Introduction to Planning

1
Introduction to Planning

• Han Yu
• University of Central Florida

2
Outline

• What is planning?
• Formal definitions of planning problems
• Basic planning algorithms
• Recent development

3
What is Planning?

• Planning is a search problem that requires to
  find an efficient sequence of actions that
  transform a system from a given starting state to
  the goal state

4
Whats Given?

• Initial state of the problem
• Goal state of the problem
• A finite set of actions
• pre-conditions a finite set of conditions for
  the action to be performed
• post-conditions a finite set of conditions that
  will be changed after the action is performed
• cost

5
Whats Output?

• A sequence of actions that meet the following
  criteria
• every action matches the current system state
• can transform system from initial state to goal
  state
• the total cost of the actions is below a
  specified value

6
Planning as a Real-World Problem

• Planning problem has a wide range of applications
  in the real world
• planning in daily life
• game world
• workflow management

7
Planning a Trip
Begin
Preparation
Airline Reservation
Hotel Reservation
Rental Car Reservation
Confirm Reservations
End
8
Towers of Hanoi
9
Problem Analysis

• Optimal solution contains 2n-1 actions, where n
  is the number of disks
• Up to 6 possible actions in each system state
• The upper bound on search space to find an
  optimal solution will be 6(2n-1)

10
Sliding-Tile Puzzle
11
Planning in Workflow Management
Install Video Card
Start Assembly
Insert Modem
Plug in CD
Examine Order
Install Network Card
Plug in Battery
Gather Components
Install Internal Disk
Test
Assembly Box and Motherboard
Install Motherboard
End Assembly
12
Partial-Order Plan versus Total-Order Plan

• Partial-order plan
• consists partially ordered set of actions
• sequence constraints exist on these actions
• plan generation algorithm can be applied to
  transform partial-order plan to total-order plan
• Total-order plan
• consists totally ordered set of actions

13
Partial-Order Plan
Get brush
Paint ceiling
Finish
Start
Get ladder
14
Total-Order Plan
Start
Get brush
Get ladder
Paint ceiling
Finish
Start
Get brush
Get ladder
Paint ceiling
Finish
15
Features of Planning Problems

• Large search space
• Action is associated with system states
• Restrictions on the action sequence
• Valid solution may not exist
• Optimization requirement

16
STRIPS Planning System

• A tuple T (P, O, I, G), where
• P is a finite set of ground literals, the
  conditions
• O is a finite set of operators
• I is the initial state, a subset of P
• G is the goal state, a subset of P

17
STRIPS Operators

• Each operator O has the following attributes
• PC, a set of ground literals, defines the
  precondition of the operator
• D, a set of ground literals, defines the
  conditions that will be removed after the
  operation is executed
• A, a set of ground literals, defines the
  conditions that will be added after the operation
  is executed
• C, the cost of the operation

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A Simple Example - Blocks World
A
C
B
B
A
C
Goal State
Initial State
On(C, A) Clear(Fl) On(A, Fl) Clear(B) On(B,
Fl) Clear(C)
On(A, B) On(B, C) On(C, Fl) Clear(A) Clear(Fl)
19
Graphical Representation of Initial State
Clear(Fl)
On(A, Fl)
On(B, Fl)
On(C, A)
Clear(B)
Clear(C)
Start
T
20
Graphical Representation of Goal State
Nil
finish
On(A, B)
On(B, C)
On(C, Fl)
Clear(A)
Clear(Fl)
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Block World - Operator

• Move(x, y, z)
• Move block x that is above y to above z
• PC On(x,y), Clear(x), Clear(z)
• D Clear(z), On(x, y)
• A On(x,z), Clear(y), Clear(Fl)

22
Graphical Representation of Operator
A
Clear(y)
On(x, z)
Clear(Fl)
Move(x, y, z)
Operator
PC
On(x, y)
Clear(x)
Clear(z)
23
Forward Chaining

• Search from the initial state
• Expand the search tree by finding the set of all
  applicable operators from the current state
• applicable means the precondition of the operator
  is a subset of current state
• Update current state
• For every state that is reached, record the
  shortest path (or path with lowest cost) from the
  initial state to this state
• If the goal state is reached, stop the algorithm

24
Forward Chaining
On(C, Fl) Clear(Fl) On(A, Fl) Clear(B) On(B,
Fl) Clear(C) Clear(A)
Move(C, A, Fl)
On(C, A) Clear(Fl) On(A, Fl) Clear(B) On(B,
Fl) Clear(C)
On(B, C) Clear(Fl) On(C, A) Clear(B) On(A, Fl)
Move(B, Fl, C)
25
Backward Chaining

• Search backward from the goal state
• Expand the search tree by finding the set of all
  applicable operators that can reach the current
  state
• applicable means set A of the operator is a
  subset of current state
• Update the state
• If the initial state is reached, stop the
  algorithm
• The solution is a partial-ordered plan
• Constraints in action ordering may be violated

