Solving Non-deterministic Planning Problems with Pattern Database Heuristics. KI 2009, Paderborn

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1 Solving Non-deterministic Planning Problems with Pattern Database Heuristics Pascal Bercher Institute of Artificial Intelligence University of Ulm, Germany Robert Mattmüller Department of Computer Science University of Freiburg, Germany KI 2009, Paderborn Solving Non-Deterministic Planning Problems with Pattern Database Heuristics Sept 2009, KI 09 1 / 18

2 Formalization Given: Desired: A non-deterministic planning problem. (Informally: Initial state, actions, goal states. Nondeterminism: actions may have several outcomes.) A solution to that problem. (Informally: How to reach a goal state, using the actions?) Solving Non-Deterministic Planning Problems with Pattern Database Heuristics Sept 2009, KI 09 2 / 18

3 Formalization Given: Non-deterministic planning problem P = (Var, A, s 0, G) with: Var, finite set of state variables. S = 2 Var is the state space. A, finite set of actions a = pre(a), eff(a) and: pre(a) Var and eff(a) = { add i, del i add i, del i Var and i {1,..., n} }. Its application (if pre(a) s) leads to: app(s, a) = { (s\del) add add, del eff(a) } s 0 S, the initial state. G Var, the goal description. A state s S is a goal state iff s G. Solving Non-Deterministic Planning Problems with Pattern Database Heuristics Sept 2009, KI 09 3 / 18

4 Formalization (Example) Let s = {x, y, z} S be a state and a A be an action with: a = pre(a), eff(a) and pre(a) = {x, y} s, eff(a) = { {z}, {x, y},, {t, z} }. {x, y, z} add: z del: x,y a add: ε del: t,z {z} {x, y} Solving Non-Deterministic Planning Problems with Pattern Database Heuristics Sept 2009, KI 09 4 / 18

5 Formalization Desired: Strong plan. (Success, regardless of non-deterministic outcome.) Solving Non-Deterministic Planning Problems with Pattern Database Heuristics Sept 2009, KI 09 5 / 18

6 Formalization Desired: Strong plan. (Success, regardless of non-deterministic outcome.) Solving Non-Deterministic Planning Problems with Pattern Database Heuristics Sept 2009, KI 09 5 / 18

7 Formalization Desired: Strong plan. (Success, regardless of non-deterministic outcome.) Solving Non-Deterministic Planning Problems with Pattern Database Heuristics Sept 2009, KI 09 5 / 18

8 Formalization Desired: Strong plan. (Success, regardless of non-deterministic outcome.) Solving Non-Deterministic Planning Problems with Pattern Database Heuristics Sept 2009, KI 09 5 / 18

9 Search Algorithm, modification of AO* s 0, c=4 s 0, c=5 c 0 c 1 Expansion c 0 c 1 s 1 h=2 s 2 h=3 s 3 h=4 s 4 h=3 s 1 h=2 s 2 c=5 s 3 h=4 c 3 s 4 h=3 s 5 h=2 s 6 h=0 s 7 h=4 Solving Non-Deterministic Planning Problems with Pattern Database Heuristics Sept 2009, KI 09 6 / 18

10 Idea Use abstraction to simplify the problem: S S 1 S 2... S m Map the search space S to abstract search spaces S i with S i S. Compute h(s), s S, on basis of all h i (s i ). Calculation of the h i is done before the search. Solving Non-Deterministic Planning Problems with Pattern Database Heuristics Sept 2009, KI 09 7 / 18

11 Formalization Idea: Disregard some (or rather most of the) state variables. The abstraction P i = (Var i, A i, s i 0, Gi ) is the planning problem P, restricted to the pattern P i Var: Var i Var P i = P i, For var Var let var i var P i. Then: a i pre(a) i, { add i, del i add, del eff(a) } for a A. Now, A i { a i a A }. s i 0 s 0 P i G i G P i. Solving Non-Deterministic Planning Problems with Pattern Database Heuristics Sept 2009, KI 09 8 / 18

