Introduction To Planning

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Introduction To Planning State-Space Search

• G51IAI Introduction to AI
• Andrew Parkes
• http//www.cs.nott.ac.uk/ajp/

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UCS for Graph Search

• From last lecture
• Uniform Cost Search finds a min-cost path to a
  goal in a graph
• Search pattern
• Search all nodes of cost c before those of cost
  c1
• Method
• store the cost g of the path used to reach the
  node
• use such costs in deciding the ordering in the
  queue

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Uniform Cost Search (UCS)

• Explicitly store the cost of a node with that
  node convention g is the path-cost
• Queue Processing Always remove the smallest cost
  node first

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Uniform Cost Search in Graph

• fringe ? MAKE-EMPTY-QUEUE()
• fringe ? INSERT( root_node ) // with g0
• loop
• if fringe is empty then return false // finished
  without goal
• node ? REMOVE-SMALLEST-COST(fringe)
• if node is a goal
• print node and g
• return true // that found a goal
• Lg ? EXPAND(node) // Lg is set of neighbours with
  their g costs //
  NOTE do not check Lg for goals here!!
• fringe ? INSERT-IF-NEW(Lg, fringe ) // ignore
  revisited nodes // unless is with new
  better g

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Properties of UCS

• Completeness
• If there is a path to a goal then UCS will find
  it
• If there is no path, then UCS will eventually
  report that the goal is unreachable
• Optimality
• UCS will report a minimum cost path (there might
  be many)
• Systematic
• With appropriate code to prevent re-visiting
  nodes UCS will only expand each node once

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Complexity of UCS on a Graph

• Suppose graph has n nodes and e edges
• e.g. n cities, and e roads
• Worst case (e.g. no goal is reachable)
• nodes expanded n
• work per node O(n) (check that not already
  visited)
• edges visited O(e) O(n2)
• overall O(n2)
• So UCS is polynomial time in size of graph
• But AI is hard because of problems for which
  time needed is exponential in the size of the
  problem
• Paradox?

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When is UCS exponentially hard?

• The graph itself can be exponentially sized!
• The size of the natural description of the
  problem might be a number S
• but the graph itself has size exponential in S
• the graph itself is not given directly
• only get a set of rules to generate the graph
• Planning is an example of this
• First an easier example the hypercube

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Example of Implicit Graph

• Hypercube in n dimensions
• nodes
• any binary (0s and 1s) string with n digits
• e.g. 1000101 for n7
• edge between nodes n1 and n2 if and only if they
  differ only in one digit
• e.g. 1000101 has an edge to 1000111 but
  not to 1000011

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Hypercube Example Cube

• Example n3, a cube,
• nodes 000, 001, 010, etc for xyz coordinates
• e.g. 001 means (x,y,z)(0,0,1)
• edges node 000 has edges to 001,010,100

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Example of Implicit Graph

• Hypercube in n dimensions
• node binary number with n digits
• edge between nodes n1 and n2 iff they differ only
  in one digit
• Size of description just the space to write the
  value of n
• Number of nodes 2n
• Number of edges 2n 3 / 2
• The graph itself is exponentially bigger than our
  description of it.
• On discussing polynomial or exponential in the
  size we need to be careful what we mean by the
  size
• Similar things do happen with planning problems
  puzzles

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Planning Puzzles

• These are search problems in which we have some
  system and
• the system can be in many different states
• want to transform it from one state into another
• by applying a sequence of actions
• Examples
• Towers of Hanoi
• Tile Puzzles
• Coursework TWO

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Planning Problem - Towers of Hanoi
State The positions of the
ringsAction Move one ring at a timeRules
No ring can ever be on top of a smaller
ringGoal Move the tower to peg 2
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Planning Problem - Towers of Hanoi
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Planning Problem - Towers of Hanoi
Note it was illegal to move the green on top of
the red
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Planning Problem - Towers of Hanoi
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Planning Problem - Towers of Hanoi
Note finally we get to move blue to its final
place
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Planning Problem - Towers of Hanoi
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Planning Problem - Towers of Hanoi
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Planning Problem - Towers of Hanoi
Note SUCCESS in 7 moves
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Planning Problem - Towers of Hanoi

• Analysis of the problem shows that the lower
  bound for the number of moves is
• 2N-1
• Since N appears as the exponent we have an
  exponential function
• The number of reachable states is at least this
  large

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Planning Problem Definition - 1

• Initial State
• The initial state of the problem, defined in some
  suitable manner
• Operator
• a.k.a. action, or move
• A set of actions that moves the problem from one
  state to another

