ku nlp structures and strategies for state space search33 3.2.3depth-first and breadth-first search...

Structures and Strategies for State Space Search

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3.2.3Depth-First and Breadth-First Search

A search algorithm must determine the order in

which states are examined.

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)

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)


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Search Example (Fig. 3.13 p100)


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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

endend

return(failure)end.


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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=[]


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Breadth first search (3)


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Breadth-First Search (4)

OPEN lists states that have been generated but whose children have not been examined.

CLOSED records states that have already been examined.

OPEN is maintained as a queue, FIFO data structure. States are added to the right of the list and removed from the left

Breadth-first search is guaranteed to find the shortest path from the start state to the goal.

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)]


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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

endend

return(failure)end.


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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]


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OPEN is maintained as a stack, or LIFO data

structure. The state are both added and removed from the left end of

OPEN.

is not guaranteed to find the shortest path.


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Depth first search (4)


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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.


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3.3 Using the State Space to Represent Reasoning with the Predicate Calculus

3.3.1 State Space Description of a Logical System

3.3.2 AND/OR Graphs

3.3.3 Further Examples and Applications MACSYMA (integration) Where is Fred? The Financial Advisor English Grammar


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3.3.1 State Space Description of a Logical System (p107)

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.

Inference rules can be used to create and

describe the arcs between states.

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.


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Example 3.3.1 propositional calculus (1)

A set of assertions : qp; rp; vq; sr; tr;

su; s; t;

State space graph of a set of implications

p

q r u

v

the arcs correspond to logical implications () propositions given true (s and t) correspond to the given dat

a of the problem

t s


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Example 3.3.1 propositional calculus (2)

Propositions that are logical consequences of the

given set of assertions correspond to the nodes t

hat 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 sr yields r.r and rp yields p.

Determining whether a given proposition is a logi

cal consequence of a set of propositions become

s a problem of finding a path from a boxed node t

o the goal node.


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3.3.2 And/Or graphs (1)

If the premises of an implication are connected b

y an operator, they are called AND nodes, and t

he arcs from this node are joined by a curved link

.

qrp is represented by

And/Or graph is actually a specialization of a type

of graph known as a hypergraph, which connects

nodes by sets of arcs.


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3.3.2 And/Or graphs (2)

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.


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3.3.2 AND/OR graphs (3)


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And/Or graph Search

operator(and nodes) indicates a problem decom

position in which the problem is broken into subp

roblems 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 proble

m-solving strategies.


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Example 3.3.3: Integration (1)

One example of an and/or graph is a program for symbolically integrating mathematical functions. (Fig. 3.22, p112, tp53)

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 b

e represented by AND nodes in the graph.

Simplification of an expression can be done by various algebraic substitutions A number of different substitutions are represented by OR nodes

of the graph.

An obvious example of Goal-directed search


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Example 3.3.3: Integration (2)


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Example 3.3.4: “Where is fred?”(1)

Given facts and rules1. 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).


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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)

Goal driven search of the AND/OR graph


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Example 3.3.5: Investment Advice

an example of goal directed depth first search wit

h backtracking

Goal: investment(X) try investment(savings). must prove savings_account(inadequate). amount_saved(X) dependents(Y) greater(X, minsaving

s(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.


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Example 3.3.5:InvestmentAdvice


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Example 3.3.6: English Grammar(1)

A set of rewrite rules for parsing sentences can

be represented by AND/OR graph.

The rules are used to parse sequence of words,

i.e. to determine whether they are well-formed

sentences.

Simple subset of English grammar:1. Sentence NP VP 6. ART a2. NP N 7. ART the3. NP ART N 8. N man4. VP V 9. N dog5. VP V NP 10. V likes11. V bites


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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.

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.

A date-driven parsing algorithm parse the input

sentence constructing the parse tree. “The dog bites the man.” (Fig. 3.26, p119, tp62)

Rewrite rules are used to generate legal sentences.


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Example 3.3.6:EnglishGrammar(4)

ku nlp structures and strategies for state space search33 3.2.3depth-first and breadth-first search...

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