Artificial Intelligence 15381 Heuristic Search Methods

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Artificial Intelligence 15-381Heuristic Search
Methods

• Jaime Carbonell
• jgc_at_cs.cmu.edu
• 17 January 2002
• Today's Agenda
• Island Search Complexity Analysis
• Heuristics and evaluation functions
• Heuristic search methods
• Admissibility and A search
• B search (time permitting)
• Macrooperators in search (time permitting)

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Complexity of Search

• Definitions
• Let depth d length(min(s-Path(S0, SG)))-1
• Let branching-factor b Ave(Succ(Si))
• Let backward branching B Ave(Succ-1(Si))
  usually bbb, but not always
• Let C(ltmethodgt,b,d) max number of Si visited
• C(ltmethodgt,b,d) worst-case time complexity
• C(ltmethodgt,b,d) lt worst-case space complexity

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Complexity of Search

• Breadth-First Search Complexity
• C(BFS,b,d) ?i0,dbi O(bd)
• C(BBFS,b,d) ?i0,dBi O(Bd)
• C(BiBFS,b,d) 2?i0,d/2bi O(bd/2), if bB
• Suppose we have k evenly-spaced islands in
  sPath(S0, SG), then
• C(IBFS,b,d) (k1) ?i0,d/(k1)bi O(bd/(k1))
• C(BiIBFS,b,d) 2 (k1) ?i0,d/(2k2)bi
  O(bd/(2k2))

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Heuristics in AI Search

• Definition
• A Heuristic is an operationally-effective nugget
  of information on how to direct search in a
  problem space. Heuristics are only approximately
  correct. Their purpose is to minimize search on
  average.

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Common Types of Heuristics

• "If-then" rules for state-transition selection
• Macro-operator formation discussed later
• Problem decomposition e.g. hypothesizing islands
  on the search path
• Estimation of distance between Scurr and SG.
  (e.g. Manhattan, Euclidian, topological distance)
• Value function on each Succ(Scurr)
• cost(path(S0, Scurr)) Ecost(path(Scurr,SG))
• Utility value(S) cost(S)

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Heuristic Search

• Value function E(o-Path(S0, Scurr), Scurr, SG)
• Since S0 and SG are constant, we abbreviate
  E(Scurr)
• General Form
• Quit if done (with success or failure), else
• s-Queue F(Succ(Scurr),s-Queue)
• Snext ArgmaxE(s-Queue)
• Go to 1, with Scurr Snext

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Heuristic Search

• Steepest-Ascent Hill-Climbing
• F(Succ(Scurr), s-Queue) Succ(Scurr)
• No history stored in s-Queue, hence Space
  complexity max(b) O(1) if b is bounded
• Quintessential greedy search
• Max-Gradient Search
• "Informed" depth-first search
• Snext ArgmaxE(Succ(Scurr))
• But if Succ(Snext) is null, then backtrack
• Alternative backtrack if E(Snext)ltE(Scurr)

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Beyond Greedy Search

• Best-First Search
• BestFS(Scurr, SG, s-Queue)
• IF Scurr SG, return SUCCESS
• For si in Succ(Scurr)
• Insertion-sort(ltSi, E(Si)gt, s-Queue)
• IF s-Queue Null, return FAILURE
• ELSE return
• BestFS(FIRST(s-Queue), SG,
  TAIL(s-Queue))

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Beyond Greedy Search

• Best-First Search (cont.)
• F(Succ(Scurr)), s-Queue)
  Sort(Append(Succ(Scurr), Tail(s-Queue)),E(si))
• Full-breadth search
• "Ragged"fringe expansion
• Does BestFS guarantee optimality?

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Beyond Greedy Search

• Beam Search
• Best-few BFS
• Beam-width parameter
• Uniform fringe expansion
• Does Beam Search guarantee optimality?

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A Search

• Cost Function Definitions
• Let g(Scurr) actual cost to reach Scurr from S0
• Let h(Scurr) estimated cost from Scurr to SG
• Let f(Scurr) g(Scurr) h(Scurr)

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A Search Definitions

• Optimality Definition
• A solution is optimal if it conforms to the
  minimal-cost path between S0 and SG. If
  operators cost is uniform, then the optimal
  solution shortest path.
• Admissibility Definition
• A heuristic is admissible with respect to a
  search method if it guarantees finding the
  optimal solution first, even when its value is
  only an estimate.

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A Search Preliminaries

• Admissible Heuristics for BestFS
• "Always expand the node with min(g(Scurr))
  first." If Solution found, expand any Si in
  s-Queue where g(Si) lt g(SG)
• Find solution any which way. Then Best FS(Si)
  for all intermediate Si in solution as follows
• If g(S(1curr) gt g(SG) in previous, quit
• Else if g(S(1G lt g(SG), SolSol1, redo (1).

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A Better Admissible Heuristics

• Observations on admissible heuristics
• Admissible heuristics based only on look-back
  (e.g. on g(S)) can lead to massive inefficiency!
• Can we do better?
• Can we look forward (e.g. beyond g(Scurr)) too?
• Yes, we can!

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A Better Admissible Heuristics

• The A Criterion
• If h(Scurr) always equals or underestimates the
  true remaining cost, then f(Scurr) is admissible
  with respect to Best-First Search.
• A Search
• A Search BestFS with admissible f g h
  under the admissibility constraints above.

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A Optimality Proof

• Goal and Path Proofs
• Let SG be optimal goal state, and s-path (S0, SG)
  be the optimal solution.
• Consider an A search tree rooted at S0 with S1G
  on fringe.
• Must prove f(SG2) gt f(SG) and g(path(S0, SG)) is
  minimal (optimal).
• Text proves optimality by contradiction.

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A Optimality Proof

• Simpler Optimality Proof for A
• Assume s-Queue sorted by f.
• Pick a sub-optimal SG2 g(SG2) gt g(SG)
• Since h(SG2) h(SG) 0, f (SG2) gt f (SG)
• If s-Queue is sorted by f, f(SG) is selected
  before f(SG2)

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B Search

• Ideas
• Admissible heuristics for mono- and bi-polar
  search
• "Eliminates" horizon problem in game-trees more
  later
• Definitions
• Let Best(S) Always optimistic eval fn.
• Let Worst(S) Always pessimistic eval fn.
• Hence Worst(S) lt True-eval(S) lt Best(S)

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Basic B Search

• Basic B Search
• B(S) is defined as
• If there is an Si in SUCC(Scurr)
• s.t. For all other Sj in SUCC(Scurr), W(Si) gt
  B(Sj)
• Then select Si
• Else ProveBest (SUCC(Scurr)) OR DisproveRest
  (SUCC(Scurr)
• Difficulties in B
• Guaranteeing eternal pessimism in W(S) (eternal
  optimism is somewhat easier)
• Switching among ProveBest and DisproveRest
• Usually W(S) ltlt True-eval(S) ltlt B(S) (not
  possible to achieve W(Si) gt B(Sj)

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Macrooperators in Search

• Linear Macros
• Cashed sequence of instantiated operators
• If S0 ---opi? S1 ---opj?S2
• Then S0 opi,j? S2
• Alternative notation
• if opj(opi(S0)) S2, Then opi,j(S0) S2
• Macros can have any length, e.g. oi,j,k,l,m,n
• Key question do linear macoros reduce search?

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Macrooperators in Search

• Disjunctive Macros
• Iterative Macros

op2
op3
op1
op4
op7
op5
op6
opk,l,m,n
Cond (s-Hist,SG)
opi,j
YES
NO
opo,p,q