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Introduction Search Heuristic Results A Planning Graph Heuristic for Forward-Chaining Adversarial Planning Pascal Bercher and Robert Mattmüller Institute for Computer Science University of Freiburg ECAI 2008, Patras, Greece Wednesday, July 23, 2008 A Planning Graph Heuristic for Forward-Chaining Adversari al Planning University of Freiburg

Introduction Search Heuristic Results Introduction Motivation ◮ Given: Adversarial planning problem (extensive two-player game) ◮ Desired: Strong plan (winning strategy) Technically ◮ Two players taking turns ◮ STRIPS-style state and action encoding ◮ Full observability ◮ Reduces to evaluation of AND/OR graph over physical states A Planning Graph Heuristic for Forward-Chaining Adversarial Planning University of Freiburg

Introduction Search Heuristic Results Example AND/OR Graph and Solution . . . Problem . . . ◮ Logistics-like problem ◮ Pilot and co-pilot have . . different capabilities . (loading, unloading, flying, re-fuelling, no-ops) ◮ Co-pilot wants to sabotage transport task . . . . . . . . . A Planning Graph Heuristic for Forward-Chaining Adversarial Planning University of Freiburg

Introduction Search Heuristic Results Search ◮ Alternatives: ◮ Symbolic regression search (cf. MBP) ◮ Heuristically guided explicit-state progression search ◮ Here: Variant of AO* algorithm ◮ Search over AND/OR graph ◮ Elimination of duplicate nodes ◮ Approximative updates of cost estimates ◮ How to initialize cost estimates at leaf nodes? Variant of FF heuristic. A Planning Graph Heuristic for Forward-Chaining Adversarial Planning University of Freiburg

Introduction Search Heuristic Results Heuristic: Example ◮ Variables: v 1 , . . . , v 8 ◮ Rules in relaxed problem: � v 1 → v i +1 � , r i = i = 1 , 2 , 3 , 4 , 5 � v j → v j +1 � , r j = j = 6 , 7 ◮ Rules controlled by protagonist: { r 1 , r 2 , r 3 , r 4 , r 5 , r 7 } ◮ Rules controlled by antagonist: { r 1 , r 2 , r 6 } ◮ Current state: { v 1 } ◮ Goal: { v 1 , . . . , v 8 } A Planning Graph Heuristic for Forward-Chaining Adversarial Planning University of Freiburg

Introduction Search Heuristic Results Heuristic: Relaxed Planning Graph F p A p F p A p F p F a A a F a A a 0 0 1 1 0 0 1 1 2 v 1 v 1 v 1 v 1 v 1 v 2 r 1 v 2 r 1 v 2 r 1 v 2 r 1 v 2 v 3 r 2 v 3 r 2 v 3 r 2 v 3 r 2 v 3 v 4 r 3 v 4 v 4 r 3 v 4 v 4 v 5 r 4 v 5 v 5 r 4 v 5 v 5 v 6 r 5 v 6 v 6 r 5 v 6 v 6 v 7 v 7 r 6 v 7 v 7 r 6 v 7 v 8 r 7 v 8 v 8 r 7 v 8 v 8 A Planning Graph Heuristic for Forward-Chaining Adversarial Planning University of Freiburg

Introduction Search Heuristic Results Heuristic: Relaxed Planning Graph F p A p F p A p F p F a A a F a A a 0 0 1 1 0 0 1 1 2 v 1 v 1 v 1 v 1 v 1 v 2 r 1 v 2 r 1 v 2 r 1 v 2 r 1 v 2 v 3 r 2 v 3 r 2 v 3 r 2 v 3 r 2 v 3 v 4 r 3 v 4 v 4 r 3 v 4 v 4 v 5 r 4 v 5 v 5 r 4 v 5 v 5 v 6 r 5 v 6 v 6 r 5 v 6 v 6 v 7 v 7 r 6 v 7 v 7 r 6 v 7 v 8 r 7 v 8 v 8 r 7 v 8 v 8 A Planning Graph Heuristic for Forward-Chaining Adversarial Planning University of Freiburg