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Planning Problem Definition - 1

• Neighbourhood (Successor Function)
• The set of all possible states reachable from a
  given state with one move
• State Space
• The set of all legal states

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Planning Problem Definition - 2

• Goal Test
• A test applied to a state which returns if we
  have reached a state that solves the problem
• Path Cost
• How much it costs to take a particular path

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Planning Problem Definition - 2

• Plan
• A sequence of actions applied to a state to
  transform it into another state
• Takes us from the initial state to the goal state
• Optimal Plan
• A mincost plan, i.e. a plan with minimal path cost

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Problem Definition Tile Puzzle Example
Initial State
Goal State
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Problem Definition - Example

• States
• A description of each of the eight tiles in each
  location that it can occupy. It is also useful to
  include the blank
• Operators
• The blank moves left, right, up or down

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Problem Definition - Example

• Goal Test
• The current state matches a certain state (e.g.
  one of the ones shown on previous slide)
• Path Cost
• Each move of the blank costs 1

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Planning Graphs 1

• How does the tile puzzle relate to previous
  lectures on searching a graph?
• Regard each puzzle state as a node of a graph
• Actions, move blank, correspond to legal moves
  between the states.
• actions generate edges between the nodes

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Planning Graphs 2

• Actions, move blank, correspond to legal moves
  between the states.
• actions generate edges between the nodes
• Finding a plan ...
• find a legal sequence of actions from initial to
  final state
• ... becomes find a path to a goal
• find a path from initial to final node
• Find an optimal plan, becomes
• find a mincost path through the planning graph

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Tile Puzzle - History

• Why is the puzzle so popular?
• Invented by Sam Loyd in 1870s
• On the 4x4 version, the 15-puzzle, he offered a
  prize of 1000 prize to transform a particular
  initial state to a particular final state
• Trick Question
• The problem was unsolvable
• The state space splits into two disconnected
  pieces
• The planning graph is not connected
• Only half the potential states are reachable from
  any state
• The reachable state space is 9!/2 181440

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Planning Representations 1

• Paradox? Consider the 24-puzzle.
• In very little space we managed to describe a
  planning problem for which the graph has 25!
  nodes and about 425! edges
• This takes far too much space to write down
  explicitly
• Instead
• we are describing the state logically
• val(2,2) 9 means the 9 tile is at coords
  (2,2)
• these are function of time
• That is we can describe the state sequence by
  means of val(x,y,t)

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Planning Representations 2

• Actions are described by rules for changing the
  state
• move blank from (1,1) at time t to (1,2) at time
  t1 becomes
• if val(1,1,t) B then permit
• val(1,2,t1) B
• val(1,1,t1 ) val(1,2,t)
• And nothing else changed
• Note that such a move applies independently of
  the values for the other squares and so describes
  many edges in the graph

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Planning Representations 3

• In general representing the effects of change in
  the real-world is very hard
• E.g. Ramification problem
• e.g. if I drive to derby some things will
  automatically change, but not others
• cell phone tower
• but not cell phone number
• in real-world then this can be hard to represent
• Puzzles and simple planning domains do not have
  this problem

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Planning Representations 4

• The standard way in AI to formally express a
  planning problem is using the STRIPS notation
• STRIPS contains
• fluents the logical quantities that describe
  the state and change as a function of time
• Actions are described using
• preconditions circumstances in which the action
  applies
• add delete rules for the changes to the
  fluents

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Planning Representations 5

• The standard way in AI to formally express a
  planning problem is using the STRIPS notation
• A lot of work has gone into solving planning
  problems expressed in STRIPS notation
• There are general purpose solvers that take any
  problem expressed in STRIPS
• STRIPS forms an important part of AI history and
  current usage
• You do not need to know how to use STRIPS for
  this course just how to do actions by hand
• But (clue!), a part of the history of STRIPS is
  relevant to coursework TWO

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Why are planning/puzzles hard?

• The associated graph is exponential in the
  number of tiles
• We can still use the same blind search methods
• BFS, UCS
• On realistic size problems blind search will use
  far too much memory and time
• Need to improve the search methods
• planning problems, and puzzles such as tile
  puzzle, have driven a lot of progress in search
  methods
• improved methods are heuristic and use
  information about the likely location of goals

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Summary

• Planning problems
• States
• Operators (a.k.a. actions, successor, etc)
• Can be thought of as defining implicitly a graph
• node state
• edge action
• But the graphs are much bigger than maps, mazes,
  etc
• These drive the need for better search methods
• Next Lecture Informed search methods

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Questions?