Extending Graphplan to handle Resources Presenter: Pham Van Cuong Department of Computer Science

1. Extending Graphplan to handle Resources Presenter: Pham Van Cuong Department of Computer Science New Mexico State University

2. Motivation • Planning with Resources is ubiquitous in real life. • Actions in the real world often need resources to execute.

3. Motivation • Our approach to planning with Resources is based on Graphplan, a well-known planning algorithm. • Techniques that make Graphplan attractive: • Polynomial time construction of planning Graph. • Use of mutexes to enhance the search for a plan.

4. Outline Graphplan background STRIPS language Planning Graph & Mutexes. Planning with Resources. GPR- A Graphplan with Resources Input language (PDDL 2.1 level 2) Data structures Mutexes Algorithm Experimental Results.

5. STRIPS language • A STRIPS action a is specified by an expression of the form action a :Pre Pre(a) :Add Add(a) :Del Del(a) For example, Action Drive(Car,LC,EP) :Pre {At(Car,LC),Has-fuel(Car)} :Add {At(Car,EP)} :Del {At(Car, LC),Has-fuel(Car)}

6. STRIPS language • The result of executing an action a in a state s is Res(a,s) = (s Add(a)) \ Del(a ) if a is executable, Res(a,s) = if otherwise. • The result of executing a sequence of actions [a1, a2 …, an] in a state s is • Res([ ],s)=s • Res([a1, a2 …, an],s) = Res(an,Res([a1, a2 …, an-1],s)), where Res(a, ) = for every a.

7. STRIPS language • A planning problem is a tuple <P,A,I,G>, where P is a finite set of fluents, A is a finite set of actions, I (the initial state) is a set of fluents, and G (the goal) is a finite set of fluent literals. • Given a planning problem Q=<P,A,I,G>, a sequence of actions [a1, a2 …, an] is a solution (plan) to Q if Res([a1, a2 …, an],I) is defined and G holds in Res([a1, a2 …, an],I).

8. Graphplan – Planning Graph • a directed, leveled graph with a set of nodes and a set of edges. • The levels alternate between proposition levels and action levels. • The proposition levels contain proposition nodes, each of which is labeled with a fluent literal. • The action levels contain action nodes, each has an action as its label.

9. Graphplan – Planning Graph • An edge presents the relation between an action and a proposition. • At time t, action nodes are connected to: • their preconditions in the proposition level t by precondition edges. • their add–effects and del-effects in proposition level t+1 by add-edges and del-edges, respectively.

10. Graphplan – mutex • Two actions A and B are mutex each other if: • action A deletes a precondition or an add-effects of B or vice versa. • a precondition of action A and a precondition of action B are mutex in the previous proposition level. • Two propositions p and q are mutex if: all ways of creating p are mutex with all ways of creating q.

11. Current approaches to Planning with Resources- some characteristics • State based search (Metric FF, LGP…). • Using heuristic function to guide search (Sapa, TP4 …) • Forward chaining approach (TLPlan …) • No existing planner uses mutexes of planning Graph to guide the search.

12. GPR- A Graphplan with Resources • Based on Graphplan algorithm. • Use of mutexes to direct the search for a plan. • generate a concurrent plan.

13. GPR- Input language (syntax) • F=FBU FN where FB is the set of boolean fluents and FN is the set of numeric fluents. • An assignment is of the form f=v where f F and v Df A set δ of assignments is: • consistent if for every fluent f F there exists at most one assignment of the form f=v in δ. • complete if for every fluent f F there exists at least one assignment of the form f=v in δ.

14. GPR- Input language (syntax) • A numeric constraint is a triple (exp1, comp, exp2) where comp {>,=,<,≥, ≤} is a comparator. • A numeric effect is a tuple of the form (f, aop, exp) where f FN , aop {assign,increase, decrease,scale-up,scale-down} is an assignment operator. • A condition con is a pair (b(con),n(con)) where b(con) FB and n(con) is a set of numeric constraints.

15. GPR- Input language (syntax) An action a is a pair (Pre(a),Eff(a)), where • pre(a) is a condition • eff(a) is a triple (b-add(eff(a)), b-del(eff(a)) ,n(eff(a))); b-add(eff(a)), b-del(eff(a)) FB; n(eff(a)) is a set of numeric effects which does not contain two numeric effects (f,aop,exp) and (f,aop’,exp’). For example, action FLY (plane,EP,LAX) Pre: ( {At(plane,EP)}, {(> (fuel plane) 300)}) Eff: ({At(plane,LAX)} , {At(plane,EP) }, {(decrease (fuel plane) 200)} ).

