Extending Graphplan to handle Resources

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• Extending Graphplan to handle Resources
• Presenter Pham Van Cuong
• Department of Computer Science
• New Mexico State University

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Motivation

• Planning with Resources is ubiquitous in real
  life.
• Actions in the real world often need resources to
  execute.

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Motivation

• Our approach to planning with Resources is based
  on Graphplan, a well-known planning algorithm.
• Techniques that make Graphplan attractive
• Polynomial time construction of planning Graph.
• Use of mutexes to enhance the search for a plan.

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Outline

• Graphplan background
• STRIPS language
• Planning Graph Mutexes.
• Planning with Resources.
• GPR- A Graphplan with Resources
• Input language (PDDL 2.1 level 2)
• Data structures
• Mutexes
• Algorithm
• Experimental Results.

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STRIPS language

• A STRIPS action a is specified by an expression
  of the form
• action a Pre Pre(a)
• Add Add(a)
• Del Del(a)
• For example,
• Action Drive(Car,LC,EP)
• Pre At(Car,LC),Has-fuel(Car)
  Add At(Car,EP)
• Del At(Car, LC),Has-fuel(Car)

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STRIPS language

• The result of executing an action a in a state s
  is
• Res(a,s) (s Add(a)) \ Del(a ) if a is
  executable, Res(a,s) if otherwise.
• The result of executing a sequence of actions
  a1, a2 , an in a state s is
• Res( ,s)s
• Res(a1, a2 , an,s) Res(an,Res(a1, a2 ,
  an-1,s)), where Res(a, ) for
  every a.

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STRIPS language

• A planning problem is a tuple ltP,A,I,Ggt, where P
  is a finite set of fluents, A is a finite set of
  actions, I (the initial state) is a set of
  fluents, and G (the goal) is a finite set of
  fluent literals.
• Given a planning problem QltP,A,I,Ggt, a
  sequence of actions a1, a2 , an is a solution
  (plan) to Q if Res(a1, a2 , an,I) is defined
  and G holds in Res(a1, a2 , an,I).

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Graphplan Planning Graph

• a directed, leveled graph with a set of nodes and
  a set of edges.
• The levels alternate between proposition levels
  and action levels.
• The proposition levels contain proposition
  nodes, each of which is labeled with a fluent
  literal.
• The action levels contain action nodes, each has
  an action as its label.

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Graphplan Planning Graph

• An edge presents the relation between an action
  and a proposition.
• At time t, action nodes are connected to
• their preconditions in the proposition level t
  by precondition edges.
• their addeffects and del-effects in
  proposition level t1 by add-edges and del-edges,
  respectively.

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Graphplan mutex

• Two actions A and B are mutex each other if
• action A deletes a precondition or an add-effects
  of B or vice versa.
• a precondition of action A and a precondition of
  action B are mutex in the previous proposition
  level.
• Two propositions p and q are mutex if
• all ways of creating p are mutex with all
  ways of creating q.

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Current approaches to Planning with Resources-
some characteristics

• State based search (Metric FF, LGP).
• Using heuristic function to guide search (Sapa,
  TP4 )
• Forward chaining approach (TLPlan )
• No existing planner uses mutexes of planning
  Graph to guide the search.

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GPR- A Graphplan with Resources

• Based on Graphplan algorithm.
• Use of mutexes to direct the search for a plan.
• generate a concurrent plan.

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GPR- Input language (syntax)

• FFB U FN where FB is the set of boolean
  fluents and FN is the set of numeric fluents.
• An assignment is of the form fv where
• f F and v Df
• A set d of assignments is
• consistent if for every fluent f F there
  exists at most one assignment of the form fv in
  d.
• complete if for every fluent f F there
  exists at least one assignment of the form fv in
  d.

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GPR- Input language (syntax)

• A numeric constraint is a triple (exp1, comp,
  exp2) where comp gt,,lt,, is a comparator.
  
• A numeric effect is a tuple of the form (f, aop,
  exp) where f FN , aop assign,increase,
  decrease,scale-up,scale-down is an assignment
  operator.
• A condition con is a pair (b(con),n(con)) where
  b(con) FB and n(con) is a set of numeric
  constraints.

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GPR- Input language (syntax)

• An action a is a pair (Pre(a),Eff(a)), where
• pre(a) is a condition
• eff(a) is a triple (b-add(eff(a)),
  b-del(eff(a)) ,n(eff(a)))
• b-add(eff(a)), b-del(eff(a)) FB
  n(eff(a)) is a set of numeric effects which
  does not contain two numeric effects (f,aop,exp)
  and (f,aop,exp).
• For example,
• action FLY (plane,EP,LAX)
• Pre ( At(plane,EP), (gt (fuel plane)
  300))
• Eff (At(plane,LAX) , At(plane,EP) ,
  (decrease (fuel plane) 200) ).

