Inference for Propositional Logic and Review

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Inference for Propositional Logic and Review

• Reading C. 8
• New TA Vijay Anand Nagarajan
• Office Hours Fridays 5-7pm in TA Room

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Inference Properties

• Inference method A is sound (or truth-preserving)
  if it only derives entailed sentences
• Inference method A is complete if it can derive
  any sentence that is entailed
• A proof is a record of the progress of a sound
  inference algorithm.

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Other Types of Inference

• Model Checking
• Forward chaining with modus ponens
• Backward chaining with modus ponens

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Model Checking

• Enumerate all possible worlds
• Restrict to possible worlds in which the KB is
  true
• Check whether the goal is true in those worlds or
  not

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Wumpus Reasoning

• Percepts nothing in 1,1 breeze in 2,1
• Assume agent has moved to 2,1
• Goal where are the pitss?
• Construct the models of KB based on rules of
  world
• Use entailment to determine knowledge about pits

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Constructing the KB
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Properties of Model Checking

• Sound because it directly implements entailment
• Complete because it works for any KB and sentence
  to prove a and always terminates
• Problem there can be way too many worlds to
  check
• O(2n) when KB and a have n variables in total

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Inference as Search

• State current set of sentences
• Operator sound inference rules to derive new
  entailed sentences from a set of sentences
• Can be goal directed if there is a particular
  goal sentence we have in mind
• Can also try to enumerate every entailed sentence

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Example
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Complexity

• N propositions M rules
• Every possible fact can be establisehd with at
  most N linear passes over the database
• Complexity O(NM)
• Forward chaining with Modus Ponens is complete
  for Horn logic

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Example
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Review for Midterm

• Concepts you should know
• Search algorithms
• Depth-first, breadth-first, iterative deepening,
  A, greedy, hill-climbing, beam
• Constraint propagation
• Game playing
• Propositional logic and inference

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Midterm format

• Multiple choice
• Short answer questions
• Problem solving
• Essay
• An example midterm will be posted under links

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Concepts

• Any words in yellow or light blue or pink on
  slides

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Uninformed Search

• Depth-first
• Breadth-first
• Iterative Deepening

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Formulating Problems as Search

• Given an initial state and a goal, find the
  sequence of actions leading through a sequence of
  states to the final goal state.
• Terms
• Successor function given action and state,
  returns action, successors
• State space the set of all states reachable from
  the initial state
• Path a sequence of states connected by actions
• Goal test is a given state the goal state?
• Path cost function assigning a numeric cost to
  each path
• Solution a path from initial state to goal state

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A different view of the same problem for
visualization Breadth first

• OPEN start node CLOSED empty
• While OPEN is not empty do
• Remove leftmost state from OPEN, call it X
• If X goal state, return success
• Put X on CLOSED
• SUCCESSORS Successor function (X)
• Remove any successors on OPEN or CLOSED
• Put remaining successors on right end of OPEN
• End while

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A different view of the same problem for
visualizationDepth-first

• OPEN start node CLOSED empty
• While OPEN is not empty do
• Remove leftmost state from OPEN, call it X
• If X goal state, return success
• Put X on CLOSED
• SUCCESSORS Successor function (X)
• Remove any successors on OPEN or CLOSED
• Put remaining successors on left end of OPEN
• End while

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Can we combine benefits of both?

• Depth limited
• Select some limit in depth to explore the problem
  using DFS
• How do we select the limit?
• Iterative deepening
• DFS with depth 1
• DFS with depth 2 up to depth d

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Complexity Analysis

• Completeness is the algorithm guaranteed to find
  a solution when there is one?
• Optimality Does the strategy find the optimal
  solution?
• Time How long does it take to find a solution?
• Space How much memory is needed to perform the
  search?
• Is this notion of completeness the same as
  completeness in logic?

