1. Control Algorithms 2 Chapter 6 Control Algorithms 2 Chapter 6 Production Systems

2. A Model of Computation Emil Post (40’s): production systems as a formal theory of computation. Equivalent to a Turing machine. Set of rewrite rules for strings. Newell and Simon (60’s, 70’s, 80’s): General Problem Solver John Anderson, Newell and Simon (80’s): learning models, ACT*, SOAR Everyone (80’s): Expert systems

3. Components 1. Set of rewrite rules S  NP VP LHS: Condition Part RHS: Action Part

4. Components 2. Working Memory --Contains the current state of the world --Contains pattern that is matched against the condition of the production --When a match occurs, an action is performed

5. Components 3. Recognize-Act Cycle --Isolate a subset of productions whose conditions match patterns in working memory: conflict set --Choose one of them ---Fire ---Change contents of working memory --Stop when there are no matches

6. Example: Production system to generate the set of palindromes over the alphabet {0,1} Productions 1. N  0N0 2. N  1N1 3. N  0 4. N  1 5. N  λ IterationWorking MemoryConflict SetFired 0N1,2,3,4,51 10N01,2,3,4,51 200N001,2,3,4,52 3001N1001,2,3,4,53 40010100

7. Knight’s Tour As a Production System Given a 3X3 matrix What squares can a knight land on What values of X, Y satisfy mv(X,Y) X,Y are elements of {1,2,…,9) 12 3 456 7 89 1. mv(1,8) 7. mv(4,9) 13. mv(8,3) 2. mv(1,6) 8. mv(4,3) 14. mv(8,1) 3. mv(2,9) 9. mv(6,1) 15. mv(9,2) 4. mv(2,7) 10. mv(6,7) 16. mv(9,4) 5. mv(3,4) 11. mv(7,2) 6. mv(3,8) 12. mv(7,6)

8. The General Case

9. Changes 1. Every expression of the form mv(x,y) becomes on(x)  on(y) 2. Use no path expression 3. Working memory is the current state and goal state 4. Conflict set is the set of rules that match the current state 5. Apply all rules until the current state equals the goal state

10. Productions 1. mv(1,8) 7. mv(4,9) 13. mv(8,3) 2. mv(1,6) 8. mv(4,3) 14. mv(8,1) 3. mv(2,9) 9. mv(6,1) 15. mv(9,2) 4. mv(2,7) 10. mv(6,7) 16. mv(9,4) 5. mv(3,4) 11. mv(7,2) 6. mv(3,8) 12. mv(7,6) 1. on(1) -> on(8)7. on(4) -> on(9)13. on(8) -> on(3) 2. on(1) -> on(6)8. on(4) -> on(3)14. on(8) -> on(1) 3. on(2) -> on(9)9. on(6) -> on(1)15. on(9) -> on(2) 4. on(2) -> on(7)10. on(6) -> on(7)16. on(9) -> on(4) 5. on(3) -> on(4)11. on(7) -> on(2) 6. on(3) -> on(8)12. on(7) -> on(6)

11. Can We Get from 1 to 2? Iteration--Working Memory-- Conflict Set Fired CurrentGoal 0121,21 18213,1413 2325,65 3427,87 49215,1615 522Halt

12. Pattern Search path(1,2) {1/x,2/y} mv(1,z)^path(z,2) {8/z} mv(1,8)^path(8,2) mv(8,z)^path(z,2) {3/z} mv(8,3)^path(3,2) mv(3,z)^path(z,2) {4/z} mv(3,4)^path(4,2) mv(4,z)^path(z,2) {9/z} mv(4,9)^path(9,2) mv(9,z)^path(z,2) {2/z} mv(9,2)^path(2,2) t Now look at working memory in the production system

13. Equivalences Production SystemPattern Search Productionsmv working memorypath(X,Y) Fire lowest numbered production Choose first rule that unifies Conclusion: Production Systems and pattern search are almost equivalent

14. Almost? Production systems have no loop detection mechanism Solution: Invent two new productions 1. assert(X) causes X to be stored in WM 2. been(X) is T if X has been visited 3. assert(been(X)) records in wm that we’ve already visited X

15. Can be expressed in PC notation like this

16. Can We Get from 1 to 7? Iteration--Working Memory-- Conflict Set Fired Current Goal been 01 711,21 18 7813,1413 23 735,65 34 747,87 49 7915,1615 52 723,43 (firing 3 causes been(9) to fail) 2724427244 777777 Notice that this search is data driven

17. Can Also Be Goal Driven Instead of starting with current state=1 and goal = 7 Start with current state = 7 and goal = 1

