2. KU NLP Structures and Strategies for State Space Search33 3.2.3Depth-First and Breadth-First Search q A search algorithm must determine the order in which states are examined. q Breadth-first search explores the space  A, B, C, D, E, F, G, H, I, J, K, L, M, N, O, P, Q, R, S, T, U (Figure 3.13, p100, tp34)  Figure 3.15 (p 103, tp37) q Depth-first search goes deeper into the search space whenever this is possible.  A, B, E, K, S, L, T, F, M, C, G, N, H, O, P, U, D, I, Q, J, R (Figure 3.13, p100, tp34)  Figure 3.17 (p 105, tp42)

3. KU NLP Structures and Strategies for State Space Search34 Search Example (Fig. 3.13 p100)

4. KU NLP Structures and Strategies for State Space Search35 Breadth-First Search (1) Procedure breadth_first_search; begin open := [Start]; closed := []; while open  [] do begin remove leftmost state from open, call it X; if X is a goal then return(success) else begin generate children of X; put X on closed; /* loop check */eliminate children of X on open or closed; /* queue */ put remaining children on right end of open end return(failure) end.

5. KU NLP Structures and Strategies for State Space Search36 Breadth-First Search(2) 1. open=[A]; closed=[] 2. open=[B,C,D]; closed=[A] 3. open=[C,D,E,F]; closed=[B,A] 4. open=[D,E,F,G,H]; closed=[C,B,A] 5. open=[E,F,G,H,I,J]; closed=[D,C,B,A] 6. open=[F,G,H,I,J,K,L]; closed=[E,D,C,B,A] 7. open=[G,H,I,J,K,L,M] (as L is already on open); closed=[F,E,D,C,B,A] 8. open=[H,I,J,K,L,M,N]; closed=[G,F,E,D,C,B,A] 9. And so on until either U is found or open=[]

6. KU NLP Structures and Strategies for State Space Search37 Breadth first search (3)

7. KU NLP Structures and Strategies for State Space Search38 Breadth-First Search (4) q OPEN lists states that have been generated but whose children have not been examined. q CLOSED records states that have already been examined. q OPEN is maintained as a queue, FIFO data structure.  States are added to the right of the list and removed from the left q Breadth-first search is guaranteed to find the shortest path from the start state to the goal. q If the path is required for a solution, we can store ancestor information along with each state.  open = [(D,A), (E,B), (F,B), (G,C), (H,C)]  closed = [(C,A), (B,A), (A,nil)]

8. KU NLP Structures and Strategies for State Space Search39 Depth-First Search (1) Procedure depth_first_search; begin open := [Start]; closed := []; while open  [] do begin remove leftmost state from open, call it X; if X is a goal then return(success) else begin generate children of X; put X on closed; eliminate children of X on open or closed; put remaining children on left end of open end return(failure) end.

9. KU NLP Structures and Strategies for State Space Search40 Depth-First Search (2) 1. open=[A]; closed=[] 2. open=[B,C,D]; closed=[A] 3. open=[E,F,C,D]; closed=[B,A] 4. open=[K,L,F,C,D]; closed=[E,B,A] 5. open=[S,L,F,C,D]; closed=[K,E,B,A] 6. open=[L,F,C,D]; closed=[S,K,E,B,A] 7. open=[T,F,C,D]; closed=[L,S,K,E,B,A] 8. open=[F,C,D]; closed=[T,L,S,K,E,B,A] 9. open=[M,C,D], as L is already on closed; closed=[F,T,L,S,K,E,B,A] 10. open=[C,D]; closed=[M,F,T,L,S,K,E,B,A] 11. open=[G,H,D]; closed=[C,M,F,T,L,S,K,E,B,A]

10. KU NLP Structures and Strategies for State Space Search41 Depth-First Search (3) q OPEN is maintained as a stack, or LIFO data structure.  The state are both added and removed from the left end of OPEN. q is not guaranteed to find the shortest path.

11. KU NLP Structures and Strategies for State Space Search42 Depth first search (4)

12. KU NLP Structures and Strategies for State Space Search43 Depth-First vs. Breadth-First  The choice depends on the specific problem being solved.  Significant features include the importance of finding the shortest path, the branching of the state space, the available time and space resources, the average length of paths to a goal node, and whether we want all solutions or only the first solution.  Breadth-First  always finds the shortest path to a goal node.  If there is a bad branching factor, the combinatorial explosion may prevent the algorithm from finding a solution.  Depth-First  may not waste time searching a large number of shallow state in the graph.  can get lost deep in a graph missing shorter paths to goal.

13. KU NLP Structures and Strategies for State Space Search44 3.3 Using the State Space to Represent Reasoning with the Predicate Calculus q 3.3.1 State Space Description of a Logical System q 3.3.2 AND/OR Graphs q 3.3.3 Further Examples and Applications  MACSYMA (integration)  Where is Fred?  The Financial Advisor  English Grammar

14. KU NLP Structures and Strategies for State Space Search45 3.3.1 State Space Description of a Logical System (p107) q Predicate calculus can be used as the formal specification language for making nodes distinguishable as well as for mapping the nodes of a graph onto the state space. q Inference rules can be used to create and describe the arcs between states. q Problems in the predicate calculus, such as determining whether a particular expression is a logical consequence of a given set of assertions, may be solved using search.

