Symmetry in Planning Problems

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Symmetry in Planning Problems

• Derek Long

University of Strathclyde, Glasgow, UK
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Symmetries in Search

• Symmetries are automorphisms on the structures
  that characterise a search problem.
• The mappings must respect constraints and,
  therefore, preserve solutions.
• Symmetries represent redundancy in the structure
  of a problem definition and this implies the
  potential for corresponding redundancy in search.
• But note
• Solution density can be more important than
  absolute size of search space
• Organisation of the solutions within a search
  space can matter more than the size of the space.

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Planning

• Given
• a description of available actions,
• a current state and
• some goal conditions,
• find an organised collection of actions that
• can be executed from the current state
• achieves the goal conditions

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Examples of Planning Problems

• Logistical operations
• Transportation of materials between multiple
  sites using multiple transporters
• Starting up a chemical processing plant
• Restoring power supply on electricity grid
• Efficiently moving planes between gates and
  runways
• Gathering science data using a remote planetary
  lander and communicating it to earth

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Planning as State-Space Search

• States collection of facts describing what is
  true (in that state)
• Actions can be seen as state-transitions
• Goal is a characterisation of a set of states
• Planning becomes a search for a path in the graph
  of the state-space, from an initial state to
  closest goal state

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Planning as State Space Search
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Scale of the Problem

• State spaces for planning can contain gtgt10100
  states (although problems with much smaller state
  spaces can prove at least as hard)
• Explicit state-space search is not an option
• Classical planning is PSPACE hard
• but for useful plans it is NP-hard

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Beyond Classical Planning

• Classical planning restricts us to
• sequential plans
• purely propositional conditions and effects
• optimisation of plan length
• Recent advances consider
• plans as temporal structures, exploiting
  concurrency
• numeric conditions and effects
• using plan metrics other than length (eg make
  span)

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Symmetry of objects
Destination
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Problem stack the crates to reach the top level.
Without symmetry breaking 8! failed plans With
symmetry breaking Just 1 failed plan to consider
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Symmetric Configurations
Destination Q
Destination P
Map ltA,Pgt to ltB,Qgt and vice versa.
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Plan Symmetry
1 Load A into X 2 Drive X from Start to P 3
Unload A from X 4 Load B into Y 5 Drive Y from
Start to Q 6 Unload B from Y
4 Load B into Y
1 Load A into X
2 Drive X from Start to P
5 Drive Y from Start to Q
6 Unload B from Y
3 Unload A from X
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Dynamic Symmetries

• Many planning problems exhibit dynamic
  symmetries objects gain and lose symmetries as a
  plan is developed.
• For example, consider planning the automatic
  repair of some machine
• The screws holding on the back of the casing will
  begin all symmetric (all screwed in place).
• During the unscrewing process the symmetry
  decays some screws will be still in place, some
  already removed and one will be partially
  unscrewed.
• Finally, all will be unscrewed and symmetric
  again.
• Dynamic roles of objects arise because of nature
  of planning problem (reflecting dynamic use of
  objects), but also because of plan development
  process itself.

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Automatic Symmetry Detection

• In planning, several efforts have been made to
  automate symmetry detection
• Joslin and Roy used NAUTY to find automorphisms
  on the graph representing initial and final
  states.
• Fox and Long have used inferred type structure
  and behaviours to find object symmetries.
• Key problems
• Symmetries are hard to find in general, so
  automated methods are restricted.
• Not all symmetries offer a good trade-off between
  the cost of managing exploitation and the savings
  in search.
• Automatic methods are sensitive to encodings.

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Breaking Symmetries

• The exploitation of symmetries in a search
  problem is often performed by adding artificial
  constraints that break the symmetries.
• Important to break symmetries in a way that
  tackles the size of the search problem.
• For example, in n-queens problem, adding a
  constraint that the queen in column 1 cannot be
  in row 1 breaks all but one of the symmetries,
  but doesnt simplify the problem.

• Handling dynamic symmetries through symmetry
  breaking by the addition of static constraints
  can be a poor trade-off.

• The alternative is to modify the search
  algorithm identifying dynamic symmetries during
  search can also be expensive compared with
  potential savings.

