Complexity of Classical Planning

1. Complexity of Classical Planning Megan Smith Lehigh University CSE 497, Spring 2007

2. Outline • Review Classical Planning • Complexity Background • Computational Complexity • Main Results

3. Review: Classical Representation • Function-free first-order language L • Statement of a classical planning problem: P = (s0, g, O) • s0: initial state - a set of ground atoms of L • g: goal formula - a set of literals • Operator: (name, preconditions, effects) • Classical planning problem: P= (,s0,Sg) Dana Nau: Lecture slides for Automated PlanningLicensed under the Creative Commons Attribution-NonCommercial-ShareAlike License: http://creativecommons.org/licenses/by-nc-sa/2.0/

4. Review: Set-Theoretic Representation • Like classical representation, but restricted to propositional logic • State: a set of propositions - these correspond to ground atoms • {on-c1-pallet, on-c1-r1, on-c1-c2, …, at-r1-l1, at-r1-l2, …} • No operators, just actions take-crane1-loc1-c3-c1-p1 precond: belong-crane1-loc1, attached-p1-loc1, empty-crane1, top-c3-p1, on-c3-c1 delete: empty-crane1, in-c3-p1, top-c3-p1, on-c3-p1 add: holding-crane1-c3, top-c1-p1 • Weaker representational power than classical representation • Problem statement can be exponentially larger Dana Nau: Lecture slides for Automated PlanningLicensed under the Creative Commons Attribution-NonCommercial-ShareAlike License: http://creativecommons.org/licenses/by-nc-sa/2.0/

5. Review: State-Variable Representation • A state variable is like a record structure in a computer program • Instead of on(c1,c2) we might write cpos(c1)=c2 • Equivalent power to classical representation • Each representation requires a similar amount of space • Each can be translated into the other in low-order polynomial time • Classical representation is more popular, mainly for historical reasons • In many cases, state-variable representation is more convenient Dana Nau: Lecture slides for Automated PlanningLicensed under the Creative Commons Attribution-NonCommercial-ShareAlike License: http://creativecommons.org/licenses/by-nc-sa/2.0/

6. Review: State-Variable Representation (cont…) • Load and unload operators: Dana Nau: Lecture slides for Automated PlanningLicensed under the Creative Commons Attribution-NonCommercial-ShareAlike License: http://creativecommons.org/licenses/by-nc-sa/2.0/

7. Decidability • Decidable • An algorithm can be written that will return “yes” if the problem is solvable and “no” if it is not • Semidecidable • An algorithm can be written that will return “yes” if the problem is solvable but will not return if it is not • Undecidable • Neither decidable nor semidecidable

8. Language Recognition Problems • Utilized in complexity analyses • Definitions: • L – finite set of characters • L* - set of all possible strings of characters in L • ℒ - a language, given L and L*, ℒ may be any subset of L*

9. Language Recognition Problems • Recognition procedure for ℒ : • computational procedure p such that for every s∈L*, • p(s) returns yes if s∈ℒ and • p(s) does not return yes if s∉ℒ • may return “no” or not at all

10. Language Recognition Problems (LR) • Worst case running time: Tmax(p,n,ℒ) = max{T(p,s)|s∈ℒand |s| = n} • Worst case running space: Smax(p,n,ℒ) = max{S(p,s)|s∈ℒand |s| = n} • reduction of ℒ1 to ℒ2: • deterministic procedure r: ℒ1*→ℒ2* • such that s ∈ℒ1 iff r(s) ∈ ℒ2

11. Complexity Classes • Time • P • NP • EXPTIME • NEXPTIME • Space • NLOGSPACE • PSPACE • EXPSPACE

12. Bounded by Time • P – Set of all languages ℒ such that ℒ has a deterministic recognition procedure whose worst-case running time is polynomially bounded

13. Bounded by Time • NP – Set of all languages ℒ such that ℒ has a nondeterministic recognition procedure whose worst-case running time is polynomially bounded • Examples: Boolean-SAT, knapsack problem, clique problem, vertex cover problem, etc.

14. Bounded by Time • EXPTIME – Set of all languages ℒ such that ℒ has a deterministic recognition procedure whose worst-case running time is exponentially bounded

15. Bounded by Space • NLOGSPACE - Set of all languages ℒ such that ℒ has a nondeterministic recognition procedure whose worst-case space requirement is logarithmically bounded

16. Bounded by Space • PSPACE - Set of all languages ℒ such that ℒ has a recognition procedure whose worst-case space requirement is polynomially bounded

17. Bounded by Space • EXPSPACE - Set of all languages ℒ such that ℒ has a recognition procedure whose worst-case space requirement is exponentially bounded

18. Complexity Classes • Given a complexity class C and a language ℒ • ℒ is C-hard if every language in C is reducible to ℒ in polynomial time • ℒ is C-complete if ℒ is C-hard and ℒ ∈C • Some sets known to be unequal • Unknown whether all are unequal • P = NP anyone? NLOGSPACE ⊆ P ⊆NP ⊆PSPACE ⊆EXPTIME ⊆NEXPTIME ⊆EXPSPACE

