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Artificial Intelligence 15-381 Heuristic Search Methods

Artificial Intelligence 15-381 Heuristic Search Methods. Jaime Carbonell jgc@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)

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Artificial Intelligence 15-381 Heuristic Search Methods

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  1. Artificial Intelligence 15-381Heuristic Search Methods Jaime Carbonell jgc@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)

  2. 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 b=bb, but not always • Let C(<method>,b,d) = max number of Si visited C(<method>,b,d) = worst-case time complexity C(<method>,b,d) < worst-case space complexity

  3. Complexity of Search • Breadth-First Search Complexity • C(BFS,b,d) = i=0,dbi = O(bd) • C(BBFS,b,d) = i=0,dBi = O(Bd) • C(BiBFS,b,d) = 2i=0,d/2bi = O(bd/2), if b=B • Suppose we have k evenly-spaced islands in s=Path(S0, SG), then: C(IBFS,b,d) = (k+1) i=0,d/(k+1)bi = O(bd/(k+1)) C(BiIBFS,b,d) = 2 (k+1) i=0,d/(2k+2)bi = O(bd/(2k+2))

  4. Heuristics in AI Search • Definition A Heuristicis 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.

  5. 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)) + E[cost(path(Scurr,SG))] • Utility: value(S) – cost(S)

  6. 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:= Argmax[E(s-Queue)] • Go to 1, with Scurr:= Snext

  7. 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:= Argmax[E(Succ(Scurr))] • But if Succ(Snext) is null, then backtrack • Alternative: backtrack if E(Snext)<E(Scurr)

  8. Beyond Greedy Search • Best-First Search BestFS(Scurr, SG, s-Queue) IF Scurr = SG, return SUCCESS For si in Succ(Scurr) Insertion-sort(<Si, E(Si)>, s-Queue) IF s-Queue = Null, return FAILURE ELSE return BestFS(FIRST(s-Queue), SG, TAIL(s-Queue))

  9. 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?

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

  11. 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)

  12. A* Search Definitions • Optimality Definition A solution isoptimalif 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.

  13. 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) < g(SG) • Find solution any which way. Then Best FS(Si) for all intermediate Si in solution as follows: • If g(S(1curr) >= g(SG) in previous, quit • Else if g(S(1G < g(SG), Sol:=Sol1, & redo (1).

  14. 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!

  15. 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.

  16. 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 S1Gon fringe. • Must prove f(SG2) >= f(SG) and g(path(S0, SG)) is minimal (optimal). • Text proves optimality by contradiction.

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

  18. 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) < True-eval(S) < Best(S)

  19. 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) > 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) << True-eval(S) << B(S) (not possible to achieve W(Si) > B(Sj)

  20. 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?

  21. Macrooperators in Search • Disjunctive Macros • Iterative Macros op2 op3 op1 op4 op7 op5 op6 opk,l,m,n opi,j Cond (s-Hist,SG) YES NO opo,p,q

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