Pushing the Envelope: Planning, Propositional Logic, and Stochastic Search

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Pushing the Envelope Planning, Propositional
Logic, and Stochastic Search

• By Henry Kautz and Bart Selman of ATT
  Laboratories.
• Presentation by Richard Smiley.

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First, some definitions

• Propositional logic I have a ladder and Im at
  my house.
• Stochastic search A random, but directed search
  (more detail to come).
• Fluent A time-varying condition, such as the
  location of a box.
• SAT Propositional satisfiability problem.
• These problems scale much better than First-Order
  Logic problems.

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Simulated Annealing

• An example of Stochastic Search
• Simulated Annealing exploits an analogy between
  the way a metal cools and freezes into a minimal
  energy crystalline structure (the annealing
  process) and the search for a minimised goal
  state in a search process.
• Unlike hill-climbing approaches (which always
  follow a steady descent in minimization
  problems), SA is much less likely to become
  trapped in local minima.
• The SA algorithm employs a random search that not
  only accepts changes that decrease the energy
  function (as expected this function might
  represent the cost of a trip between two cities,
  say), but also some changes that increase the
  function value, thus allowing SA to jump out of
  local minima.

Copied from http//www.compapp.dcu.ie/tonyv/CA3-A
I/AI-Lectures/ai3.rtf
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A New Style of Encoding

• Include time in the operators
• pickup(A,3)
• each action takes 1 tick on the clock. Action
  will be finished at time 4.
• on(A,B,3)
• whether A is on B at time 3.
• Before execution, the length is determined by a
  binary search of execution possibilities.
• If optimal plan is of length 7, search proceeds
  through plans of length 2, 4, 8 (plan found), 6
  (no plan found), and finally 7.

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A Little More on SAT Encoding

• Compiler takes plan, guesses a plan length, and
  generates a propositional logic formula.
• Simplifier uses various techniques to shrink the
  CNF forumula.
• Solver uses systematic or stochastic methods to
  find an assignment, which the decoder turns into
  a solution plan.

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Representing Arbitrary Constraints

• pickup(x,i) ??y.stack(x,y,i1)
• Every pickup is immediately followed by a stack.
• clear(x,i)?y.on(y,x,i)
• Saying x is clear is equivalent to saying that
  there doesnt exist any y that is on x.

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Encoding Problems The Graphplan Method

• Graphplan works by converting a STRIPS-style
  specification into a planning graph.
• A solution is a subset of the planning graph that
  contains both the start and goal conditions, and
  contains no two operators in the same level that
  conflict.
• Each fact at layer i implies the disjunction of
  all the operators at level i-1 that have it as an
  add-effect.
• in(A,R,3)?(load(A,R,L,2)?load(A,R,P,2)?maintain(in
  (A,R),2))
• Operators imply their preconditions.
• load(A,R,L,2)?(at(A,L,1)?at(R,L,1))
• Conflicting actions are mutually exclusive
• load(A,R,L,2)?move(R,L,P,2)

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A Sample Graph
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Linear Encodings

• An action implies both its preconditions and
  effects
• Exactly one action occurs at each time instant
• The initial state is completely specified
• Classical frame conditions (if an action doesnt
  change the truth condition of a fact, then the
  fact remains true or remains false when the
  action occurs)
• Replace predicates that take 2 or more arguments
  (plus a time argument) with ones that take a
  single argument (plus time).
• move(x,y,z,i)(object(x,i)?source(y,i)?destination
  (z,i)
• Yields O(3n²) propositions rather than O(n4)

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State-Based Encodings

• Has the advantages of the previous two encodings,
  plus some extra refinements
• (in(x,y,i)?in(x,y,i1))??z.load(x,y,z,i)
• (in(a,r,2)?in(a,r,3))?(load(a,r,l,2)?
  load(a,r,p,2))
• So far, this hasnt helped anything. But now,
  the states have some built-in consistency checks.
  We can eliminate propositions and simplify
  axioms to the point that we can completely
  eliminate propositions that refer to actions, so
  that only fluents are used.

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Now, instead of the load axioms, we have

• at(obj,loc,i) ? at(obj,loc,i1)
  ??x?truck?airplane. in(obj,x,i1)? at(x,loc,i)?
  at(x,loc,i1)
• In English If an object is at a location, it
  either remains at that location or goes into some
  truck or plane that is parked at that location.
• This replaces load-airplane and load-truck.
• This method also eliminates the need for
  drive-truck and fly-airplane. State validity
  checks make these happen automatically.

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Benefits of State-Based Encoding

• A solution to a problem is a sequence of states.
• The missing actions are derived from the gaps
  between states. Finding them is easy it only
  involves finding an unordered plan of length 1.
• Can solve state-based encodings of problems that
  no other domain-independent planner can solve.
• No explicit frame axioms, preconditions, or
  conflicts everything is represented in relations
  between fluents.

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The Results

• Both Systematic and Stochastic methods used for
  testing the encodings.
• Systematic Tableau, one of the fastest and most
  complete SAT procedures.
• Stochastic Walksat, a randomized greedy local
  search
• Method Pick a random truth assignment, and flip
  the variable that minimizes the number of clauses
  that are satisfied by the current assignment, but
  which would become unsatisfied if the variable
  were flipped.
• Stochastic because this method is only applied
  half the timethe other half, a random variable
  is flipped to true.

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The Results
To prove optimality, you must perform a
systematic search of all plans of length less
than the found plan, and show that no shorter
plans exist.