Introduction Search Heuristic Results Heuristic: Relaxed Planning Graph F p A p F p A p F p F a A a F a A a 0 0 1 1 0 0 1 1 2 v 1 v 1 v 1 v 1 v 1 v 2 r 1 v 2 r 1 v 2 r 1 v 2 r 1 v 2 v 3 r 2 v 3 r 2 v 3 r 2 v 3 r 2 v 3 v 4 r 3 v 4 v 4 r 3 v 4 v 4 v 5 r 4 v 5 v 5 r 4 v 5 v 5 v 6 r 5 v 6 v 6 r 5 v 6 v 6 v 7 v 7 r 6 v 7 v 7 r 6 v 7 v 8 r 7 v 8 v 8 r 7 v 8 v 8 A Planning Graph Heuristic for Forward-Chaining Adversarial Planning University of Freiburg

Introduction Search Heuristic Results Heuristic: Relaxed Planning Graph F p A p F p A p F p F a A a F a A a 0 0 1 1 0 0 1 1 2 v 1 v 1 v 1 v 1 v 1 v 2 r 1 v 2 r 1 v 2 r 1 v 2 r 1 v 2 v 3 r 2 v 3 r 2 v 3 r 2 v 3 r 2 v 3 v 4 r 3 v 4 v 4 r 3 v 4 v 4 v 5 r 4 v 5 v 5 r 4 v 5 v 5 v 6 r 5 v 6 v 6 r 5 v 6 v 6 v 7 v 7 r 6 v 7 v 7 r 6 v 7 v 8 r 7 v 8 v 8 r 7 v 8 v 8 A Planning Graph Heuristic for Forward-Chaining Adversarial Planning University of Freiburg

Introduction Search Heuristic Results Heuristic: Relaxed Planning Graph F p A p F p A p F p F a A a F a A a 0 0 1 1 0 0 1 1 2 v 1 v 1 v 1 v 1 v 1 v 2 r 1 v 2 r 1 v 2 r 1 v 2 r 1 v 2 v 3 r 2 v 3 r 2 v 3 r 2 v 3 r 2 v 3 v 4 r 3 v 4 v 4 r 3 v 4 v 4 v 5 r 4 v 5 v 5 r 4 v 5 v 5 v 6 r 5 v 6 v 6 r 5 v 6 v 6 v 7 v 7 r 6 v 7 v 7 r 6 v 7 v 8 r 7 v 8 v 8 r 7 v 8 v 8 A Planning Graph Heuristic for Forward-Chaining Adversarial Planning University of Freiburg

Introduction Search Heuristic Results Heuristic: Relaxed Planning Graph F p A p F p A p F p F a A a F a A a 0 0 1 1 0 0 1 1 2 v 1 v 1 v 1 v 1 v 1 v 2 r 1 v 2 r 1 v 2 r 1 v 2 r 1 v 2 v 3 r 2 v 3 r 2 v 3 r 2 v 3 r 2 v 3 v 4 r 3 v 4 v 4 r 3 v 4 v 4 v 5 r 4 v 5 v 5 r 4 v 5 v 5 v 6 r 5 v 6 v 6 r 5 v 6 v 6 v 7 v 7 r 6 v 7 v 7 r 6 v 7 v 8 r 7 v 8 v 8 r 7 v 8 v 8 A Planning Graph Heuristic for Forward-Chaining Adversarial Planning University of Freiburg

Introduction Search Heuristic Results Heuristic: Relaxed Planning Graph F p A p F p A p F p F a A a F a A a 0 0 1 1 0 0 1 1 2 v 1 v 1 v 1 v 1 v 1 v 2 r 1 v 2 r 1 v 2 r 1 v 2 r 1 v 2 v 3 r 2 v 3 r 2 v 3 r 2 v 3 r 2 v 3 v 4 r 3 v 4 v 4 r 3 v 4 v 4 v 5 r 4 v 5 v 5 r 4 v 5 v 5 v 6 r 5 v 6 v 6 r 5 v 6 v 6 v 7 v 7 r 6 v 7 v 7 r 6 v 7 v 8 r 7 v 8 v 8 r 7 v 8 v 8 A Planning Graph Heuristic for Forward-Chaining Adversarial Planning University of Freiburg

