Acknowledgements Many of the slides used in todays lecture are - PowerPoint PPT Presentation

Acknowledgements Many of the slides used in today’s lecture are modifications of Heuristic Search for Planning slides developed by Malte Helmert, Bernhard Nebel, and Jussi Rintanen. Sheila McIlraith Some material comes from papers by Daniel Bryce and Rao Kambhampati. University of Toronto I would like to gratefully acknowledge the contributions of these Fall 2010 researchers, a nd thank them for generously permitting me to use aspects of their presentation material. S. McIlraith Heuristic Search for Planning 1 / 50 S. McIlraith Heuristic Search for Planning 2 / 50 Outline A simple heuristic for deterministic planning 1 How to obtain a heuristic The STRIPS heuristic STRIPS (Fikes & Nilsson, 1971) used the number of state variables Relaxation and abstraction that differ in current state s and a STRIPS goal l 1 ∧ · · · ∧ l n : 2 Towards relaxations for planning: Positive normal form h ( s ) := |{ i ∈ { 1 , . . . , n } | s ( a ) �| = l i }| . Motivation Definition & algorithm Intuition: more true goal literals � closer to the goal Example � STRIPS heuristic (properties?) 3 Relaxed planning tasks Note: From now on, for convenience we usually write heuristics as Definition functions of states (as above), not nodes. Greedy algorithm Node heuristic h ′ is defined from state heuristic h as Optimality h ′ ( σ ) := h ( state ( σ )) . Discussion Towards better relaxed plans S. McIlraith Heuristic Search for Planning 3 / 50 S. McIlraith Heuristic Search for Planning 4 / 50

Criticism of the STRIPS heuristic Outline 1 How to obtain a heuristic The STRIPS heuristic What is wrong with the STRIPS heuristic? Relaxation and abstraction quite uninformative: the range of heuristic values in a given task is small; 2 Towards relaxations for planning: Positive normal form typically, most successors have the same estimate Motivation Definition & algorithm very sensitive to reformulation: Example can easily transform any planning task into an equivalent one where h ( s ) = 1 for all non-goal states 3 Relaxed planning tasks ignores almost all problem structure: Definition heuristic value does not depend on the set of operators! Greedy algorithm � need a better, principled way of coming up with heuristics Optimality Discussion Towards better relaxed plans S. McIlraith Heuristic Search for Planning 5 / 50 S. McIlraith Heuristic Search for Planning 6 / 50 Coming up with heuristics in a principled way Relaxing a problem How do we relax a problem? General procedure for obtaining a heuristic Example (Route planning for a road network) The road network is formalized as a weighted graph over points in Solve an easier version of the problem. the Euclidean plane. The weight of an edge is the road distance between two locations. Two common methods: A relaxation drops constraints of the original problem. relaxation: consider less constrained version of the problem abstraction: consider smaller version of real problem Example (Relaxation for route planning) � | x 1 − y 1 | 2 + | x 2 − y 2 | 2 as a heuristic Use the Euclidean distance Both have been very successfully applied in planning. for the road distance between ( x 1 , x 2 ) and ( y 1 , y 2 ) We consider both in this course, beginning with relaxation. This is a lower bound on the road distance ( � admissible). � We drop the constraint of having to travel on roads. S. McIlraith Heuristic Search for Planning 7 / 50 S. McIlraith Heuristic Search for Planning 8 / 50

