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# Complexity of Classical Planning - PowerPoint PPT Presentation

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

• 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)

• 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

• 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

• 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

• 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