15. KU NLP Structures and Strategies for State Space Search46 Example 3.3.1 propositional calculus (1) q A set of assertions : q  p; r  p; v  q; s  r; t  r; s  u; s; t; q State space graph of a set of implications p qr u v  the arcs correspond to logical implications (  )  propositions given true (s and t) correspond to the given data of the problem ts

16. KU NLP Structures and Strategies for State Space Search47 Example 3.3.1 propositional calculus (2) q Propositions that are logical consequences of the given set of assertions correspond to the nodes that may be reached along a directed path from a state representing a true proposition.  [s,r,p] corresponds to the sequence of inferences: s and s  r yields r. r and r  p yields p. q Determining whether a given proposition is a logical consequence of a set of propositions becomes a problem of finding a path from a boxed node to the goal node.

17. KU NLP Structures and Strategies for State Space Search48 3.3.2 And/Or graphs (1)  If the premises of an implication are connected by an  operator, they are called AND nodes, and the arcs from this node are joined by a curved link.  q  r  p is represented by q And/Or graph is actually a specialization of a type of graph known as a hypergraph, which connects nodes by sets of arcs.

18. KU NLP Structures and Strategies for State Space Search49 3.3.2 And/Or graphs (2) q Definition : HYPERGRAPH  A hypergraph consists of : N, a set of nodes. H, a set of hyperarcs.  Hyperarcs are also known as k-connections, where K is the cardinality of the set of descendant nodes. If k=1, OR node, If k>1, AND nodes.

19. KU NLP Structures and Strategies for State Space Search50 3.3.2 AND/OR graphs (3)

20. KU NLP Structures and Strategies for State Space Search51 And/Or graph Search   operator(and nodes) indicates a problem decomposition in which the problem is broken into subproblems such that all of the subproblems must be solved to solve the original problem.   operator indicates a selection, a point at which a choice may be made between alternative problem-solving strategies.

21. KU NLP Structures and Strategies for State Space Search52 Example 3.3.3: Integration (1) q One example of an and/or graph is a program for symbolically integrating mathematical functions. (Fig. 3.22, p112, tp53) q In performing integrations, break an expression into sub-expressions that may be integrated independently and then combine the results algebraically into one solution expression.  Decomposition of a problem into independent subproblems can be represented by AND nodes in the graph. q Simplification of an expression can be done by various algebraic substitutions  A number of different substitutions are represented by OR nodes of the graph. q An obvious example of Goal-directed search

22. KU NLP Structures and Strategies for State Space Search53 Example 3.3.3: Integration (2)

23. KU NLP Structures and Strategies for State Space Search54 Example 3.3.4: “Where is fred?”(1) q Given facts and rules 1. collie(fred). 2. master(fred, sam). 3. day(saturday). 4.  (warm(saturday)). 5. trained(fred). 6.  X[spaniel(X)  (collie(X)  trained(X))  gooddog(X)] 7.  (X,Y,Z)[goodog(X)  master(X,Y)  location(Y,Z)  location(X,Z)] 8. day(saturday)  warm(saturday)  location(sam, park). 9. day(saturday)   (warm(saturday))  location(sam, museum).

24. KU NLP Structures and Strategies for State Space Search55 Example 3.3.4: “Where is fred?”(2)

25. KU NLP Structures and Strategies for State Space Search56 Example 3.3.4: “Where is fred?”(3) q To determine Fred’s location  caluse 7: location(fred, Z)  The premises of the rule 7 must be proved.  gooddog(fred)  master(fred,Y)  location(Y,Z).  rule 6 & fact 1 & fact 5  gooddog(fred) is proved.  master(fred,sam) by fact 2. {sam/Y}  location(sam, Z) must be proved.  Rule 7 is failed (because gooddog(sam) is not true)  rule 8 is failed. (because warm(saturday) is not true)  rule 9 & fact 3 & fact 4  location(sam, museum)  Location(fred, museum) q Goal driven search of the AND/OR graph

26. KU NLP Structures and Strategies for State Space Search57 Example 3.3.5: Investment Advice q an example of goal directed depth first search with backtracking q Goal: investment(X)  try investment(savings).  must prove savings_account(inadequate).  amount_saved(X)  dependents(Y)   greater(X, minsavings(Y))  (given X=20,000, Y=2)  greater(X,minsavings(Y)) is false.  backtrack to investment(stocks).  savings_account(adequate)  income(adequate).  both are proved true.

27. KU NLP Structures and Strategies for State Space Search58 Example 3.3.5: Investment Advice

28. KU NLP Structures and Strategies for State Space Search59 Example 3.3.6: English Grammar(1) q A set of rewrite rules for parsing sentences can be represented by AND/OR graph. q The rules are used to parse sequence of words, i.e. to determine whether they are well-formed sentences. q Simple subset of English grammar: 1. Sentence  NP VP6. ART  a 2. NP  N7. ART  the 3. NP  ART N8. N  man 4. VP  V9. N  dog 5. VP  V NP10. V  likes 11. V  bites

29. KU NLP Structures and Strategies for State Space Search60 Example 3.3.6: English Grammar(2)

30. KU NLP Structures and Strategies for State Space Search61 Example 3.3.6: English Grammar(3) q AND/OR graph(Fig. 3.25, tp60) define rewrite rules  AND nodes correspond to RHS of the rule.  Multiple rules with the same conclusion from the OR node. q An expression is well formed in a grammar if it consists entirely of terminal symbols and there is a series of substitutions in the expression using rewrite rules that reduce it to the SENTENCE symbol. q A date-driven parsing algorithm parse the input sentence constructing the parse tree.  “The dog bites the man.” (Fig. 3.26, p119, tp62) q Rewrite rules are used to generate legal sentences.

31. KU NLP Structures and Strategies for State Space Search62 Example 3.3.6: English Grammar(4)