1. 1 State-Space representation and Production Systems Introduction: what is State-space representation? (E.Rich, Chapt.2) Basis search methods. (Winston, Chapt.4 + Russel&Norvig) Optimal-path search methods. (Winston, Chapt.5 + Russel&Norvig) Advanced variants. (Rich, Chapt.3 + Russel&Norvig + Nilsson) Game Playing (Winston, Chapt.6 + Rich + Russel&Norvig )

2. 2 State-space representation: Introduction  What is state-space representation?  Which are the technical issues that arise in that context?

3. 3 Example: the 8-puzzle.  Given: a board situation for the 8-puzzle: 138 27 546 123 567 48  Problem: find a sequence of moves (allowed under the rules of the 8-puzzle game) that transform this board situation in a desired goal situation:

4. 4 State-space representation: general outline:  Select some way to represent states in the problem in an unambiguous way.  Formulate all actions that can be preformed in states:  including their preconditions and effects  == PRODUCTION RULES  Represent the initial state (s).  Formulate precisely when a state satisfies the goal of our problem.  Activate the production rules on the initial state and its descendants, until a goal state is reached.

5. 5 Initial issues to solve:  How to formulate production rules? (repr. 2 )  Ex.:  express how/when squares may be moved?  Or: express how/when the blank space is moved?  When is a rule applicable to a state? (matching)  How to formulate when the goal criterion is satified and how to verify that it is?  How/which rules to activate? (control) 138 27 546  How to represent states? (repr.1)  Ex.: using a 3 X 3 matrix

6. 6 The (implicit) search tree  Each state-space representation defines a search tree: 138 27 546 138 27 546 138 27 546 1 3 8 27 546 138 27 5 4 6 9!/2nodes! goal  But this tree is only IMPLICITLY available !!

7. 7 A second example: Chess  Problem: develop a program that plays chess (well). 1 2 3 4 5 6 7 8 A B C D E F G H  1. A way to represent board situations in an unambiguous way:  Ex.: List: (( king_black, 8, C), ( knight_black, 7, B), ( knight_black, 7, B), ( pawn_black, 7, G), ( pawn_black, 7, G), ( pawn_black, 5, F), ( pawn_black, 5, F), ( pawn_white, 2, H), ( pawn_white, 2, H), ( king_white, 1, E)) ( king_white, 1, E))

8. 8 Chess (2):  2. Describe the rules that represent allowed moves:  Ex.: 1 2 3 4 A B C D E F G H ( (pawn_white,2,x), (blank, 3, x), (blank, 4, x) ) add( ( pawn_white, 4, x) ), remove( (pawn_white, 2, x) )

9. 9 Chess (3):  3. Provide a way to check whether a rule is applicable to some state:  Ex.: Matching mechanism !! ( (pawn_white,2,x), (blank, 3, x), (blank, 4, x) ) add( ( pawn_white, 4, x) ), remove( (pawn_white, 2, x) ) List: (( king_black, 8, C), ( knight_black, 7, B), ( knight_black, 7, B), ( pawn_black, 7, G), ( pawn_black, 7, G), ( pawn_black, 5, F), ( pawn_black, 5, F), ( pawn_white, 2, H), ( pawn_white, 2, H), ( king_white, 1, E)) ( king_white, 1, E)) 1 2 3 4 5 6 7 8 A B C D E F G H

10. 10 Chess (4):  4. How to specify a state in which the goal is reached (= a winning state):  Ex.: win( black )  attacked( king_white ) and no_legal_move( king_white ) no_legal_move( king_white ) + similar definitions for: + similar definitions for: attacked( Piece )  … no_legal_move( Piece ) ...

11. 11 Chess (5):  5. A way to verify whether a winning state is reached.  Ex.: ?- win( black ). win( black )  attacked( king_white ) and no_legal_move( king_white ) no_legal_move( king_white )… Need a theorem prover ( e.g. Prolog) to verify that the state is a winning one.

12. 12 Chess (6).  6. The initial state. A program that is ABLE to play chess. The control problem ! Main focus of this entire chapter of the course !!  7. A mechanism that selects in each state an appropriate rule to apply.

13. 13 Chess (7). Move 1 Move 2 Move 3 Implicit search tree ~15 ~ (15) 2 ~ (15) 3 Need very efficient search techniques to find good paths in such combinatorial trees.

14. 14 Very many issues and trade-offs: 1. How to choose the rules? 2. Should we search through the implicit tree or through an implicit graph? 3. Do we need an optimal solution, or just any solution? ‘optimal path problems’ ‘optimal path problems’ 4. Can we decompose states into components on which simple rules can in an independent way? Problem reduction or decomposability Problem reduction or decomposability 5. Should we search forwards from the initial state, or backwards from a goal state?