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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

Complexity of Classical Planning

Megan Smith

Lehigh University

CSE 497, Spring 2007

outline
Outline
  • Review Classical Planning
  • Complexity Background
  • Computational Complexity
  • Main Results
review classical representation
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/

review set theoretic representation
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/

review state variable representation
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/

review state variable representation cont
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/

decidability
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
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 problems9
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
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
Complexity Classes
  • Time
    • P
    • NP
    • EXPTIME
    • NEXPTIME
  • Space
    • NLOGSPACE
    • PSPACE
    • EXPSPACE
bounded by time
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 time13
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 time14
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
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 space16
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 space17
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 classes18
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
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 problems20
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
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
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
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
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
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 cont27
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
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
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
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/

propositional strips planning
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
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
references
References
  • Automated Planning, Theory and Practice, Chapter 3
  • Tom Bylander The Computational Complexity of Propositional STRIPS Planning, in: Artificial Intelligence, 1994
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