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GPR- Input language (semantics)

• A state is a consistent and complete set of
  assignments.
• An assignment fv holds in a state s, denoted by
  s (fv), if fv s. A set of assignments d
  holds in s, denoted by s d, if for all fv
  d s (fv).
• A numeric constraint (exp1, comp, exp2) holds in
  a state s, denoted by s (exp1, comp, exp2), if
  both exp1 and exp2 are defined in s and exp1
  comp exp2 holds.

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GPR- Input language (semantics)

• A set of numeric constraints C holds in a state s
  if s c for every numeric constraint c C.
• A condition con(b(con),n(con)) holds in a state
  s (s con) if s b(con) and s n(con).
• An action a is executable in a state s if
  s Pre(a) .

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GPR- Input language (semantics)

• A state transition Res(a,s), if a is an
  executable action in s, contains the following
  assignments
• ftrue if f b-add(eff(a))
• ffalse if f b-del(eff(a))
• fs(exp) if (f,assign,exp)
  (eff(a))
• fs(f) s(exp) if (f,increase,exp)
  (eff(a))
• fs(f)- s(exp) if (f,decrease,exp)
  (eff(a))
• fs(f) s(exp) if (f,scale-up,exp)
  (eff(a))
• fs(f)/s(exp) if (f,scale-down,exp)
  (eff(a))
• fs(f) if there does not exist the fluent f in
  the left hand side in every assignment fv
  Res(a,s) .

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GPR- Input language (semantics)

• if a is not executable in s, then Res(a,s) (or
  undefined).
• For a sequence a1, a2,.., an of actions,
  Res(a1, a2,.., an,s) Res(an, Res(a1, a2,..,
  an-1,s) and Res( ,s)s, where Res(a, )
  for every a.

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GPR- Input language (semantics)

• A planning problem is a tuple (F,A,I,G), where
  F FB U FN , A is a finite set of actions, I
  (the initial state) is a set of assignments, and
  G (the goal) is a condition.
• A solution (plan) to a numeric planning problem
  is a sequence a1, a2 , an of actions if
  Res(a1, a2 , an,I) G and Res(a1, a2 ,
  an,I) is defined .
• The semantics can be extended to allow parallel
  actions.

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GPR - Planning Graph

• A directed, leveled graph with a set of nodes and
  a set of edges.
• The levels alternate between fluent levels and
  action levels.
• The action levels contain action nodes, each is
  labeled with an executable action in that level.
• The fluent levels contain fluent nodes, each of
  which is labeled with an assignment .

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GPR - Planning Graph

• An edge presents the relation between an action
  and an assignment.
• At time t, each action node a is connected
• to assignments fv which make Pre(a) hold in
  the fluent level t, denoted by Pre(a,t) (fv),
  by incoming edges.
• to assignments created by a s effect in the
  fluent level t1 by outgoing edges

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GPR mutex between actions A and B at level t

• Inconsistent effects
• An add-effect of A is negated by B or vice
  versa.
• Interference
• One of the del-effects of A is a precondition of
  B or vice versa.
• There exist two mutexed assignments f1v1 and
  f2v2 in level t and Pre(A,t)(f1v1) and
  Pre(B,t)(f2v2).

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GPR mutex between assignments

• Two assignments f1v1 and f2v2 are mutex at time
  t if
• f1 f2 and v1 ? v2 or,
• all ways of creating f1v1 are mutex with all
  ways of creating f2v2.

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GPR Algorithm description

• GPR algorithm alternates between two phases
  constructing the planning Graph and extracting a
  solution.
• The planning Graph is constructed until the
  planning Graph is leveled off or a valid plan is
  found.
• Extracting a solution phase starts whenever the
  goal is reached.

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GPR Constructing planning Graph

• The fluent level 1 consists of all assignments
  in the initial state.
• Once an executable action A is found at time t,
  GPR will do the followings.
• creates an action node in the level t and labels
  it with A.
• for each assignment fv in the fluent level t
  s.t. pre(A,t) (fv), adds an edge connecting
  it to A.
• for each assignment fv created by some effect of
  A, GPR creates a fluent node with the label fv,
  adds it to the fluent level t1, and then
  inserts an edge from A to this node.
• finds all action nodes in level t which are mutex
  with A and updates the mutex list.

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GPR- Extracting a solution

• Given a goal Gt at time t, GPR non-deterministical
  ly selects a set of actions At-1 and computes the
  goal Gt-1 as follows.
• At-1 is a set of actions in level t-1 s.t. for
  each g Gt there exists one edge from some a
  At-1 to g.
• Gt-1 is the set of assignments in level t-1 s.t.
  every action a At-1 is executable in Gt-1
• If t0 and Gt I, this indicates that a
  solution is found.

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GPR- Experimental Results

• GPR generates a concurrent plan for the Rocket
  domain in 3 time steps with.
• For the Rocket Domain with renewable Resources,
  GPR also generates a plan in good quality.
• GPR is available on www.cs.nmsu.edu/cvan

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Future works

• GPR is the first step towards creating a planner
  for domains with resources. It can be improved
  by
• Finding a plan that satisfies some constraints
  (eg. minimal resource consumption).
• Considering actions with duration.

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• Thank you !