Artificial Intelligence University Politehnica of Bucharest Adina Magda Florea

Lecture No. 2 Problem solving strategies Representing problem solution Basic search strategies Informed search strategies

1. Representing problem solution Symbolic structure Computational instruments Planning method / approach

1.1. State space representation state, state space, initial state, final state(s), operators (S i, O, S f ) Problem solution

8-puzzle

1.2 AND/OR graph representation Problem decomposition into sub-problems (P i, O, P e ) AND/OR graph Solved node Unsolvable node Problem solution

AND/OR graph OR AND

Towers of Hanoi

1.3 Equivalence of representations State space Problem decomposition

2. Basic search strategies Criteria Completeness Optimality Complexity Possibility to backtrack Informedness Conventions: unknown node, evaluated node, expanded node, OPEN, CLOSED

Computational costs of search

2.1. Uninformed search in state space Algorithm BREADTH:Breadth first search in state space 1.Init lists OPEN  {S i }, CLOSED  {} 2.if OPEN = {} then return FAIL 3.Remove first node S from OPEN and insert it in CLOSED 4.Expand node S 4.1.Generate all direct successors S j of node S 4.2.for each successor S j of S do Make link S j  S if S j is final state then i.Solution is (S j, S,.., S i ) ii.return SUCCESS Insert S j in OPEN, at the end 5.repeat from 2 end.

Features of breadth first search Previous algorithm for tree space not graph For graphs - Insert step 3’ 3’.if S  OPEN  CLOSED then repeat from 2 Depth first search in state space Depth of a node Ad(S i ) = 0, where S i is the initial state, Ad(S) = Ad(S p )+1, where S p is the predecessor node of S.

Algorithm DEPTH(AdMax): Depth first search in state space 1.Init lists OPEN  {S i }, CLOSED  {} 2.if OPEN = {} then return FAIL 3. Remove first node S from OPEN and insert it into CLOSED 3’. if Ad(S) = AdMax then repeat from 2 4.Expand node S 4.1.Generate all successors S j of node S 4.2.for each succesor S j of S do Set link S j  S if S j is final state then i.Solution is (S j,.., S i ) ii.return SUCCESS Insert S j in OPEN, at the beginning 5.repeat from 2 end.

Features of depth first search Iterative deepening for AdMax=1, Val do DEPTH(AdMax) Features of iterative deepening Bidirectional search Which strategy to choose ?

2.2. Uninformed search in AND/OR graphs Depth of a node Ad(S i ) = 0, where, S i is the node of the initial problem Ad(S) = Ad(S p ) + 1 if S p is node OR predecesor of node S, Ad(S) = Ad(S p ) if S p is node AND predecesor of node S.

Algorithm BREADTH-ANDOR:Breadth first search in AND/OR trees 1.Init lists OPEN  {S i }, CLOSED  {} 2.Remove first node S from OPEN and insert it into CLOSED 3.Expand node S 3.1.Generate all direct successors S j of node S 3.2.for each successor S j of S do Make link S j  S if S j is a set of at least 2 sub-problems then/* is an AND node*/ i.Generate all successors sub-problems S k j of S j ii.Set link S k j  S j iii.Insert nodes S k j in OPEN, at the end else Insert S j in OPEN, at the end

4.if no successor of S was generated in previous step (3) then 4.1. if S is terminal node labeled with a non-elementary problem then Label S unsolvable Label with unsolvable all nodes predecessor of S which become unsolvable because of S if node AND is unsolvable then return FAIL/* no solution */ Remove from OPEN all nodes which have unsolvable predecessors 4.2. else/* S is terminal node labeled with a non-solvable problem */ Label S solved Label with solved all nodes predecessor of S which become unsolvable because of S if node AND is solved then i. Build solution tree following the links ii. return SUCCESS/* Solution found */ Remove from OPEN all solved nodes and all nodes which have solved predecessors 5.repeat from 2 end.

2.3. Complexity of search strategies B – branching factor of the search space 8-puzzle Number of moves: 2 m for corners = 8 3 m for laterals = 12 4m for center  24 moves B = no. moves / no. pozitions of free square = 2.67 Number of moves : 1 m for corners = 4 2 m for laterals = 8 3m for center  15 moves  B = 1.67

Complexity of search strategies B - branching factor Root – B nodes, B 2 on level 2, etc. Number of states possible to be generated on a search level d is B d T – total number of states generated during a search, d – depth of solution node T = B + B 2 + … + B d = O(B d )

Number of generated nodes Breadth first search B + B 2 + … + B d + (B d+1 -B) = O(B d+1 ) B – branching factor, d – depth of solution Depth first search B - branching factor, m – maximum depth B*m+1 Iterative deepening search d*B+(d-1)*B 2 + … + (1)*B d = O(B d )

Complexity of search strategies CriterionLevelDepthLimited depth Iterative deepening Bidirectio nal TimeBdBd BmBm BlBl BdBd B d/2 SpaceBdBd B*mB*lBdBd B d/2 Optimality ? YesNo Yes Completen ess YesNo Yes if l  d Yes B – branching factor, d – solution depth, m – maximum depth of the tree, l –limit of search (AdMax)

