Production Systems and Searching

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Production Systems and Searching
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Solving Problems by Searching

• To build a system to solve a particular problem
• 1. Define the problem precisely.
• 2. Analyze the problem.
• 3. Isolate and represent the task knowledge
  needed to solve the problem.
• Choose the best problem-solving technique and
  apply it to the problem.

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Solving Problems by Searching

• Why is search necessary?
• No model of the world is complete, consistent,
  and computable. Any intelligent systems must
  encounter surprises.
• Solutions cannot be entirely precomputed Many
  problems must be solved dynamically, starting
  from observed data.
• Search provides flexibility to deal with a highly
  variable environment.
• Search can handle local ambiguity in
  interpretation of perceptual data. Global
  constraints can provide unambiguous total
  interpretation.

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Define the Problem As a State Space Search

• Playing chess
• Write a set of rules for each possible move
  (10120 possible moves).
• Or.
• Write a set of general rules for possible types
  of moves, I.E. Define the problem as moving
  around in a state space where each state
  corresponds to a legal position on the board.
  Start at initial state, then use a set of rules
  to move from one state to another, to try to end
  up at one of a set of final states.

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Define the Problem As a State Space Search

• State space representation corresponds to the
  structure of the problem
• A. It allows for a formal definition of a
  problem as the need to convert a given situation
  into a desired situation using a set of
  permissible operators.
• B. It permits definition of the process of
  solving the problem as combination of known
  techniques (each represented as a rule defining a
  single step in the space) and search, the general
  technique of exploring the space to try to find
  some path from the current state to a goal state.

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Water Jug Problem A State Space Search

• Water jug problem you are given two jugs with
  no measuring marks, a 4-gallon one and a 3-gallon
  one. There is a pump to fill the jugs with water.
  How can you get exactly 2 gallons of water into
  the 4-gallon jug?
• State space set of ordered pairs of integers
  (x, y) such at x 0,1,2,3, or 4 for amount of
  water in 4-gallon jug, and y 0, 1, 2, or 3 for
  amount of water in the 3-gallon jug. The start
  state is (0,0). The goal state is (2,n) for any
  value of n. The operators are shown in list,
  represented as rules whose left sides are matched
  against the current state and whose right rules
  describe the new state that results from applying
  the rule. Explicit assumptions required we
  can fill a jug from the pump, we can pour water
  out of a jug onto the ground, we can pour water
  from one jug to the other, and there are no other
  measuring devices available.

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Water Jug Problem A State Space Search

• Control structure needed that loops through a
  cycle in which a rule whose left side matches the
  current state is chosen, an appropriate change is
  made to the state as described in the right side
  of the rule, and the resulting state is checked
  to see if it corresponds to a goal state.

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Formal Description of a Problem

• Define a state space that contains all the
  possible configurations of the relevant objects.
  Doesnt have to enumerate all states.
• Specify one or more states within that space that
  describe possible states from which the the
  problem-solving can start. These states are
  called the initial states.
• Specify one or more states that will be
  acceptable as solutions to the problem. These are
  called goal states.

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Formal Description of a Problem (Contd)

• 4. Specify a set of rules that describe the
  actions (operators) available.
• What unstated assumptions are present in the
  informal problem description?
• How general should the rules be?
• How much of the work required to solve the
  problem should be precomputed and represented in
  the rules?
• What is the goal test?
• What is the path cost for each action?

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Formal Description of a Problem (Contd)

• Measure the performance.
• Does it find a solution?
• Is it a good solution with a low path cost?
• What is the search cost wrt time and memory
  needs?
• Total cost of a search is the sum of the path
  cost and the search cost.

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State-space Representationand Production Systems

• Select some way to represent states in the
  problem in an unambiguous way.
• Formulate all actions that can be preformed in
  states
• Including their preconditions and effects.
• Production rules.
• Represent the initial state (s).
• Formulate precisely when a state satisfies the
  goal of our problem.
• Activate the production rules on the initial
  state and its descendants, until a goal state is
  reached.

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• AI uses state space search to find solutions to
  problems
• State space search uses state space graphs to
  represent problems
• This representation may be used instead of or in
  addition to representation in predicate calculus
• Graph theory allows us to analyze the structure
  and complexity of problems and search strategies

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• A graph is a set of nodes and a set of arcs that
  connect pairs of nodes
• In state space search, a node in a graph
  represents a state of the world or a state in the
  solution of a problem
• An arc represents a transition between states
• A chess move (arc) changes the game board from
  one state to another
• A robot arm moving a block (arc) changes the
  blocks world from one state to another
• A rule (arc) like if its raining, take an
  umbrella, changes the state of the known world

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BLIND Search Methods

• Depth-first
• Breadth-first
• Non-deterministic search
• Iterative deepening
• Bi-directional search

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Depth-first Search

• Let L be a list of start states (the frontier) -
  initialised to root state
• While L is not empty do
• Let N be the first state in L
• If N is a goal state then return success
• Else remove N from front of L
• Add all states reachable from N to the front
  of L
• Return failure