16. GPR- Input language (semantics) • Astate is a consistent and complete set of assignments. • An assignment f=v holds in a state s, denoted by s ╞(f=v), if f=v s. A set of assignments δholds in s, denoted by s ╞δ, if for all f=v δ s ╞ (f=v). • A numeric constraint (exp1, comp, exp2) holds in a state s, denoted by s ╞(exp1, comp, exp2), if both exp1 and exp2 are defined in s and exp1 comp exp2 holds.

17. GPR- Input language (semantics) • A set of numeric constraints C holds in a state s if s ╞ c for every numeric constraint c C. • A condition con=(b(con),n(con)) holds in a state s (s ╞ con) if s ╞ b(con) and s ╞ n(con). • An action a is executable in a state s if s ╞ Pre(a) .

18. GPR- Input language (semantics) • A state transition Res(a,s), if a is an executable action in s, contains the following assignments: • f=true if f b-add(eff(a)) • f=false if f b-del(eff(a)) • f=s(exp) if (f,assign,exp) (eff(a)) • f=s(f)+ s(exp) if (f,increase,exp) (eff(a)) • f=s(f)- s(exp) if (f,decrease,exp) (eff(a)) • f=s(f)* s(exp) if (f,scale-up,exp) (eff(a)) • f=s(f)/s(exp) if (f,scale-down,exp) (eff(a)) • f=s(f) if there does not exist the fluent f in the left hand side in every assignment f=v Res(a,s) .

19. GPR- Input language (semantics) • if a is not executable in s, then Res(a,s)= (or undefined). • For a sequence [a1, a2,.., an] of actions, Res([a1, a2,.., an],s) = Res(an, Res([a1, a2,.., an-1],s) and Res([ ],s)=s, where Res(a, )= for every a.

20. GPR- Input language (semantics) • A planning problem is a tuple (F,A,I,G), where F = FBU FN , A is a finite set of actions, I (the initial state) is a set of assignments, and G (the goal) is a condition. • A solution (plan) to a numeric planning problem is a sequence [a1, a2 …, an] of actions if Res([a1, a2 …, an],I) ╞G and Res([a1, a2 …, an],I) is defined . • The semantics can be extended to allow parallel actions.

21. GPR - Planning Graph • A directed, leveled graphwith a set of nodes and a set of edges. • The levels alternate between fluent levels and action levels. • The action levels contain action nodes, each is labeled with an executable action in that level. • The fluent levels contain fluent nodes, each of which is labeled with an assignment .

22. GPR - Planning Graph An edge presents the relation between an action and an assignment. At time t, each action node a is connected: • to assignments f=v which make Pre(a) hold in the fluent level t, denoted by Pre(a,t) ╞(f=v), by incoming edges. • to assignments created by a ’s effect in the fluent level t+1 by outgoing edges

23. GPR– mutex between actions A and B at level t • Inconsistent effects: • An add-effect of A is negated by B or vice versa. • Interference : • One of the del-effects of A is a precondition of B or vice versa. • There exist two mutexed assignments f1=v1 and f2=v2 in level t and Pre(A,t)╞(f1=v1) and Pre(B,t)╞(f2=v2).

24. GPR – mutex between assignments • Two assignments f1=v1 and f2=v2 are mutex at time t if: • f1 = f2 and v1≠ v2 ; or, • all ways of creating f1=v1 are mutex with all ways of creating f2=v2.

25. GPR – Algorithm description • GPR algorithm alternates between two phases: constructing the planning Graph and extracting a solution. • The planning Graph is constructed until the planning Graph is leveled off or a valid plan is found. • Extracting a solution phase starts whenever the goal is reached.

26. GPR – Constructing planning Graph • The fluent level 1 consists of all assignments in the initial state. • Once an executable action A is found at time t, GPR will do the followings. • creates an action node in the level t and labels it with A. • for each assignment f=v in the fluent level t s.t. pre(A,t) ╞(f=v), adds an edge connecting it to A. • for each assignment f=v created by some effect of A, GPR creates a fluent node with the label f=v, adds it to the fluent level t+1, and then inserts an edge from A to this node. • finds all action nodes in level t which are mutex with A and updates the mutex list.

27. GPR- Extracting a solution • Given a goal Gt at time t, GPR non-deterministically selects a set of actions At-1 and computes the goal Gt-1 as follows. • At-1 is a set of actions in level t-1 s.t. for each g Gt there exists one edge from some a At-1 to g. • Gt-1 is the set of assignments in level t-1 s.t. every action a At-1 is executable in Gt-1 • If t=0 and Gt I, this indicates that a solution is found.

28. GPR- Experimental Results • GPR generates a concurrent plan for the Rocket domain in 3 time steps with. • For the Rocket Domain with renewable Resources, GPR also generates a plan in good quality. • GPR is available on www.cs.nmsu.edu/~cvan

29. Future works • GPR is the first step towards creating a planner for domains with resources. It can be improved by: • Finding a plan that satisfies some constraints (eg. minimal resource consumption…). • Considering actions with duration.

30. Thank you !