Introduction Search Heuristic Results Heuristic: Relaxed Planning Graph F p A p F p A p F p F a A a F a A a 0 0 1 1 0 0 1 1 2 v 1 v 1 v 1 v 1 v 1 v 2 r 1 v 2 r 1 v 2 r 1 v 2 r 1 v 2 v 3 r 2 v 3 r 2 v 3 r 2 v 3 r 2 v 3 v 4 r 3 v 4 v 4 r 3 v 4 v 4 v 5 r 4 v 5 v 5 r 4 v 5 v 5 v 6 r 5 v 6 v 6 r 5 v 6 v 6 v 7 v 7 r 6 v 7 v 7 r 6 v 7 v 8 r 7 v 8 v 8 r 7 v 8 v 8 A Planning Graph Heuristic for Forward-Chaining Adversarial Planning University of Freiburg

Introduction Search Heuristic Results Heuristic: Relaxed Planning Graph F p A p F p A p F p F a A a F a A a 0 0 1 1 0 0 1 1 2 v 1 v 1 v 1 v 1 v 1 v 2 r 1 v 2 r 1 v 2 r 1 v 2 r 1 v 2 v 3 r 2 v 3 r 2 v 3 r 2 v 3 r 2 v 3 v 4 r 3 v 4 v 4 r 3 v 4 v 4 v 5 r 4 v 5 v 5 r 4 v 5 v 5 v 6 r 5 v 6 v 6 r 5 v 6 v 6 v 7 v 7 r 6 v 7 v 7 r 6 v 7 v 8 r 7 v 8 v 8 r 7 v 8 v 8 A Planning Graph Heuristic for Forward-Chaining Adversarial Planning University of Freiburg

Introduction Search Heuristic Results Heuristic: Relaxed Planning Graph F p A p F p A p F p F a A a F a A a 0 0 1 1 0 0 1 1 2 v 1 v 1 v 1 v 1 v 1 v 2 r 1 v 2 r 1 v 2 r 1 v 2 r 1 v 2 v 3 r 2 v 3 r 2 v 3 r 2 v 3 r 2 v 3 v 4 r 3 v 4 v 4 r 3 v 4 v 4 v 5 r 4 v 5 v 5 r 4 v 5 v 5 v 6 r 5 v 6 v 6 r 5 v 6 v 6 v 7 v 7 r 6 v 7 v 7 r 6 v 7 v 8 r 7 v 8 v 8 r 7 v 8 v 8 A Planning Graph Heuristic for Forward-Chaining Adversarial Planning University of Freiburg

Introduction Search Heuristic Results Heuristic: Relaxed Plan Extraction F p A p F p A p F p F a A a F a A a 0 0 1 1 0 0 1 1 2 v 1 v 1 v 1 v 1 v 1 v 2 r 1 v 2 r 1 v 2 r 1 v 2 r 1 v 2 v 3 r 2 v 3 r 2 v 3 r 2 v 3 r 2 v 3 v 4 r 3 v 4 v 4 r 3 v 4 v 4 v 5 r 4 v 5 v 5 r 4 v 5 v 5 v 6 r 5 v 6 v 6 r 5 v 6 v 6 v 7 v 7 r 6 v 7 v 7 r 6 v 7 v 8 r 7 v 8 v 8 r 7 v 8 v 8 A Planning Graph Heuristic for Forward-Chaining Adversarial Planning University of Freiburg

