Midterm Review

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Midterm Review

• cmsc421 Fall 2005

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Outline

• Review the material covered by the midterm
• Questions?

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Subjects covered so far

• Search Blind Heuristic
• Constraint Satisfaction
• Adversarial Search
• Logic Propositional and FOL

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and subjects to be covered

• planning
• uncertainty
• learning
• and a few more

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Search
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Stating a Problem as a Search Problem
S

• State space S
• Successor function x ? S ? SUCCESSORS(x) ?
  2S
• Arc cost
• Initial state s0
• Goal test
• x?S ? GOAL?(x) T or F
• A solution is a path joining the initial to
  a goal node

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Basic Search Concepts

• Search tree
• Search node
• Node expansion
• Fringe of search tree
• Search strategy At each stage it determines
  which node to expand

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Search Algorithm

• If GOAL?(initial-state) then return initial-state
• INSERT(initial-node,FRINGE)
• Repeat
• If empty(FRINGE) then return failure
• n ? REMOVE(FRINGE)
• s ? STATE(n)
• For every state s in SUCCESSORS(s)
• Create a new node n as a child of n
• If GOAL?(s) then return path or goal state
• INSERT(n,FRINGE)

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Performance Measures

• CompletenessA search algorithm is complete if it
  finds a solution whenever one existsWhat about
  the case when no solution exists?
• OptimalityA search algorithm is optimal if it
  returns an optimal solution whenever a solution
  exists
• ComplexityIt measures the time and amount of
  memory required by the algorithm

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Blind vs. Heuristic Strategies

• Blind (or un-informed) strategies do not exploit
  state descriptions to select which node to expand
  next
• Heuristic (or informed) strategies exploits state
  descriptions to select the most promising node
  to expand

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Blind Strategies

• Breadth-first
• Bidirectional
• Depth-first
• Depth-limited
• Iterative deepening
• Uniform-Cost(variant of breadth-first)

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Comparison of Strategies

• Breadth-first is complete and optimal, but has
  high space complexity
• Depth-first is space efficient, but is neither
  complete, nor optimal
• Iterative deepening is complete and optimal, with
  the same space complexity as depth-first and
  almost the same time complexity as breadth-first

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Avoiding Revisited States

• Requires comparing state descriptions
• Breadth-first search
• Store all states associated with generated nodes
  in CLOSED
• If the state of a new node is in CLOSED, then
  discard the node

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Avoiding Revisited States

• Depth-first search
• Solution 1
• Store all states associated with nodes in current
  path in CLOSED
• If the state of a new node is in CLOSED, then
  discard the node
• Only avoids loops
• Solution 2
• Store of all generated states in CLOSED
• If the state of a new node is in CLOSED, then
  discard the node
• ? Same space complexity as breadth-first !

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Uniform-Cost Search (Optimal)

• Each arc has some cost c ? ? gt 0
• The cost of the path to each fringe node N is
• g(N) ? costs of arcs
• The goal is to generate a solution path of
  minimal cost
• The queue FRINGE is sorted in increasing cost
• Need to modify search algorithm

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Modified Search Algorithm

• INSERT(initial-node,FRINGE)
• Repeat
• If empty(FRINGE) then return failure
• n ? REMOVE(FRINGE)
• s ? STATE(n)
• If GOAL?(s) then return path or goal state
• For every state s in SUCCESSORS(s)
• Create a node n as a successor of n
• INSERT(n,FRINGE)

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Avoiding Revisited States in Uniform-Cost Search

• When a node N is expanded the path to N is also
  the best path from the initial state to STATE(N)
  if it is the first time STATE(N) is encountered.
• So
• When a node is expanded, store its state into
  CLOSED
• When a new node N is generated
• If STATE(N) is in CLOSED, discard N
• If there exits a node N in the fringe such that
  STATE(N) STATE(N), discard the node N or N
  with the highest-cost path

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Best-First Search

• It exploits state description to estimate how
  promising each search node is
• An evaluation function f maps each search node N
  to positive real number f(N)
• Traditionally, the smaller f(N), the more
  promising N
• Best-first search sorts the fringe in increasing f

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Heuristic Function

• The heuristic function h(N) estimates the
  distance of STATE(N) to a goal state Its value
  is independent of the current search tree it
  depends only on STATE(N) and the goal test
• Example
• h1(N) number of misplaced tiles 6

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Classical Evaluation Functions

• h(N) heuristic functionIndependent of search
  tree
• g(N) cost of the best path found so far between
  the initial node and NDependent on search
  tree
• f(N) h(N) ? greedy best-first search
• f(N) g(N) h(N)

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Can we Prove Anything?