18. Works great for a 3x3 matrix What about 8x8? Either enumerate all moves or encode them 8 possible situations 1. d(2),r(1)5. u(2),r(1) 2. d(2),l(1)6. u(2),l(1) 3. d(1),r(2)7. u(1),r(2) 4. d(1),l(2)8. u(1),l(2)

19. Not applicable everywhere Situation have preconditions: Pre: row <=6, col <=7 Situation 1: d(2),r(1) Requires 4 new functions sq(r,c) returns cell number, left to right, top to bottom where r is row number, c is column number plus(r,2) returns r + 2 eq(X,Y) T if X = Y lte(X,Y) T if X<=Y

20. Encoding of situation 1: d(2),r(1) mv(sq(R,C),sq(Nr,Nc))  lte(R,6)^eq(Nr,plus(R,2)) ^lte(C,7) ^eq(Nc,plus(c,1) There are 7 more just like this

21. Control Loop for Knight’s Tour

22. Strength of Production Systems 1. Said to model human cognition 2. Separation of knowledge from control 3. Natural mapping onto state space search 4. Modularity of production rules 5. Simple Tracing and explanation— compare a rule with a line of c++ code 6. Language independence

23. But Production systems are easily rendered in prolog We’ll consider several versions of the knight’s tour

24. Global Record of Squares Visited Global Record of Squares Visited

25. Put Squares Visited on a List Put Squares Visited on a List

26. Use a Stack to Display Path to Goal Use a Stack to Display Path to Goal

27. Use of Queue to Keep Track of Visited Squares Use of Queue to Keep Track of Visited Squares

28. Display Backtracking Display Backtracking Try: path(1,W) ◦1 is on the stack ◦Base case succeeds: w = 1 ◦Now path1 is invoked, Start = 1 ◦Move(1,6) succeeds, stack: [6,1] ◦Base case succeeds: w = 6, path is 1,6 ◦Now path1 invoked:, Start = 6 ◦Move (6,1) fails because 1 has been visited ◦Move(6,7) succeeds: stack:[7,6,1] ◦Base case succeeds: w = 7, path is 1,6,7 ◦This process continues until all possible states have been visited ◦Occurs when path is 1,6,7,2,9,4,3,8 ◦Now we backtrack from 8 ◦But everything has been tried until we return to second move predicate starting with 1 (move(1,8)) ◦Now we go through the entire process again

29. Cut ! ◦Always succeeds the first time it is encountered ◦When backtracked to, it causes the entire goal in which it was contained to fail Without !Without ! (4 2 path moves) With !With ! (2 2 path moves, because the first move cannot be backtracked to)

30. Farmer Problem A farmer (f) has a dog (d), a goat (g),and a cabbage (c) A river runs North and South The farmer has a boat that can hold only the farmer and one other item Without the farmer ◦The goat will eat the cabbage ◦The dog will eat the goat How does the farmer (and his cohort) cross the river

31. Define a predicate: state(F,D,G,C) Where F,D,G,C can be set to e or w indicating the side of the river each is on.

32. As State-Space st(w,w,w,w) st(e,e,w,w) st(e,w,e,w) s(e,w,w,e) st(w,w,e,w) s(e,e,e,w) st(e,w,e,e) etc.

33. Constructing a Move Predicate st(e,e,-,-)  st(w,w,-,-) Means Farmer and dog went from east to west Can be rewritten: mv(st(X,X,G,C),st(Y,Y,G,C))

34. Facts opp(e,w) opp(w,e) Giving mv(st(X,X,G,C),st(Y,Y,G,C)) :- opp(X,Y). opp(e,w). opp(w,e).

35. In Motion Suppose: st(e,e,G,C) mv(st(X,X,G,C), st(Y,Y,G,C)) :- opp(X,Y). opp(e,w). opp(w,e). w,w mv(st(e,e,G,C),st(Y,Y,G,C)) opp(X,Y) {e/X, w/Y) opp(e,w) This is one of 4 move predicates (3 items to move + return trip alone)

36. Unsafe Goat and cabbage are together ◦unsafe(st(X,D,Y,Y)) if x != y ◦unsafe(st(X,D,Y,Y)) :- opp(X,Y) Dog and goat are together ◦Unsafe(st(X,Y,Y,C) if x != y ◦Unsafe(st(X,Y,Y,C) :- opp(X,Y)

37. Never move to an unsafe state mv(st(X,X,G,C),st(Y,Y,G,C)) :- opp(X,Y), not(unsafe(st(Y,Y,G,C))).

38. Traversal Mechanism Use the stack mechanism from Knight3 Farmer Problem