨ ). We write succ ( n ) for the successors of node n ∈ N , i.e., succ ( n ) = { n ′ ∈ N | � n , n ′ � ∈ A } . Note: We draw AND nodes as squares and OR nodes as circles. M. Helmert, G. R¨ oger (Universit¨ at Basel) Planning and Optimization October 27, 2016 5 / 29 M. Helmert, G. R¨ oger (Universit¨ at Basel) Planning and Optimization October 27, 2016 6 / 29 C3. Delete Relaxation: AND/OR Graphs AND/OR Graphs C3. Delete Relaxation: AND/OR Graphs AND/OR Graphs AND/OR Graph Valuations Example: A Consistent Valuation Definition (Consistent Valuations of AND/OR Graphs) F F F Let G be an AND/OR graph with nodes N . A valuation or truth assignment of G is a valuation α : N → { T , F } , treating the nodes as propositional variables. T T F F We say that α is consistent if ◮ for all AND nodes n ∈ N : α | n ′ ∈ succ ( n ) n ′ . = � = n iff α | n ′ ∈ succ ( n ) n ′ . ◮ for all OR nodes n ∈ N : α | = n iff α | = � T F n ′ ∈∅ n ′ = ⊤ and � n ′ ∈∅ n ′ = ⊥ . Note that � M. Helmert, G. R¨ oger (Universit¨ at Basel) Planning and Optimization October 27, 2016 7 / 29 M. Helmert, G. R¨ oger (Universit¨ at Basel) Planning and Optimization October 27, 2016 8 / 29
C3. Delete Relaxation: AND/OR Graphs AND/OR Graphs C3. Delete Relaxation: AND/OR Graphs AND/OR Graphs Example: Another Consistent Valuation Example: An Inconsistent Valuation T T F F F T F � T T F F T T T T � T F T M. Helmert, G. R¨ oger (Universit¨ at Basel) Planning and Optimization October 27, 2016 9 / 29 M. Helmert, G. R¨ oger (Universit¨ at Basel) Planning and Optimization October 27, 2016 10 / 29 C3. Delete Relaxation: AND/OR Graphs Forced Nodes C3. Delete Relaxation: AND/OR Graphs Forced Nodes How Do We Find Consistent Valuations? If we want to use valuations of AND/OR graphs algorithmically, C3.2 Forced Nodes a number of questions arise: ◮ Do consistent valuations exist for every AND/OR graph? ◮ Are they unique? ◮ If not, how are different consistent valuations related? ◮ Can consistent valuations be computed efficiently? M. Helmert, G. R¨ oger (Universit¨ at Basel) Planning and Optimization October 27, 2016 11 / 29 M. Helmert, G. R¨ oger (Universit¨ at Basel) Planning and Optimization October 27, 2016 12 / 29
C3. Delete Relaxation: AND/OR Graphs Forced Nodes C3. Delete Relaxation: AND/OR Graphs Forced Nodes Some Partial Answers Forced True Nodes ◮ We already know from our previous example that consistent valuations are in general not unique. Definition (Forced True Nodes) ◮ We will now study two special kinds of valuations: Let G be an AND/OR graph. ◮ the most conservative valuation The set of nodes of G that are forced true is defined ◮ the least conservative valuation by finite application of the following rules: ◮ We show that these two valuations are always consistent ◮ If n is an AND node where all successors and can be computed efficiently. are forced true, then n is forced true. ◮ We also show that all consistent valuations ◮ If n is an OR node where at least one successor lie “in between” these two valuations. is forced true, then n is forced true. ◮ In particular, an AND/OR graph has a unique consistent valuation iff its most and least conservative valuation coincide. M. Helmert, G. R¨ oger (Universit¨ at Basel) Planning and Optimization October 27, 2016 13 / 29 M. Helmert, G. R¨ oger (Universit¨ at Basel) Planning and Optimization October 27, 2016 14 / 29 C3. Delete Relaxation: AND/OR Graphs Forced Nodes C3. Delete Relaxation: AND/OR Graphs Forced Nodes Forced False Nodes Example: Forced Nodes Definition (Forced False Nodes) Let G be an AND/OR graph. The set of nodes of G that are forced false is defined by finite application of the following rules: (2) T T (3) F (2) ◮ If n is an AND node where at least one successor is forced false, then n is forced false. ◮ If n is an OR node where all successors are forced false, then n is forced false. T (1) F (1) M. Helmert, G. R¨ oger (Universit¨ at Basel) Planning and Optimization October 27, 2016 15 / 29 M. Helmert, G. R¨ oger (Universit¨ at Basel) Planning and Optimization October 27, 2016 16 / 29
C3. Delete Relaxation: AND/OR Graphs Forced Nodes C3. Delete Relaxation: AND/OR Graphs Forced Nodes Remarks on Forced Nodes Forced Nodes and Consistent Valuations Theorem Notes: ◮ Forced nodes are well-defined because the rules Let G be an AND/OR graph with nodes N. defining them are monotonic. Let N T ⊆ N be the forced true nodes of G, ◮ They can be computed in linear time in the size of the graph. and let N F ⊆ N be the forced false nodes of G. ◮ There exists at least one forced true node Let α : N → { T , F } be a consistent valuation of G. Then: iff the graph has an AND node without successors. ◮ α ( n ) = T for all n ∈ N T ◮ There exists at least one forced false node ◮ α ( n ) = F for all n ∈ N F iff the graph has an OR node without successors. This property explains why they are called forced nodes. M. Helmert, G. R¨ oger (Universit¨ at Basel) Planning and Optimization October 27, 2016 17 / 29 M. Helmert, G. R¨ oger (Universit¨ at Basel) Planning and Optimization October 27, 2016 18 / 29 C3. Delete Relaxation: AND/OR Graphs Forced Nodes C3. Delete Relaxation: AND/OR Graphs Forced Nodes Forced Nodes and Consistent Valuations: Proof (1) Forced Nodes and Consistent Valuations: Proof (2) Proof. Proof (continued). We prove the property of forced true nodes. Case 1: n i is an AND node. The property of forced false nodes can be proved analogously. In the i -th rule application, n i is shown as forced true, Proof by contradiction: assume that there is a consistent valuation so all its successors must have been shown as forced true α and a forced true node n ∈ N T with α ( n ) � = T , i.e., α ( n ) = F . in an earlier rule application. Let N T = { n 1 , . . . , n k } , where the nodes are ordered in a way Hence all nodes in succ ( n i ) are of the form n j with j < i . that matches a possible sequence of rule applications: By the choice of n i , we have α ( n j ) = T for all these nodes. i.e., the i -th rule application derives that n i is forced true. n ′ ∈ succ ( n i ) n ′ , = � We conclude α �| = n i and α | Let n i be the first node in the order with α ( n i ) = F . which shows that α is not consistent: a contradiction. . . . Hence, α ( n j ) = T for all j < i . . . . M. Helmert, G. R¨ oger (Universit¨ at Basel) Planning and Optimization October 27, 2016 19 / 29 M. Helmert, G. R¨ oger (Universit¨ at Basel) Planning and Optimization October 27, 2016 20 / 29
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