• If the state space is finite and we discard nodes
  that revisit states, the search is complete, but
  in general is not optimal
• If the state space is finite and we do not
  discard nodes that revisit states, in general the
  search is not complete
• If the state space is infinite, in general the
  search is not complete

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Admissible Heuristic

• Let h(N) be the cost of the optimal path from N
  to a goal node
• The heuristic function h(N) is admissible if
  0 ? h(N) ? h(N)
• An admissible heuristic function is always
  optimistic !
• Note G is a goal node ? h(G) 0

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A Search(most popular algorithm in AI)

• f(N) g(N) h(N), where
• g(N) cost of best path found so far to N
• h(N) admissible heuristic function
• for all arcs 0 lt ? ? c(N,N)
• modified search algorithm is used
• ? Best-first search is then called A search

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Result 1

• A is complete and optimal

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Experimental Results

• 8-puzzle with
• h1 number of misplaced tiles
• h2 sum of distances of tiles to their goal
  positions
• Random generation of many problem instances
• Average effective branching factors (number of
  expanded nodes)

d IDS A1 A2
2 2.45 1.79 1.79
6 2.73 1.34 1.30
12 2.78 (3,644,035) 1.42 (227) 1.24 (73)
16 -- 1.45 1.25
20 -- 1.47 1.27
24 -- 1.48 (39,135) 1.26 (1,641)
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Iterative Deepening A (IDA)

• Idea Reduce memory requirement of A by applying
  cutoff on values of f
• Consistent heuristic h
• Algorithm IDA
• Initialize cutoff to f(initial-node)
• Repeat
• Perform depth-first search by expanding all nodes
  N such that f(N) ? cutoff
• Reset cutoff to smallest value f of non-expanded
  (leaf) nodes

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Local Search

• Light-memory search method
• No search tree only the current state is
  represented!
• Only applicable to problems where the path is
  irrelevant (e.g., 8-queen), unless the path is
  encoded in the state
• Many similarities with optimization techniques

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Search problems
Blind search
Heuristic search best-first and A
Variants of A
Construction of heuristics
Local search
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When to Use Search Techniques?

• The search space is small, and
• No other technique is available
• Developing a more efficient technique is not
  worth the effort
• The search space is large, and
• No other technique is available, and
• There exist good heuristics

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Constraint Satisfaction
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Constraint Satisfaction Problem

• Set of variables X1, X2, , Xn
• Each variable Xi has a domain Di of possible
  values
• Usually Di is discrete and finite
• Set of constraints C1, C2, , Cp
• Each constraint Ck involves a subset of variables
  and specifies the allowable combinations of
  values of these variables
• Goal Assign a value to every variable such that
  all constraints are satisfied

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CSP as a Search Problem

• Initial state empty assignment
• Successor function a value is assigned to any
  unassigned variable, which does not conflict with
  the currently assigned variables
• Goal test the assignment is complete
• Path cost irrelevant

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Questions

1. Which variable X should be assigned a value next?
2. Minimum Remaining Values/Most-constrained
   variable
3. In which order should its domain D be sorted?
4. least constrained value
5. How should constraints be propagated?
6. forward checking
7. arc consistency

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Adversarial Search
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Specific Setting Two-player, turn-taking,
deterministic, fully observable, zero-sum,
time-constrained game

• State space
• Initial state
• Successor function it tells which actions can be
  executed in each state and gives the successor
  state for each action
• MAXs and MINs actions alternate, with MAX
  playing first in the initial state
• Terminal test it tells if a state is terminal
  and, if yes, if its a win or a loss for MAX, or
  a draw
• All states are fully observable

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Choosing an Action Basic Idea

• Using the current state as the initial state,
  build the game tree uniformly to the maximal
  depth h (called horizon) feasible within the time
  limit
• Evaluate the states of the leaf nodes
• Back up the results from the leaves to the root
  and pick the best action assuming the worst from
  MIN
• ? Minimax algorithm

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Minimax Algorithm

• Expand the game tree uniformly from the current
  state (where it is MAXs turn to play) to depth h
• Compute the evaluation function at every leaf of
  the tree
• Back-up the values from the leaves to the root of
  the tree as follows
• A MAX node gets the maximum of the evaluation of
  its successors
• A MIN node gets the minimum of the evaluation of
  its successors
• Select the move toward a MIN node that has the
  largest backed-up value

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Alpha-Beta Pruning

• Explore the game tree to depth h in depth-first
  manner
• Back up alpha and beta values whenever possible
• Prune branches that cant lead to changing the
  final decision

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Example
The beta value of a MIN node is an upper bound
on the final backed-up value. It can never
increase
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Example
a 1
The alpha value of a MAX node is a lower bound
on the final backed-up value. It can never
decrease
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Alpha-Beta Algorithm

• Update the alpha/beta value of the parent of a
  node N when the search below N has been completed
  or discontinued
• Discontinue the search below a MAX node N if its
  alpha value is ? the beta value of a MIN ancestor
  of N
• Discontinue the search below a MIN node N if its
  beta value is ? the alpha value of a MAX ancestor
  of N

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Logical Representations and Theorem Proving
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Logical Representations

• Propositional logic
• First-order logic
• syntax and semantics
• models, entailment, etc.

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The Game

• Rules
• Red goes first
• On their turn, a player must move their piece
• They must move to a neighboring square, or if
  their opponent is adjacent to them, with a blank
  on the far side, they can hop over them
• The player that makes it to the far side first
  wins.

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Logical Inference

• Propositional truth tables or resolution
• FOL resolution unification
• strategies
• shortest clause first
• set of support

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Questions?