12 Heuristic Computation Recall: A pattern is a set of state variables P i Var. Then, a pattern collection P is a set of patterns. Compute h(s), s S, on basis of all h i (s i ), P i P, P finite pattern collection, i.e. set of patterns. How to calculate those h i (s i ), s i S i? h i (s i ) is the true cost value cost* of the planning problem P i. Calcuation is done by a complete exhaustive search. (Thus, S i and therefore P i have to be small!) (True means: prefer shallow solution graphs.) Solving Non-Deterministic Planning Problems with Pattern Database Heuristics Sept 2009, KI 09 9 / 18

13 Additivity (Theorem) How to calculate h(s), s S? By using additivity! A pattern collection P is called additive, if for all states s S: h i (s i ) cost*(s), i.e. if this sum is still admissible. P i P Known from classical planning: Theorem (textual description) If there is no action a A that affects variables in more than one pattern from P, then P is additive. Solving Non-Deterministic Planning Problems with Pattern Database Heuristics Sept 2009, KI / 18

14 Additivity (Theorem) How to calculate h(s), s S? By using additivity! A pattern collection P is called additive, if for all states s S: h i (s i ) cost*(s), i.e. if this sum is still admissible. P i P Known from classical planning: Theorem (mathematical description) If for all a A and for all patterns P i P holds: If P i effvar(a), then P j effvar(a) = for all P j P with P j P i, where effvar(a) = add,del eff(a) add del. Then P is additive. Solving Non-Deterministic Planning Problems with Pattern Database Heuristics Sept 2009, KI / 18

15 Additivity (Example) P = ({a, b, c, d, e}, A, {a}, {b, c, d, e}) with A = {a 1,..., a 9 } and: a 1 = {a}, { {b}, {a}, {c}, {a} } a 2 = {b}, { {e},, {d}, } a 3 = {c}, { {e},, {d}, } a 4 = {b, d}, { {c}, } a 5 = {c, d}, { {b}, } a 6 = {b, e}, { {c}, } a 7 = {c, e}, { {b}, } a 8 = {b, c, d}, { {e}, } a 9 = {b, c, e}, { {d}, } Solving Non-Deterministic Planning Problems with Pattern Database Heuristics Sept 2009, KI / 18

16 Additivity (Example) P = ({a, b, c, d, e}, A, {a}, {b, c, d, e}) with A = {a 1,..., a 9 } and: a 1 = {a}, { {b}, {a}, {c}, {a} } a 2 = {b}, { {e},, {d}, } a 3 = {c}, { {e},, {d}, } a 4 = {b, d}, { {c}, } a 5 = {c, d}, { {b}, } a 6 = {b, e}, { {c}, } a 7 = {c, e}, { {b}, } a 8 = {b, c, d}, { {e}, } a 9 = {b, c, e}, { {d}, } Now, consider the pattern collection P = {{a, b, c}, {d, e}}. Solving Non-Deterministic Planning Problems with Pattern Database Heuristics Sept 2009, KI / 18

17 Additivity (Example) P = ({a, b, c, d, e}, A, {a}, {b, c, d, e}) with A = {a 1,..., a 9 } and: a 1 = {a}, { {b}, {a}, {c}, {a} } a 2 = {b}, { {e},, {d}, } a 3 = {c}, { {e},, {d}, } a 4 = {b, d}, { {c}, } a 5 = {c, d}, { {b}, } a 6 = {b, e}, { {c}, } a 7 = {c, e}, { {b}, } a 8 = {b, c, d}, { {e}, } a 9 = {b, c, e}, { {d}, } Now, consider the pattern collection P = {{a, b, c}, {d, e}}. Solving Non-Deterministic Planning Problems with Pattern Database Heuristics Sept 2009, KI / 18