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Speed (Depth-first)

• In the worst case
• The (only) goal node may be on the right-most
  branch,

d
b
G
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Memory (Depth-first)

• Largest number of nodes in QUEUE is reached in
  bottom left-most node.
• Example d 3, b 3

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Breadth-first Algorithm

• 1. QUEUE
• 2. WHILE QUEUE is not empty
• AND goal is not
  reached
• DO remove the first path from the QUEUE
• Create new paths (to all children)
• Reject the new paths with loops
• Add the new paths to back of QUEUE
• 3. IF goal reached
• THEN success
• ELSE failure

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Completeness (Breadth-first)

• Would even remain complete without our
  loop-checking.
• Note ALWAYS finds the shortest path.

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Speed (Breadth-first)

• If a goal node is found on depth m of the tree,
  all nodes up till that depth are created.

Thus O(bm) note depth-first would also visit
deeper nodes.
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Memory (Breadth-first)

• Largest number of nodes in QUEUE is reached on
  the level m of the goal node.

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Assessing Search Performance

• Both strategies are blind.
• The list L contains all states generated but not
  yet expanded. On average, how long will L be?
• If the length of L exceeds the available memory,
  the search fails, even though the goal is
  reachable.
• Each "move" takes a finite amount of time to
  generate. On average, how many states will we
  expand?
• For effective performance, we must be able to
  return an answer in good time.

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Depth- and Breadth-first Comparison

• Breadth-first and depth-first take, on average,
  approximately the same time
• Depth-first requires considerably less space for
  large trees
• Depth-first is even better if the first node is
  bottom left of the tree

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Depth- and Breadth-first Comparison

• Breadth-first is better if there is a node top
  right
• Breadth-first will always find solution with
  smallest number of moves

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Traveling Salesman Problem

• List of cities to visit
• Must visit all cities exactly once
• The route to follow for the shortest possible
  round trip

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Heuristic Search

• A heuristic is a technique that improves the
  efficiency of a search process, possibly by
  sacrificing claims of completeness
• Example Nearest Neighbor heuristic

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Problem Characteristics

• Is problem Decomposable into a set of nearly
  independent smaller or easier sub problems?
• Can solution steps be ignored?
• Universe predictable?
• Is good solution obvious?
• State or path?
• Role of Knowledge
• Interaction between user and computer

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Production System Characteristics

• A monotonic production system is a production
  system in which the application of a rule never
  prevents the later application of a rule that
  could also have been applied at the time the
  first rule was selected
• A partially Commutative system has the property
  that if the application of a particular sequence
  of rules transforms state x into state y then any
  permutation of those rules also does so

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Informed Search

• Improving search by counting the cost of each
  move (i.E. The cost from the start)
• Hill-climbing
• Best-first

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Informed Search

• Improving search by estimating the cost of
  getting to the goal
• Greedy search methods
• A - an optimal search strategy

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Hill Climbing Search

• Similar to depth-first, but now ordering the
  children of the current node by their cost from
  start - nearest first.
• Let L be a list of start states (the frontier) -
  initialised to root state.
• While L is not empty do.
• Let N be the first state in L.
• If N is a goal state then return success.
• Else remove N from front of L.
• Obtain C, all states reachable from N.
• Order C by cost from start to get
  C1.
• Add C1 to the front of L.
• Return failure.

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Best-first Search

• Similar to hill-climbing, but now inserting the
  children into the list in order of cost.
• Let L be a list of start states (the frontier) -
  initialised to root state.
• While L is not empty do.
• Let N be the first state in L.
• If N is a goal state then return success.
• Else remove N from front of L.
• Obtain C, all states reachable from N.
• Insert C into L in order of cost
  from the start.
• Return failure.

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Estimating the Remaining Cost

• Hill-climbing and best-first search still have
  problems
• Hill-climbing is still just depth-first search,
  and often gets lost searching branches which will
  never lead to the goal.
• Best-first is a combination of depth-first and
  breadth-first, but still prefers states which are
  closest to the start state, without concern for
  how close they are to the goal.

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Estimating the Remaining Cost

• What we need to do is incorporate an estimate of
  how expensive it will be to get to the goal from
  a new state, and use that to decide on the order
  in which we expand states.

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Estimating the Remaining Cost

• This estimate is called a heuristic.
• A large part of the art of applying search
  methods to problems is devising appropriate
  heuristics.

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Greedy Hill-climbing Search

• Similar to hill-climbing, but now ordering the
  children by their estimated cost of getting to
  the goal.
• Let L be a list of start states (the frontier) -
  initialised to root state.
• While L is not empty do.
• Let N be the first state in L.
• If N is a goal state then return success.
• Else remove N from front of L.
• Obtain C, all states reachable from N.
• Order C by estimated cost of
  getting to the goal to get C1.
• Add C1 to the front of L.
• Return failure.

