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Complexity of Classical Planning. Megan Smith Lehigh University CSE 497, Spring 2007. Outline. Review Classical Planning Complexity Background Computational Complexity Main Results. Review: Classical Representation. Function-free first-order language L

Complexity of Classical Planning

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Complexity of Classical Planning

Megan Smith

Lehigh University

CSE 497, Spring 2007

- Review Classical Planning
- Complexity Background
- Computational Complexity
- Main Results

- 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/

- 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/

- 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/

- Load and unload operators:

- 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

- 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*

- 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

- computational procedure p such that for every s∈L*,

- 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

- Time
- P
- NP
- EXPTIME
- NEXPTIME

- Space
- NLOGSPACE
- PSPACE
- EXPSPACE

- P – Set of all languages ℒ such that ℒ has a deterministic recognition procedure whose worst-case running time is polynomially bounded

- 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.

- EXPTIME – Set of all languages ℒ such that ℒ has a deterministic recognition procedure whose worst-case running time is exponentially bounded

- NLOGSPACE - Set of all languages ℒ such that ℒ has a nondeterministic recognition procedure whose worst-case space requirement is logarithmically bounded

- PSPACE - Set of all languages ℒ such that ℒ has a recognition procedure whose worst-case space requirement is polynomially bounded

- EXPSPACE - Set of all languages ℒ such that ℒ has a recognition procedure whose worst-case space requirement is exponentially bounded

- 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

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

- 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

- 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

- 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!

- 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?

…

…

- 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

- 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

- If the operator set is fixed in advance:
- classical planning is at most in PSPACE

NLOGSPACE ⊆ P ⊆NP ⊆PSPACE ⊆EXPTIME ⊆NEXPTIME ⊆EXPSPACE

- 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

- 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

- 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

- 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

- 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

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