18 Additivity (Example) P = ({a, b, c, d, e}, A, {a}, {b, c, d, e}) with A = {a 1,..., a 9 } and: a 1 = {a}, { {b}, {a}, {c}, {a} } a 2 = {b}, { {e},, {d}, } a 3 = {c}, { {e},, {d}, } a 4 = {b, d}, { {c}, } a 5 = {c, d}, { {b}, } a 6 = {b, e}, { {c}, } a 7 = {c, e}, { {b}, } a 8 = {b, c, d}, { {e}, } a 9 = {b, c, e}, { {d}, } Now, consider the pattern collection P = {{a, b, c}, {d, e}}. Only the effect variables matter! Solving Non-Deterministic Planning Problems with Pattern Database Heuristics Sept 2009, KI / 18

19 Additivity (Example, cont d) a b a 1 a 2 a 3 c be bd ce cd a 6 a 4 a 7 a 5 bcd a 8 a 9 bce bcde Solving Non-Deterministic Planning Problems with Pattern Database Heuristics Sept 2009, KI / 18

20 Additivity (Example, cont d) a b a 1 a 2 a 3 c be bd ce cd a 6 a 4 a 7 a 5 bcd a 8 a 9 bce bcde Solving Non-Deterministic Planning Problems with Pattern Database Heuristics Sept 2009, KI / 18

21 Additivity (Example, cont d) a b a 1 c a ε a 2 a 3 be bd ce cd a 6 a 4 a 7 a 5 b a 1 1 c a 2 2 /a2 3 d e bcd a 8 a 9 bce a 1 4 /a1 6 a 1 5 /a1 7 a 2 8 a 2 9 bc de bcde Solving Non-Deterministic Planning Problems with Pattern Database Heuristics Sept 2009, KI / 18

22 Additivity (Example, cont d) a b a 1 c a ε a 2 a 3 be bd ce cd a 6 a 4 a 7 a 5 b a 1 1 c a 2 2 /a2 3 d e bcd a 8 a 9 bce a 1 4 /a1 6 a 1 5 /a1 7 a 2 8 a 2 9 bc de bcde Example: h({a}) = h 1 ({a} 1 ) + h 2 ({a} 2 ) = h 1 ({a}) + h 2 ( ) = = 4 = cost ({a}). Solving Non-Deterministic Planning Problems with Pattern Database Heuristics Sept 2009, KI / 18

23 Heuristic Calculation (cont d) Let M be a set of additive pattern collections. h M (s) := max h i (s i ). P M P i P h M (and in particular, every single h i ) is admissible. How to find M? Current research. (Here: still domain-dependent by hand.) Solving Non-Deterministic Planning Problems with Pattern Database Heuristics Sept 2009, KI / 18

24 Compared Systems Encoded two domains and compared: Our planner with the heuristic of FF. Our planner with the presented pattern database heuristics. GAMER. Important differences between GAMER and our system: Optimal solutions vs. suboptimal solutions. Regression vs. progression. Solving Non-Deterministic Planning Problems with Pattern Database Heuristics Sept 2009, KI / 18

25 Results Pattern database heuristic quality is problem dependent: Domain 1: Pattern database heuristics about 25% more node expansions than FF heuristic. Domain 2: Pattern database heuristics calculate true (perfect) cost value (as opposed to the FF heuristic). Calculation time of pattern database heuristic is much smaller than the FF heuristic s. Thus, more problems could be solved. Progression with heuristic search seems promising approach. (Note: No comparison to sub-optimal planner, yet.) Solving Non-Deterministic Planning Problems with Pattern Database Heuristics Sept 2009, KI / 18

26 Summary Presented fomalization for domain-independent pattern database heuristics in non-deterministic planning. Generalization of additivity criterion. Benchmarks look promising. Solving Non-Deterministic Planning Problems with Pattern Database Heuristics Sept 2009, KI / 18

27 Future Work Automatic pattern selection. Strong plans strong cyclic plans. Search algorithm, LAO*. Pattern database heuristics: Admissibility/Additivity? Multi-valued state variables. Solving Non-Deterministic Planning Problems with Pattern Database Heuristics Sept 2009, KI / 18

28 Thank you! Solving Non-Deterministic Planning Problems with Pattern Database Heuristics Sept 2009, KI / 18