26
Backward Chaining
On(A, B) On(B, C) On(C, Fl) Clear(A) Clear(Fl)
On(B, C) On(C, Fl) Clear(A) Clear(Fl) On(A,
Fl) Clear(B)
Move(A, Fl, B)
27
Recent Developments

• Plan Reuse
• Graphplan
• Problem-specific planning
• Evolutionary computation approach

28
Plan Reuse

• Reuse old plans for new planning problems
• Consists of two steps
• plan matching
• plan modification
• Research findings
• generally, plan reuse is even harder than plan
  from scratch
• do better only when two problems are close enough
• plan matching could be the bottleneck

29
Graphplan

• Partial-order general planner
• Constructing a planning graph before search
• plan graph contains all possible actions that can
  be taken in each time step
• actions that interfere with one another can
  coexist in the graph
• More efficient than other general planners in
  some problems

30
Problem-specific Planning

• Heuristics combined during search
• heuristics is problem dependent
• cannot apply to other problems
• Usually outperforms general planners in specific
  problems
• Example
• in sliding-tile puzzle, accurate estimation of
  the distance between current state and goal state
  can speed up the search for a plan

31
Genetic Algorithms

• Parallel search and optimization algorithm
• Inspired by the basic rule of natural selection,
  survival-of-the-fittest
• Non-deterministic algorithm
• randomness is incorporated during implementation

32
Solution Formation

• Candidate solution is encoded as a list of genes,
  called chromosome
• Start with a population of randomly generated
  chromosomes
• Population is evolved in every generation
• evaluate each chromosome
• select chromosomes to the next generation
• apply genetic operations

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Genetic Operation

• crossover

Crossover point
Before crossover
1
1
1
1
0
1
After crossover
0
1
0
1
0
0
34
Genetic Operation

• Mutation

Before mutation
1
1
0
1
0
1
After mutation
1
1
0
1
0
0
35
GA Procedure

• Initialize the population
• Evaluate each chromosome in the population
• While the stopping condition is not met
• select chromosomes to the next generation
• crossover, mutation
• evaluate newly generated chromosomes
• End while

36
Evolutionary Computation Approach

• Non-deterministic algorithm
• Starts from a set of randomized plan
• Plans are evolved during generations
• In each generation
• evaluate the performance of the plan
• select the plans to next generation, based on
  performance
• crossover, mutation to create new plans

37
Evolutionary Computation Approach

• Solution encoding
• a list of floating-point number, each number maps
  to an action in the problem
• number of actions can vary during search

38
Evolutionary Computation Approach

• State-aware crossover
• crossover points are selected based on the
  matching of states

s1
Parent 1
Match(s1, s2) true
s2
Parent 2
Child 1
Child 2
39
Evolutionary Computation Approach

• Mutation
• randomly select an action and replace it with a
  random floating-point number

Before mutation
0.8
0.5
0.3
0.2
0.5
After mutation
0.8
0.5
0.6
0.5
0.2
40
Evolutionary Computation Approach

• Fitness evaluation
• consists of two parts goal fitness fg and cost
  fitness fc
• goal fitness how the solutions reach the goal
  state
• cost fitness the total cost of the solution
• overall fitness a fg b fc

41
Experimental Results

• Towers of Hanoi
• can solve the 5 and 6-disk case with high
  probability
• state-aware crossover does not perform well
• Sliding-tile puzzle
• can solve 33 problem case with high probability
• state-aware crossover perform much better since a
  looser definition of matching state is used
• Performance scalability is not good
• Heuristics may be helpful to improve the
  performance

42
References

• Artificial Intelligence A Modern Approach.
  Stuart Russell and Peter Norvig. Prentice Hall.
• Artificial Intelligence A New Symthesis. Nils
  Nilsson. Kaufmann.
• Intelligent Planning A Decomposition and
  Abstraction Based Approach. Qiang Yang. Springer.
• Internet-Based Workflow Management Towards a
  Semantic Web. Dan C. Marinescu. Wiley, 2002.

43
References

• Plan reuse versus plan generation a theoretical
  and empirical analysis. B. Nebel, J. Koehler.
  Journal of Artificial Intelligence 76 (1995),
  427-454.
• Fast planning through planning graph analysis. A.
  L. Blum, M. L. Furst Journal of Artificial
  Intelligence 90 (1997), 281-300.
• Planning as heuristic search. B. Bonet, H.
  Geffner. Journal of Artificial Intelligence 129
  (2001), 5-33.
• What it is, What it could be, An introduction to
  the Special Issue on Planning and Scheduling. D.
  McDermott, J. Handler. Journal of Artificial
  Intelligence 76 (1995), 1-16.

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Concluding Words
Failing to plan is planning to fail --Effie Jones