Introduction Search Heuristic Results Heuristic: Relaxed Plan Extraction F p A p F p A p F p F a A a F a A a 0 0 1 1 0 0 1 1 2 v 1 v 1 v 1 v 1 v 1 v 2 r 1 v 2 r 1 v 2 r 1 v 2 r 1 v 2 v 3 r 2 v 3 r 2 v 3 r 2 v 3 r 2 v 3 v 4 r 3 v 4 v 4 r 3 v 4 v 4 v 5 r 4 v 5 v 5 r 4 v 5 v 5 v 6 r 5 v 6 v 6 r 5 v 6 v 6 v 7 v 7 r 6 v 7 v 7 r 6 v 7 v 8 r 7 v 8 v 8 r 7 v 8 v 8 A Planning Graph Heuristic for Forward-Chaining Adversarial Planning University of Freiburg

Introduction Search Heuristic Results Heuristic: Relaxed Plan Extraction F p A p F p A p F p F a A a F a A a 0 0 1 1 0 0 1 1 2 v 1 v 1 v 1 v 1 v 1 v 2 r 1 v 2 r 1 v 2 r 1 v 2 r 1 v 2 v 3 r 2 v 3 r 2 v 3 r 2 v 3 r 2 v 3 v 4 r 3 v 4 v 4 r 3 v 4 v 4 v 5 r 4 v 5 v 5 r 4 v 5 v 5 v 6 r 5 v 6 v 6 r 5 v 6 v 6 v 7 v 7 r 6 v 7 v 7 r 6 v 7 v 8 r 7 v 8 v 8 r 7 v 8 v 8 A Planning Graph Heuristic for Forward-Chaining Adversarial Planning University of Freiburg

Introduction Search Heuristic Results Heuristic: Relaxed Plan Extraction F p A p F p A p F p F a A a F a A a 0 0 1 1 0 0 1 1 2 v 1 v 1 v 1 v 1 v 1 v 2 r 1 v 2 r 1 v 2 r 1 v 2 r 1 v 2 v 3 r 2 v 3 r 2 v 3 r 2 v 3 r 2 v 3 v 4 r 3 v 4 v 4 r 3 v 4 v 4 v 5 r 4 v 5 v 5 r 4 v 5 v 5 v 6 r 5 v 6 v 6 r 5 v 6 v 6 v 7 v 7 r 6 v 7 v 7 r 6 v 7 v 8 r 7 v 8 v 8 r 7 v 8 v 8 A Planning Graph Heuristic for Forward-Chaining Adversarial Planning University of Freiburg

Introduction Search Heuristic Results Heuristic: Relaxed Plan Extraction F p A p F p A p F p F a A a F a A a 0 0 1 1 0 0 1 1 2 v 1 v 1 v 1 v 1 v 1 v 2 r 1 v 2 r 1 v 2 r 1 v 2 r 1 v 2 v 3 r 2 v 3 r 2 v 3 r 2 v 3 r 2 v 3 v 4 r 3 v 4 v 4 r 3 v 4 v 4 v 5 r 4 v 5 v 5 r 4 v 5 v 5 v 6 r 5 v 6 v 6 r 5 v 6 v 6 v 7 v 7 r 6 v 7 v 7 r 6 v 7 v 8 r 7 v 8 v 8 r 7 v 8 v 8 A Planning Graph Heuristic for Forward-Chaining Adversarial Planning University of Freiburg

Introduction Search Heuristic Results Heuristic: Relaxed Plan Extraction F p A p F p A p F p F a A a F a A a 0 0 1 1 0 0 1 1 2 v 1 v 1 v 1 v 1 v 1 v 2 r 1 v 2 r 1 v 2 r 1 v 2 r 1 v 2 v 3 r 2 v 3 r 2 v 3 r 2 v 3 r 2 v 3 v 4 r 3 v 4 v 4 r 3 v 4 v 4 v 5 r 4 v 5 v 5 r 4 v 5 v 5 v 6 r 5 v 6 v 6 r 5 v 6 v 6 v 7 v 7 r 6 v 7 v 7 r 6 v 7 v 8 r 7 v 8 v 8 r 7 v 8 v 8 A Planning Graph Heuristic for Forward-Chaining Adversarial Planning University of Freiburg