A ∗ using the Euclidean distance heuristic A ∗ using the Euclidean distance heuristic Wurzburg Wurzburg Frankfurt 100 km Frankfurt 100 km 1 1 120 km 2 120 km 2 0 0 k Nuremberg k Nuremberg m m 100 km 100 km 270 km 200 km 150 km 200 km Regensburg Regensburg Karlsruhe 100 km Karlsruhe 100 km Stuttgart Stuttgart 160 km 160 km 100 km 100 km 80 km 80 km 120 km 120 km 120 km 120 km Passau Passau 1 1 0 0 0 0 k k m m Ulm Ulm Munich Munich Freiburg Freiburg S. McIlraith Heuristic Search for Planning 9 / 50 S. McIlraith Heuristic Search for Planning 10 / 50 A ∗ using the Euclidean distance heuristic A ∗ using the Euclidean distance heuristic 100 km Wurzburg 100 km Wurzburg Frankfurt Frankfurt 420 km 420 km 180 km 180 km 1 1 120 km 120 km 2 2 0 0 k Nuremberg k Nuremberg m m 100 km 100 km 340 km 120 km Regensburg Regensburg 450 km 100 km 100 km 130 km Karlsruhe Karlsruhe Stuttgart Stuttgart 160 km 160 km 100 km 100 km 80 km 80 km 120 km 120 km 120 km 120 km Passau Passau 1 0 1 0 0 0 k m k m Ulm Ulm Munich Munich Freiburg Freiburg S. McIlraith Heuristic Search for Planning 11 / 50 S. McIlraith Heuristic Search for Planning 12 / 50

A ∗ using the Euclidean distance heuristic A ∗ using the Euclidean distance heuristic Wurzburg Wurzburg Frankfurt 100 km Frankfurt 100 km 4 4 0 1 k 0 0 m k 1 1 460 km 120 km m 2 120 km 2 0 0 k Nuremberg k Nuremberg m m 100 km 100 km Regensburg Regensburg 450 km 450 km Karlsruhe 100 km 130 km Karlsruhe 100 km 130 km Stuttgart Stuttgart 160 km 160 km 100 km 100 km 80 km 80 km 120 km 120 km 120 km 120 km Passau Passau 1 1 0 0 0 0 k k m m Ulm Ulm Munich Munich Freiburg Freiburg S. McIlraith Heuristic Search for Planning 13 / 50 S. McIlraith Heuristic Search for Planning 14 / 50 A ∗ using the Euclidean distance heuristic A ∗ using the Euclidean distance heuristic 100 km Wurzburg 100 km Wurzburg Frankfurt Frankfurt 1 1 120 km 460 km 120 km 460 km 2 2 0 0 k Nuremberg k Nuremberg m m 100 km 100 km Regensburg Regensburg 100 km 100 km Karlsruhe Karlsruhe Stuttgart 540 km Stuttgart 120 km 160 km 160 km 100 km 100 km 80 km 80 km 120 km 120 km 120 km 120 km Passau Passau 1 0 1 0 0 0 k m k m Ulm Ulm Munich Munich Freiburg Freiburg S. McIlraith Heuristic Search for Planning 15 / 50 S. McIlraith Heuristic Search for Planning 16 / 50

Outline Relaxations for planning 1 How to obtain a heuristic The STRIPS heuristic Relaxation and abstraction Relaxation is a general technique for heuristic design: 2 Towards relaxations for planning: Positive normal form Straight-line heuristic (route planning): Ignore the fact that Motivation one must stay on roads. Manhattan heuristic (15-puzzle): Ignore the fact that one Definition & algorithm cannot move through occupied tiles. Example We want to apply the idea of relaxations to planning. 3 Relaxed planning tasks Informally, we want to ignore bad side effects of applying Definition operators. Greedy algorithm Optimality Discussion Towards better relaxed plans S. McIlraith Heuristic Search for Planning 17 / 50 S. McIlraith Heuristic Search for Planning 18 / 50 What is a good or bad effect? Outline 1 How to obtain a heuristic The STRIPS heuristic Relaxation and abstraction Question: Which operator effects are good, and which are bad? 2 Towards relaxations for planning: Positive normal form Difficult to answer in general, because it depends on context: Motivation Locking the entrance door is good if we want to keep burglars Definition & algorithm out. Example Locking the entrance door is bad if we want to enter. 3 Relaxed planning tasks We will now consider a reformulation of planning tasks that makes Definition the distinction between good and bad effects obvious. Greedy algorithm Optimality Discussion Towards better relaxed plans S. McIlraith Heuristic Search for Planning 19 / 50 S. McIlraith Heuristic Search for Planning 20 / 50