Planning Methods based on Logic

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Planning Methods based on Logic
The Situation Calculus Reasoning about States and
Actions Some Difficulties Generating Plans
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Reasoning about States and Actions

• Describe goal condition using any wff in
  predicate calculus
• Describe world state using formulas
• Try to prove goal wff from formulas describing
  world state
• Predicate Calculus can be used to reason about
  states and actions
• Search is over a space of logical expressions
  not a space of models of world states

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Predicate Calculus formalization of states,
actions and the effects of actions on
states Knowledge about states and actions are
represented using formulas in FOPC Deduction
system ask questions e.g.Does there exist a
state such that it satisfies certain (goal)
properties, and if so, how can the present state
be transformed into that state by actions
? Answer Plan for achieving desired state
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State S0

• Reify States include them into the world as
  entities that exist (S0,S1,S2)
• Change atomic wffs to include a term denoting a
  state in which the intended relation holds
• Interpretation of wffs relations over states
• wffs are called fluents . True for S0
• true of all states and

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• Step 1 Reify the actions. Denote actions by
  constants/ variable symbols/functional
  expressions. Action is regarded as a function
  involving entities. Example move(B,A,Floor)
  action of moving B from A to
  the Floor.
• Step 2 Function constant do function that
  maps actions and states into states.
  maps the state-action pair into the
  state obtained by performing the action
  in the state
• Step 3 Express effects of actions by wffs
  called effect axioms

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• Two wffs for each action-fluent pair For the
  pair (On,move)
• Positive effect axiom describes how an
  action makes a fluent true
• Negative effect axiom describes how an action
  makes a fluent false

preconditions
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• Two wffs for each action-fluent pair For the
  pair (Clear,move)
• Positive effect axiom
• Negative effect axiom

No action can make the floor not clear
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On(B,A,S0)On(A,C,S0)On(C,Floor,S0)Clear(B,S0)C
lear(Floor,S0)
S0
move(B,A, Floor)
S1 do(move(B,A,Floor),S0)
Inferred using effect axioms substitution
B/x, A/y,S0/s, Floor/z
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Frame Axioms

• Two wffs for each action-fluent pair For the
  pair (move, On) describes fluent left unchanged
  by an action
• Positive frame axiom
• Negative frame axiom

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More Frame Axioms

• Two wffs for each action-fluent pair For the
  pair (move, Clear) describes fluent left
  unchanged by an action
• Positive frame axiom
• Negative frame axiom

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Mapping a State-Action Pair into a State - Frame
Axioms
On(B,A,S0)On(A,C,S0)On(C,Floor,S0)Clear(B,S0)C
lear(Floor,S0)
S0
move(B,A, Floor)
S1 do(move(B,A,Floor),S0)
Inferred using frame axioms
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Difficulties
The Frame Problem

• Two frame axioms for each action-fluent pair
• Becomes unmanageable in realistic applications
  to express things that remain constant in the
  world
• Several techniques to reduce the number of frame
  axioms but to reason about fluents that do
  not change over sequence of actions remains
  computationally cumbersome

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The Qualification Problem

• Adding preconditions how do we know when we
  have added all the pre-conditions/qualification
  s ?
• Attempts to solve this problem was made using
  nonmonotonic reasoning allows default
  conclusions in the effect axioms withdrawn
  if further qualifications is added

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The Ramification Problem

• In complex domains statements are often
  deduced based on domain knowledge e.g. a
  packet that a robot is carrying is in a room
  if the robot itself is in the room
• Can use different mechanisms to reason e.g.
  first about the location of the robot, then
  about the location of the package
• How do we prevent the frame axioms from deducing
  that the package is in a different room than
  the robot (due to perceived consistencies ?
• This problem is therefore how to keep track of
  which derived formulas survive subsequent
  state transformations

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Generating Plans

• To generate a plan that achieves some goal
  , attempt to prove and extract
  the state as a function of the nested actions
  that produce it
• Proof effort becomes much too large for such a
  simple plan
• Frame problem made most attempts to use
  situation calculus impractical