Exploiting Symmetry in Planning

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Exploiting Symmetry in Planning
Maria Fox Durham Planning Group University of
Durham, UK
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Introduction

• Why is symmetry important?
• Leads to redundant search.
• Examples
• n-queens and other geometric puzzles
• many planning problems
• many mathematical problems
• Symmetry can be characterised in terms of
  permutations (symmetric groups).
• Managing symmetry in search.
• Future work.

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Brief CSP Review

• A CSP problem consists of
• a collection of variables, Vars
• for each variable, v, a domain of possible
  values, Dv
• a collection of constraints that restrict value
  assignments.
• A CSP solution is built up by selecting an
  assignment for each variable in turn.
• At each stage in the construction, the state of
  the solution will be a partial assignment,
  identifying the values of a subset of the
  variables.

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Why is symmetry important?

• A symmetry can be seen as defining an equivalence
  class of solutions and partial solutions.
• If vi is symmetric to vj then failure of choice
  vi occurs if and only if choice vj leads to
  failure.
• Thus, if v1 and v2 are symmetric, failure of v1
  obviates need to consider v2.

S
Next variable v
v4
v1
Choice of values v1,..,v4 for assignment to
variable v.
v3
v2
S1
S2
S3
S4
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Planning and CSP

• Planning can be seen as a CSP problem (Eg van
  Beek, Kambhampati et al).
• Variables are goal propositions and values are
  possible achievers.
• Dummy values are required to provide for
  propositions that are not activated as
  preconditions of assigned action values.
• Constraints express the delete effects of actions
  and incompatibilities between actions.
• It is not necessary to actually perform this
  mapping in order to exploit symmetry in planning.

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What is a Symmetry?

• In CSP terms "a 11 function on full
  assignments... that is solution preserving"
  (Gent and Smith, 1999).
• A slightly more restricted definition
• A symmetry for a state, S, is a composition of a
  permutation on variables and a permutation on
  values (either of which might be the identity),
  sym, such that
• S sym(S) and
• for all partial assignments, e, that extend S,
  sym(e) satisfies the constraints of the problem
  if and only if e does.
• Note that this definition is with respect to a
  state a symmetry can exist in a state that is
  not a symmetry in the initial state.

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Symmetry Constraint

• If g is a 11 mapping on partial assignments, and
  A is a partial assignment to a collection of
  variables, we can write
• A g(A) var ? val Þ g(var ? val)
• (Gent and Smith 1999)
• to mean that, if A and g(A) are identical, and
  the assignment of val to var fails, then any
  assignment symmetric to var val (under g) will
  also fail.

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An example

• One encoding of the n-queens problems uses n
  variables, q1...qn, each with domain 1...n.
• Each variable qi corresponds to the queen in
  column i, and takes the value of the row for that
  queen.
• The symmetry corresponding to vertical reflection
  is qi q(n1-i) for each variable qi.
• The symmetry for horizontal reflection is i
  (n1-i) for each value i.
• This representation does not allow diagonal
  reflection as a symmetry (in our terms) - columns
  and rows are treated differently, so the symmetry
  is lost.

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Recovering Symmetry

• A modified uniform representation can allow
  recovery of the symmetry
• Variables qRi, qCi, for i 1...n, for the row
  and column of each queen.
• Values r1...rn for the row values and c1...cn for
  the columns.
• Diagonal reflection along leading diagonal is
  now
• qRi qCi, ri ci for each i 1...n
• Horizontal reflection is ri r(n1-i) and
  vertical reflection is ci
  c(n1-i)

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Exploiting Symmetry

• Given a state, S, and a set of symmetries, Syms,
  on S
• Suppose var is variable selected for assignment
  and val is the next value tried for var var
  val.
• Remove from Syms all symmetries, sym, such that
  sym(var) ? var or sym(val) ? val (all symmetries
  broken by this assignment).
• Continue search recursively from new state, S'
  S I var val.
• If search fails, restore Syms and, for each sym,
  remove sym(val) from the domain of sym(var).
• At the symmetries available for exploitation
  can be supplemented with new symmetries on S'.

11
V
V
D
D
D
D
D
H
V
X
X
X
H
X
X
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Previous Exploitation

• Gent and Smith (1999) exploit a version of this
  algorithm in ILOG-solver to exploit hand-coded
  symmetries for the initial state, with no new
  symmetries introduced during execution.
• In Fox and Long (1999) we exploit a version of
  the algorithm in STAN3 using automatically
  induced symmetries for the initial state, but
  using no newly introduced symmetries during
  search.
• Other work exploiting symmetries
• Puget (1993) used symmetry breaking in solution
  of SAT problems.
• Brown, Finkelstein and Purdom (1998)
• Roy and Pachet (1998)

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Symmetry in Planning

• Fox and Long (1999) define two objects, o1 and
  o2, to be symmetric if they have identical
  initial and final state propositions true of
  them. This analysis yields disjoint sets of
  symmetric objects.
• symmetries are induced for values and variables
  from these sets -two action or proposition
  instances are symmetric if they are equal up to
  symmetric objects.
• Eg if o1 and o2 are symmetric then P(o1,x) and
  P(o2,x) are symmetric.
• symmetries remaining in a state are recorded
  using a simple array brokenSyms recording the
  status of each object.
• future symmetric selections are excluded using a
  special matrix recording which symmetric groups
  of values have been tried.
• The approach applies equally to forward planners,
  partial-order planners and graphplan planners.

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Restoring Symmetry

• A critical limitation is that symmetry is rapidly
  lost as objects are selected for special roles,
  and never restored, even though in many planning
  problems objects converge on new symmetry states
  during search.
• Eg In gripper, as balls move from one room to
  another they lose the original symmetry, but gain
  a new symmetry status as they enter a new room.
• Results demonstrate order of magnitude
  improvements but underlying exponential growth if
  new symmetries are not identified.
• Gent and Smith (1999) results for n-queens show
  similar trend but new symmetries unlikely to make
  exponential difference to performance.

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Further Work

• Try to exploit group-theoretic properties of
  symmetries (eg identify which permutations are
  needed to generate all symmetries)
• Extend symmetry management to include restoration
  - could lead to exponential improvement in
  performance in many interesting cases in
  planning.
• Explore conjecture that expansion required to
  express 11 solution-preserving mappings as
  permutations is polynomial.
• Extend notion of symmetry in planning problems
• eg almost symmetry arises when objects have
  effectively indistinguishable initial and final
  states.