19. Planning Problems as LR Problems • PLAN-EXISTENCE – set off all strings s⊆L* such that s is the statement of a solvable problem • Also known as PLANSAT

20. Planning Problems as LR Problems • PLAN-LENGTH – set of all strings of the form (s,k) such that s is the statement of a solvable planning problem, k is a nonnegative integer, and s has a solution plan that contains no more than k actions

21. Decidability Results • For classical, set-theoretic, & state-variable planning, PLAN-EXISTENCE is decidable • If we extend classical or state-variable planning to allow terms to contain function symbols, PLAN-LENGTH is still decidable • If we extend classical or state-variable planning to allow terms to contain function symbols, PLAN-EXISTENCE is semidecidable

22. Proof of Semidecidability • Let P = (O,s0, g) • O has no negative preconditions, no negative effects, and at most one positive effect • Set of Horn clauses Hp: • ∀a ∈ s0; Hp includes {true ⇒ a} • Hp includes {true ⇒g} • ∀o∈O let precond(o) = {p1,…,pn} and effects(o)={e}; Hp includes {e ⇒ p1,…,pn} • P has solution plan iff Hp is consistent SEMIDECIDABLE!

23. EXPSPACE Turing Machine • Turing Machine M = (K,∑,Γ,δ,q0,F) • K = {q0,…, qm} is a finite set of states • F⊆K is set of final states • Γ is finite set of allowable symbols • ∑ ⊆ Γ is the set of allowable input symbols • q0 ∈ K is the start state • δis a mapping from K× Γ to K× Γ× {Left,Right} • Given a Turing Machine M that uses at most an exponential number of tape cells in terms of the length of its input and an input string x, does M accept the string x? … …

25. Complexity Results • No restrictions, the size of any state is at most exponential: • PLAN-EXISTENCE is in EXPSPACE • No negative effects: • PLAN-EXISTENCE is reduced to NEXPTIME • No negative effects & no negative preconditions (ordering now doesn’t matter) • PLAN-EXISTENCE is now reduced to EXPTIME NLOGSPACE ⊆ P ⊆NP ⊆PSPACE ⊆EXPTIME ⊆NEXPTIME ⊆EXPSPACE

26. Complexity Results (cont…) • Even with both these restrictions, PLAN-LENGTH remains NEXPTIME • If each operator has at most one precondition: • Both PLAN-EXISTENCE & PLAN-LENGTH are now in PSPACE • Restricting all atoms to be ground lowers complexity by one level NLOGSPACE ⊆ P ⊆NP ⊆PSPACE ⊆EXPTIME ⊆NEXPTIME ⊆EXPSPACE

27. Complexity Results (cont…) • If the operator set is fixed in advance: • classical planning is at most in PSPACE NLOGSPACE ⊆ P ⊆NP ⊆PSPACE ⊆EXPTIME ⊆NEXPTIME ⊆EXPSPACE

28. Interesting Properties • Extending planning operators to allow conditional effects doesn’t affect these complexity results • In most cases, the complexity of PLAN-EXISTENCE in classical planning is one level harder than in set-theoretic planning NLOGSPACE ⊆ P ⊆NP ⊆PSPACE ⊆EXPTIME ⊆NEXPTIME ⊆EXPSPACE

29. Interesting Properties (cont…) • If negative effects are allowed, PLAN-EXISTENCE is EXPSPACE-complete but PLAN-LENGTH is only NEXPTIME-complete • PLAN-LENGTH has same complexity regardless of allowing/disallowing negated preconditions • Negative effects are more powerful than negative preconditions NLOGSPACE ⊆ P ⊆NP ⊆PSPACE ⊆EXPTIME ⊆NEXPTIME ⊆EXPSPACE

30. STRIPS Review • π  the empty plan • do a modified backward search from g • instead of -1(s,a), each new set of subgoals is just precond(a) • whenever you find an action that’s executable in the current state, then go forward on the current search path as far as possible, executing actions and appending them to π • repeat until all goals are satisfied π = a6, a4 s = ((s0,a6),a4) g6 satisfied in s0 g1 a1 a4 a6 g4 g2 g a2 g3 a5 g5 a3 a3 g3 current search path Dana Nau: Lecture slides for Automated PlanningLicensed under the Creative Commons Attribution-NonCommercial-ShareAlike License: http://creativecommons.org/licenses/by-nc-sa/2.0/

31. Propositional STRIPS Planning • Initial state is finite set of ground atomic formuals • Preconditions & postconditions of an operator are ground literals • Goals are ground literals • Operators don’t have any variables or indirect side effects

32. Extended Propositional STRIPS Planning • Augments propositional STRIPS planning with a “domain theory” • domain theory = set of ground formulas • infers additional effects • requires preference ordering of all literals to avoid ambiguity

33. References • Automated Planning, Theory and Practice, Chapter 3 • Tom Bylander The Computational Complexity of Propositional STRIPS Planning, in: Artificial Intelligence, 1994