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Greedy Best-first Search

• Similar to best-first, but now ordering all
  states by the estimated cost of getting to the
  goal.
• Let L be a list of start states (the frontier) -
  initialised to root state.
• While L is not empty do.
• Let N be the first state in L.
• If N is a goal state then return success.
• Else remove N from front of L.
• Obtain C, all states reachable from N.
• Insert C into L in order of
  estimated distance from the goal.
• Return failure.

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Combining the Informed Methods

• Greedy search methods can be misled by the
  heuristic.
• The main problem is that they forget the work
  they have done to get to the current state, and
  are only able to look ahead.
• What we need to do is combine the two methods.

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Combining the Informed Methods

• Let g(n) be the true cost of getting from the
  start to the state N.
• Let h(n) be the true cheapest cost of getting to
  the goal from the state N.

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Combining the Informed Methods

• Let h'(n) be the estimated cost of getting to the
  goal from the state N.
• Then f(n) g(n) h'(n) is our estimate of the
  cheapest path from the start to the goal which
  visits N.

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A Search

• Similar to best-first, but now ordering all nodes
  by the estimated total cost of getting from start
  to goal.
• Let L be a list of start states - initialised to
  root state.
• While L is not empty do.
• Let N be the first state in L.
• If N is a goal state the return success.
• Else remove N from front of L.
• Obtain C, all states reachable from N.
• Obtain total estimate for all
  states in C by adding cost from start to
  estimated cost to goal.
• Insert C into L in order of
  estimated total cost.
• Return failure.

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A Finds the Cheapest Path

• Theorem.
• If h'(n) ? h(n) for all N, then A will always
  return the lowest cost path to the goal, i.E if
  the distance measure is an underestimate.
• Proof
• Let G be the lowest cost goal state.
• Let G' be a higher cost goal state .
• For all goal states, h(g) 0.
• Lower cost means g(g) ? g(g'), so f(g) ? f(g').
• Suppose A has G' at the front of its list L -
  i.E. G' is the state we will take next (and
  therefore choose a higher cost state).
• Suppose G itself is in L.
• If this were so, then G would be in front of G
  (since f(g)
• So G cant be on L.

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A Finds the Cheapest Path
Start
L (frontier)
N
G
G
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A Finds the Cheapest Path

• Proof (ctd.)
• Let N be any state on the path to G.
• For any states N and N.'
• G(n) h(n) g(n') h(n') g(g) h(g).
• F(n) g(n) h'(n) ? g(n) h(n)
  g(g) h(g) f(g).
• H'(n) ? h(n).
• H(g) 0.
• So f(n) ? f(g).
• But if G' is at the front of the list, then f(g')
  ? f(n).
• Therefore f(g') ? f(g) - contradiction.
• Therefore, A can never have G' at the front of
  its list.
• Therefore, A can never return a higher cost
  goal.

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The Benefits of A

• A is complete
• If every state has a finite number of successors
  and there is no path of infinite length but
  finite cost, then A will return a solution.
• A is optimal
• A will return the cheapest solution
• A is optimally efficient
• No other optimal search method which works by
  expanding a path from the start state is
  guaranteed to expand fewer states than A

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But ...

• A is not the answer to all search problems.
• As the path to the best solution gets longer, A
  takes exponentially longer to get there.
• A's memory requirement grows exponentially.
• For many real problems, systematic search of this
  sort is infeasible. To make progress we need to
  consider non-systematic techniques, or, more
  fundamentally, we need to incorporate more
  knowledge and experience about the problem
  domain.

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Knowledge Representation
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Prepositional Calculus

• The prepositional calculus is based on statements
  which have truth values ( true or false).
• The calculus provides a means of determining the
  truth values associated with statements formed
  from atomic'' statements. An example
• If p stands for fred is rich'' and q for fred
  is tall'' then we may form statements such as

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• Note that the first 4 are all binary connectives.
  They are sometimes referred to, respectively, as
  the symbols for disjunction, conjunction,
  implication and equivalence. Also not is unary
  and is the symbol for negation.
• If prepositional logic is to provide us with the
  means to assess the truth value of compound
  statements from the truth values of the building
  blocks' then we need some rules for how to do
  this.
• For example, the calculus states that p q is true
  if either p is true or q is true (or both are
  true). Similar rules apply for all the ways in
  which the building blocks can be combined.

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First Order Predicate Calculus

• The predicate calculus includes a wider range of
  entities. It permits the description of relations
  and the use of variables. It also requires an
  understanding of quantification.
• The last two are known as quantifiers.
• The non-logical constants include both the
  names' of entities that are related and the
  names' of the relations. For example, the
  constant dog might be a relation and the constant
  fido an entity.

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• Note that the word and'' used in the left hand
  column is used to suggest that we have more than
  one formula for combination ---and not
  necessarily a conjunction.
• In the last two examples, dog(x)'' contains a
  variable which is said to be free while the X''
  in X.Dog(x)'' is bound.
• Sentence.
• ---A formula with no free variables is a
  sentence.
• Two informal examples to illustrate
  quantification follow.