Complexity of Classical Planning

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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

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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
• 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)

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

• Weaker representational power than classical representation
• Problem statement can be exponentially larger

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

Review: State-Variable Representation (cont…)

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
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*
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
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
Complexity Classes
• Time
• P
• NP
• EXPTIME
• NEXPTIME
• Space
• NLOGSPACE
• PSPACE
• EXPSPACE
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
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.
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
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
Bounded by Space
• PSPACE - Set of all languages ℒ such that ℒ has a recognition procedure whose worst-case space requirement is polynomially bounded
Bounded by Space
• EXPSPACE - Set of all languages ℒ such that ℒ has a recognition procedure whose worst-case space requirement is exponentially bounded
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

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
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
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
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!

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?

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

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

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

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

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

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

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
Extended Propositional STRIPS Planning
• Augments propositional STRIPS planning with a “domain theory”
• domain theory = set of ground formulas