3. Informed search strategies Use heuristic knowledge to incraese efficiency of search: Select which node to expand nex during search While expanding a node decide which successors to generate and which to ignore Remove from the search space some nodes that have previously been generated – prune the search space

3.1 Best-first search Evaluate the information that can be obtained by expanding a node and ist importance in guiding the search The quality of a node is estimated by the heuristic search function w(n) for node n hill climbing strategy best-first strategy

Algorithm BFS:Best-first in state space 1.Init lists OPEN  {S i }, CLOSED  {} 2.Compute w(S i ) and associate this value to S i 3.if OPEN = {} then return FAIL 4. Remove node S with minimum w(S) from OPEN and insert it in CLOSED 5.if S is final state then i.Solution is (S,.., S i ) ii.return SUCCESS 6.Expand node S 6.1. Generate all successors S j of node S 6.2. for each succesor S j of S do 6.2.1Compute w(S j ) and associate it to S j Set link S j  S

6.2.3.if S j  OPEN  CLOSED then insert S j in OPEN with associated w(S j ) else i. Be S’ j the copy of S j from OPEN or CLOSED ii. if w(S j ) < w(S’ j ) then - Remove link S’ j  S p, with S p pred. of S’ j - Set link S’ j  S, and update cost of S’ j to w(S j ) - if S’ j is in CLOSED - then remove S’ j from CLOSED and insert S’ j in OPEN iii. else ignore node S j 7.repeat from 3 end.

Cases Best-first strategy is a generalization of uninformed search strategies - Breadth first search w(S) = Ad(S) - Depth first search w(S) = -Ad(S) Uniform cost strategy Minimize the search effort – heuristic search w(S) = heuristic function

3.2 Optimal solutions: A*Algorithm w(S) becomes f(S) with 2 components: g(S), estimates the real cost g*(S) of the search path from S i to S, h(S), estimates the real cost h*(S) of the search path from S to S f. f(S) = g(S) + h(S) f*(S) = g*(S) + h*(S)

Components of A*

Computing f(S) Computation of g(S) Computation of h(S) Must be admissible A heuristic function is called admissible if for any state S, h(S)  h*(S). This definition gives the admissibility condition of function h and is used to define the property of admissibility of an algorithm A*.

A* admissibility Consider an algorithm A* which uses g and h compenents of f if (1)h satisfies the admissibility condition (2) for any 2 states S, S', where c > 0 is a constant and the cost is finite then A* algorithm is admissible, i.e., is guaranteed to find the path of minimal cost to solution. Completeness

Implementation of A* Modify the algorithm correspnding to the "best-first" strategy in state space as such: … 2.Compute w(S i )=g(S i ) + h(S i ) and associate this value to S i 3.if OPEN = {} then return FAIL - unmodified 4. Remove node S with w(S) minimum from OPEN and insert it into CLOSED - unmodified … else i. Be S’ j the copy of S j from OPEN or CLOSED ii. if g(S j ) < g(S’ j ) then …

The heuristic of A* Consider 2 algorithms A*, A1 and A2, with evaluation functions h 1 and h 2 admissibile, g 1 =g 2 It is said that A2 is more informed than A1 if for any state S with S  S f We say that h2 dominates h1 monotony

How we compute f 8-puzzle Traveling salesman h2(S) = the cost of the minimum spanning tree of unvisited cities until S

Missionaries and cannibals

Relaxing the admissibility condition of A* An heuristic function h is called  -admissible if with  > 0 An A* algorithm which uses an evaluation function f with a component h  -admissible will find a solution which has a cost greater than the cost of the optimal solution with at most . such an algorithm - an  -admissible algorithm and the solutions is called  -optimal solution.

Relaxing the admisibility condition of A* 8-puzzle

Recursive Best First Best first with linear space Recursive implementation Remembers the value f of the best alternate path which starts from any precedent node of the current node Finds the minimum cost solution if h is admissible but has a space complexity of O(B*d)

S1 S3 S8 S5 S2 S6 S7 S4 S9 S10S inf S1 S3 S8 S5 S2 S6 S7 S4 S12S inf

S1 S3 S8 S5 S2 S6 S7 S4 S9 S10S inf S14S S BestFR(S) Recursive Best First strategy /* returns the solution or FAIL */ BFR(Si, inf)

Algorithm BFR(S, f_lim): Recursive Best First strategy /* Returns a solution or FAIL and a new limit f_limit */ 1. if S final state then return S, f_lim 2. Generate all successors S j of S 3. if there are no successors then return FAIL, inf 4. for each successor Sj do f(S j )  max(g(S j ) + h(S j ), f(S)) 5. Best  S j min, node with minimal value f(S j ) among successors 6. if f(Best) > f_lim then return FAIL, f(Best) 7. Alternat  f(S j min2 ), the 2 nd smallest value f(S j ) 8. Rez, f(Best)  BFR(Best, min(f_lim, Alternat) 9. if Rez  FAIL then return Rez, f(Best) 10. repeat from step 5 end.