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Cost variables

• Time number of nodes generated
• Space maximum number of nodes stored in memory
• Branching factor b
• Maximum number of successors of any node
• Depth d
• Depth of shallowest goal node
• Path length m
• Maximum length of any path in the state space

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ComplexityBFS vs. DFS

• Not optimal
• Time 1bb2b3bmO(bm)
• Space O(bm)Space is linear
• Complete in finite spaces fails in infinite depth

• Optimal
• Time 1bb2b3bd(bd1-1) O(bd1)
• Space O(bd1)Space is the big problem
• Complete if b is finite

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Informed Search

• Best-first
• A
• Greedy
• Hill climbing
• Variants
• Randomness, Simulated annealing, Local beam
  search,
• Online search will not be on midterm

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Greedy Search

• OPEN start node CLOSED empty
• While OPEN is not empty do
• Remove leftmost state from OPEN, call it X
• If X goal state, return success
• Put X on CLOSED
• SUCCESSORS Successor function (X)
• Remove any successors on OPEN or CLOSED
• Compute heuristic function for each node
• Put remaining successors on either end of OPEN
• Sort nodes on OPEN by value of heuristic function
• End while

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A Search

• Try to expand node that is on least cost path to
  goal
• Evaluation function f(n)
• f(n)g(n)h(n)
• h(n) is heuristic function cost from node to
  goal
• g(n) is cost from initial state to node
• f(n) is the estimated cost of cheapest solution
  that passes through n
• If h(n) is an underestimate of true cost to goal
• A is complete
• A is optimal
• A is optimally efficient no other algorithm
  using h(n) is guaranteed to expand fewer states

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Admissable heuristics

• A heuristic that never overestimates the cost to
  the goal
• h1 and h2 are admissable heuristics
• Consistency the estimated cost of reaching the
  goal from n is no greater than the step cost of
  getting to n plus estimated cost to goal from n
• h(n) ltc(n,a,n)h(n)

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Local Search Algorithms

• Operate using a single current state
• Move only to neighbors of the state
• Paths followed by search are not retained
• Iterative improvement
• Keep a single current state and try to improve it

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Steepest Ascent
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Problems for hill climbing

• When the higher the heuristic function the
  better maxima (objective fns) when the lower
  the function the better minima (cost fns)
• Local maxima A local maximum is a peak that is
  higher than each of its neighboring states, but
  lower than the global maximum
• Ridges a sequence of local maxima
• Plateaux an area of the state space landscape
  where the evaluation function is flat

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Some solutions

• Stochastic hill-climbing
• Chose at random from among the uphill moves
• First-choice hill climbing
• Generates successors randomly until one is
  generated that is better than current state
• Random-restart hill climbing
• Keep restarting from randomly generated initial
  states, stopping when goal is found
• Simulated annealing
• Generate a random move. Accept if improvement.
  Otherwise accept with continually decreasing
  probability.
• Local beam search
• Keep track of k states rather than just 1

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Constraint Propagation
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CSP algorithm

• Depth-first search often used
• Initial state the empty assignment all
  variables are unassigned
• Successor fn assign a value to any variable,
  provided no conflicts w/constraints
• All CSP search algorithms generate successors by
  considering possible assignments for only a
  single variable at each node in the search tree
• Goal test the current assignment is complete
• Path cost a constant cost for every step

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Local search

• Complete-state formulation
• Every state is a compete assignment that might or
  might not satisfy the constraints
• Hill-climbing methods are appropriate

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General purpose methods for efficient
implementation

• Which variable should be assigned next?
• in what order should its values be tried?
• Can we detect inevitable failure early?
• Can we take advantage of problem structure?

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Order

• Choose the most constrained variable first
• The variable with the fewest remaining values
• Minimum Remaining Values (MRV) heuristic
• What if there are gt1?
• Tie breaker Most constraining variable
• Choose the variable with the most constraints on
  remaining variables

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Order on value choice

• Given a variable, chose the least constraining
  value
• The value that rules out the fewest values in the
  remaining variables

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Forward Checking

• Keep track of remaining legal values for
  unassigned variables
• Terminate search when any variable has no legal
  values

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Game Playing

• Minimax
• Alpha-beta pruning
• Evaluation function (what is the difference
  between a cost function, a utility function, a
  heuristic function, an evaluation function?)

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Logic

• Model checking
• Forward and backward chaining
• And-or trees
• Resolution theorem proving

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Knowledge Representation

• Semantic nets
• Frames
• KL-one
• Ontological commitments
• Inheritance
• Properties you want to represent generics vs
  instants