Complexity of classical planning l.jpg
This presentation is the property of its rightful owner.
Sponsored Links
1 / 33

Complexity of Classical Planning PowerPoint PPT Presentation


  • 218 Views
  • Updated On :
  • Presentation posted in: General

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

Related searches for Complexity of Classical Planning

Download Presentation

Complexity of Classical Planning

An Image/Link below is provided (as is) to download presentation

Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author.While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.


- - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - -

Presentation Transcript


Complexity of classical planning l.jpg

Complexity of Classical Planning

Megan Smith

Lehigh University

CSE 497, Spring 2007


Outline l.jpg

Outline

  • Review Classical Planning

  • Complexity Background

  • Computational Complexity

  • Main Results


Review classical representation l.jpg

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 l.jpg

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 l.jpg

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 l.jpg

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 l.jpg

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 l.jpg

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 l.jpg

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 l.jpg

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 l.jpg

Complexity Classes

  • Time

    • P

    • NP

    • EXPTIME

    • NEXPTIME

  • Space

    • NLOGSPACE

    • PSPACE

    • EXPSPACE


Bounded by time l.jpg

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 l.jpg

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 l.jpg

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 l.jpg

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 l.jpg

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 l.jpg

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 l.jpg

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 l.jpg

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 l.jpg

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 l.jpg

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 l.jpg

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 l.jpg

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 l.jpg

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 l.jpg

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 l.jpg

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 l.jpg

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 l.jpg

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 l.jpg

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 l.jpg

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 l.jpg

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 l.jpg

References

  • Automated Planning, Theory and Practice, Chapter 3

  • Tom Bylander The Computational Complexity of Propositional STRIPS Planning, in: Artificial Intelligence, 1994


  • Login