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)