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Control Algorithms 1Chapter 6
Pattern Search
So Far
◦Problem solving: search through states◦Predicate calculus: medium for describing
states◦Sound inference: method for generating new
states◦Formal search techniques: BT,DF,BF◦Reducing search space
Heuristic search AB pruning Passed over stochastic methods: will be explored
in machine learning units
Next
◦Further search techniques that are part of AI Pattern search Production systems
Problem with Predicate Calculus
No mechanism for applying rules
Task--Develop a general search procedure for
predicate calculus by applying recursive search to a space of logical inferences
--Basis for prolog
Knight’s Tour
Given a 3X3 matrixOne move paths (place on board)
1 2 3
4 5 6
7 8 9
mv(1,8)mv(4,9) mv(8,3)
mv(1,6)mv(4,3) mv(8,1)
mv(2,9)mv(6,1) mv(9,2)
mv(2,7)mv(6,7) mv(9,4)
mv(3,4)mv(7,2)
mv(3,8)mv(7,6)
Two-Move Paths
),(2),(),(( yxpathyzmvzxmvzyx
(Place on board)
Three Move Paths
There is a three move path from x to y if there is a 1 move path from x to some state z and a two move path from z to y
),(3),(2),(( yxpathyzpathzxmvzyx
(Place on board)
Example: path3(1,4)
path3(1,4) {1/x,4/y) mv(1,z)^path2(z,4) prove 1st conjunct {8/z}mv(1,8)^path2(8,4) 1st conjunct is T, prove 2nd conjunct {8/x,4/y}
mv(8,z)^mv(z,4) prove 1st conjunct {3/z}mv(8,3)^mv(3,4) both conjuncts are TT
TT
What if mv(8,1) appeared before mv(8,3)?
path3(1,4) {1/x,4/y)mv(1,z)^path2(z,4) {8/z}mv(1,8)^path2(8,4) {8/x,4/y}
mv(8,z)^mv(z,4) {1/z} mv(8,1)^mv(1,4)
f backtrack mv(8,z)^mv(z,4) {3/z} mv(8,3)^mv(3,4) T
TT
To Generalize
We have 1 move paths, 2 move paths, three move paths and, in general,
),(),(),(( yxpathyzpathzxmvzyx
But how do we stop?
Base Case
)),(( xxpathx
Produces an endless loop
Suppose, again, that mv(8,1) appears before mv(8,3)
path(1,4)mv(1,z)^path(z,4) {8/z}mv(1,8)^path(8,4)
mv(8,z)^path(z,4) {1/z} mv(8,1)^path(1,4)
Solution
Global Closed List: If a path has been tried, don’t retry
path(1,4)mv(1,z)^path(z,4) {8/z}mv(1,8)^path(8,4)
mv(8,z)^path(z,4) {1/z} mv(8,1)^path(1,4)
Backtrack mv(8,z)^path(z,4) {3/z}
mv(8,3)^path(3,4) mv(3,z)^path(z,4) {4/z} mv(3,4)^path(4,4) T
T
TT
Leads To: Pattern Search
Goal DirectedDepth First Control StructureBasis of PrologGoal: return the substitution set that will
render the expression true.
Here’s the Surprise
Found on pp. 198-99We’ve been running the algorithm
informally all along
Six Cases: 1 - 3
1. If Current Goal is a member of the closed list--return F, Backtrack
2. If Current Goal unifies with a fact--Current Goal is T
3. If Current Goal unifies with a rule conclusion--apply unifying substitutions to the premise--try to prove premise
--if successful, T
Six Cases: 4 - 6
4. Current Goal is a disjunction--Prove each disjunct until you exhaust them or find one that is T.
5. If Current Goal is a conjunction--try to prove each conjunct--if successful, apply substitutions to other conjuncts--if unsuccessful, backtrack, trying new substitutions until they are exhausted
6. If Current Goal is negated (~p)--Try to prove p--If successful, current goal is F--If unsuccessful, current goal is T--In the algorithm, returned substitution set is {} when ~p is true, because the algorithm failed to find a substitution set that would make p true (i.e., ~p is T only when p is F)
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