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Exploitation of Symmetry in Planning

• Joslin and Roy used symmetry breaking constraints
  in a CSP formulation of the planning problem
• Fox and Long examined functional equivalence,
  both static and dynamic, using specialised
  mechanisms to break symmetry during search
• Rintannen uses symmetry breaking constraints
• Fox and Long have considered plan permutation
  symmetry using specialised mechanisms during
  search

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Exploiting Symmetry in Planning

• Consider sets of functionally equivalent objects.
• Symmetries are induced on actions and
  propositions two actions or propositions are
  symmetric if they are equal up to symmetric
  objects.
• Eg if o1 and o2 are symmetric at layer k then
  propositions Pk(...,o1,...) and Pk(...,o2,...)
  are symmetric.
• If o1 and o2 are symmetric at layer k then
  actions Ak(...,o1,...) and Ak(...,o2,...) are
  symmetric.
• Permutations on variables/values (as
  transpositions) can be constructed from these
  induced symmetric relationships.

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Results SymmetricStan versus Stan version
3(Many new symmetry groups arise during search)
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Results SymmetricStan versus Stan version 3(few
symmetry groups arise during search)
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Symmetry and Abstraction

• Symmetries arise from abstractions.
• The representation of a problem abstracts details
  from the situation it describes.
• Not describing properties that are irrelevant to
  the problem makes components interchangeable.
• For example, in the 8-queens problem
• 90-degree rotational symmetry is supported by the
  abstraction that ignores square colours.
• All symmetries are supported by the abstraction
  that ignores any link to a physical board and
  placement of physical queens onto it.
• Implication for automatic symmetry detection
  formulation might not make the abstractions that
  support identification of a possible symmetry.

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With or without symmetry breaking 8! failed
plans The problem here is a question of
relevance can we determine that colour is
irrelevant?
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• Suppose crates can be painted by stripping the
  old paint, preparing the surface and respraying,
  but now we constrain crates to stack only if they
  have the same colour?
• Colour is not irrelevant.
• However, the crates remain effectively symmetric.
• These actions increase the size of the search
  space without adding useful new branches.

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Almost symmetric objects

• What if they start stacked in some structure?
• The crates are almost symmetric
• An appropriate abstraction (eg an action unstack
  to floor) would restore symmetry
• Otherwise, crates are symmetric up to some
  number of actions (the actions that would
  convert the properties of one into those of
  another)

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Inducing Symmetries from Inside a Problem

• In many planning problems objects begin in
  similar, but not quite identical initial
  positions.
• These differences can often be eliminated by
  adding small plan prefixes that rearrange the
  objects into symmetric configurations.
• For example, vehicles could all be driven to a
  common vehicle pool.
• Resources could all be restored to a common
  reference point.

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Inducing Symmetries from Outside a Problem

• Symmetry can be induced in a problem by
  abstracting the constraints or properties that
  differentiate components.
• Easiest abstractions are of properties that are
  irrelevant to the problem.
• For example the registration numbers of the
  vehicles in a transportation problem are
  irrelevant to their use.
• More subtle irrelevance can arise if the
  vehicles have difference capacities, but the
  values exceed the total of all possible loads,
  then these values are irrelevant.
• Abstraction can also be used to remove problem
  structure that is relevant. Solutions to the
  abstracted problem could then be refined into
  concrete solutions to the original problem.

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Automatic Problem Abstraction

• Attempt to identify relations within problem
  structure that break symmetry and then remove
  them (abstraction)
• In contrast to plan-prefixes, this approach
  removes state structure
• Solving abstracted problem requires abstracted
  actions (reduced preconditions)
• Harder problem mapping solution to abstracted
  problem to real solution
• Possible relaxation strategy?

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Using Symmetry Positively

• Pruning search is good! But there are other uses
• Propose solutions to symmetric parts of the
  problem (within-problem exploitation)
• Propose solutions to symmetric versions of old
  problems (across-problem exploitation)

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Conclusions

• Planning problems often exhibit symmetry.
• Planning problems are naturally dynamic and
  therefore often exhibit dynamic symmetries.
• Planning problems can often begin with
  configurations in which objects are almost
  symmetric adding plan prefixes can allow
  symmetries to be exploited.
• Symmetries arise as a consequence of
  abstractions
• It is possible to apply abstractions to a
  planning problem in order to induce symmetries.
• The application of abstractions to induce
  symmetries is relevant to all search problems
• the automatic analysis of graph structures to
  find limited classes of abstractions that then
  yield automorphisms appears to be general
  strategy.