Logic Programming (Control and Backtracking)

1. Logic Programming (Control and Backtracking)

2. Contents • Logic Programming • sans Control • with Backtracking • Streams

3. Example • (append empty x x) ← • (append (cons w x) y (cons w z)) ← (append x y z) • (append q r (cons 1 (cons 2 empty))) • Unify: (append empty x x) • { q |→ empty, r |→ (cons 1 (cons 2 empty))

4. Example • (append empty x x) ← • (append (cons w x) y (cons w z)) ← (append x y z) • (append q r (cons 1 (cons 2 empty))) • Unify: (append (cons w x) y (cons w z)) • (append x r (cons 2 empty))

5. Example • (append empty x x) ← • (append (cons w x) y (cons w z)) ← (append x y z) • (append x r (cons 2 empty)) • Unify: (append empty x x) • { x |→ empty, r |→ (cons 2 empty) }

6. Example • (append empty x x) ← • (append (cons w x) y (cons w z)) ← (append x y z) • (append x r (cons 2 empty)) • Unify: (append (cons w x) y (cons w z)) • (append x r empty)

7. Example • (append empty x x) ← • (append (cons w x) y (cons w z)) ← (append x y z) • (append x r empty) • Unify: (append empty x x) • { x |→ empty, r |→ empty }

8. Example • (append empty x x) ← • (append (cons w x) y (cons w z)) ← (append x y z) • (append x r empty) • !Unify: (append (cons w x) y (cons w z))

9. Logic Programming • solve • Input: List of Goals • KB: List of Rules • Output: List of mappings • Problem: • Keeping variables straight

10. Logic Programming • Keeping variables straight • Variable renaming • Replace: • (append empty x x) ← • With: • (append empty v0 v0) ← • Instantiate as needed

11. Logic Programming (define solve (lambda (goals) (if (null? goals) <then-expression> <else-expression>)))

12. Logic Programming (define solve (lambda (goals) (if (null? goals) <found a solution> <try to find rule for first goal>))) • Fake Problem • Output solution when found

13. Logic Programming (define solve (lambda (goals soln) (if (null? goals) (add-to-answer soln) <try to find all rules for first goal>)))

14. Logic Programming (define solve (lambda (goals soln) (if (null? goals) (add-to-answer soln) (for-each (lambda (x) (let ((u (unify (car goals) (head x)))) (if u <call to solve>)))))))

15. Logic Programming (define solve (lambda (goals soln) (if (null? goals) (add-to-answer soln) (for-each (lambda (x) (let ((u (unify (car goals) (head x)))) (if u (solve (append (cdr goals) (inst (tail x) u)) (append soln u)))))))))

16. Backtracking • Problem • Want to return solution

17. Backtracking • Problem • Want to return solution (define solve (lambda (goals soln k) (if (null? goals) (k soln) …)))

18. Backtracking • Problem • Want to return solution *and be able to continue* (define solve (lambda (goals soln k) (if (null? goals) (backtrack k soln) …)))

19. Backtracking • Problem • Want to return solution *and be able to continue* (define backtrack (lambda (k soln) (call-with-current-continuation (lambda (x) …))))

20. Backtracking • Problem • Want to return solution *and be able to continue* (define backtrack (lambda (k soln) (call-with-current-continuation (lambda (x) (k (cons soln x))))))

21. Backtracking • Problem • Want to return solution, be able to continue *and return other solutions* • Current “solution” will use the old continuation to return answers

22. Backtracking (define solve-k (lambda (x) x)) (define solve (lambda (goals soln) (if (null? goals) (backtrack soln) …))) (define backtrack (lambda (soln) (call-with-current-continuation (lambda (x) (solve-k (cons soln x))))))

23. Backtracking • Problem • Need to return 0 argument function • When executed, function should get the current-continuation (for return continuation) • Must be able to overwrite old return continuation

24. Backtracking (define backtrack (lambda (soln) (set! solve-k (call-with-current-continuation (lambda (x) …)))))

25. Backtracking (define backtrack (lambda (soln) (set! solve-k (call-with-current-continuation (lambda (x) (solve-k (cons soln …)))))))

26. Backtracking (define backtrack (lambda (soln) (set! solve-k (call-with-current-continuation (lambda (x) (solve-k (cons soln (lambda () (call-with-current-continuation (lambda (y) (x y)))))))))))

27. Backtracking • Want to return answer ASAP • Pass your answer forward • Need return continuation • Will need to be changed with every continue • Make it global • Remember to return continuation • (cons <answer> <continue>)

28. Backtracking • Book • Control entirely handled by continuations • This • Mixed control: Backtracking handled by continuations, function control flow handled by Scheme call stack

29. Backtracking • Book • sk (success continuation) is solve-k • What to do with answers • fk (failure continuation) has no direct equivalent • Essentially says, here’s what to do next • Is related to for-each statements in code

30. Streams • Infinite Lists • (define x (cons 1 1)) • (set-cdr! x x) • (eq? x (cdr x)) → #t • (car x) → 1

31. Streams • Infinite List • Consecutive numbers • Primes • Digits of pi • Random numbers

32. Streams • stream = (cons <value> <func:→ stream>) • Very similar to backtracking returns • (car x) = (car x) • (cdr x) = ((cdr x))