Is there a search plan for the highest safe rung, with q tests in the worst case?

1. 2 HSR(n, k, q) Is there a search plan for the highest safe rung, with q tests in the worst case? 1 2 ... k n ... 2 1 0 (safe)

2. Search Plan A search plan is a tree made up of two kinds of nods x? yes no x Test node; do jars break when thrown from rung x? Conclusion; x is the highest safe rung

3. 2 minHSR(n, k, q) Is q the worst case minimum number of tests to be performed by a search plan for the highest safe rung? 1 2 ... k n ... 2 1 0 (safe)

4. 2 minHSR(n, 1, n) n rungs 1,3,..., n must be sequentially (a.k.a. linearly) tested. ... 2 1 0 (safe)

5. Linear Search Plan3 rungs, 3 jars 1? no yes 2? 3? q = 3 2 3 0 1

6. 2 minHSR(n, ⌈lg n⌉, lg n) n binary search is feasible. ... 2 1 0 (safe)

7. Binary Search Plan7 rungs, ⌈lg 7⌉ = 3 jars 4? yes no q = 3 6? 2? 7? 5? 3? 1? 6 7 4 5 2 3 0 1

8. 2 minHSR(n, k, q) 1 2 ... k n If 1<k<lg n,what is the search plan with the minimum number of tests? ... 2 1 0 (safe)

10. 2 1 2 ... K N What is the highest rung a jar can be safely thrown from? ... 2 1 (safe)

11. 2 N If K=1,rungs 2,3,..., N must be sequentially tested. ... 2 1 (safe)

12. 2 N If K>lg N,binary search is feasible. ... 2 1 (safe)

13. 2 N If1<K<lg N,what is the search plan with the minimum number of tests? ... 2 1 (safe)

14. Search Plan (a.k.a. Decision Tree) A search plan is a tree made up of two kinds of nods x? yes no x Test node; do jars break when thrown from rung x? Conclusion; x is the highest safe rung

15. Binary Search Plan8 rungs, lg 8 = 3 jars 5? Jar1 breaks Depth = 3 tests 7? 3? Jar 2 breaks 8? 6? 4? 2? Jar 3 breaks 7 8 5 6 3 4 1 2

17. 2 HSR(n, k, q) HSR(n, k, q) = ∃ p ∈ SearchPlans: correct(p, n, k, q). 1 2 ... k n ... 2 1 (safe)

19. What is the Formal Science Wikipedia (FSW) and How to Use it and Contribute to it?

20. Introduction to the Formal Science Wikipedia (FSW)

21. Agenda • What is FSW? • How to use FSW? • How to contribute to FSW?

22. What is FSW?

23. FSW vs. Wikipedia Lab (parameterized logical statement) Page(informal text) Develop an Avatar Edit Pages Code the lab Create a page

24. How to use FSW?

25. Investigation Cycle

26. How to contribute to FSW?(as a user)

27. 3 Toy Labs AxEyIneq(c ∈ [0,2]) := ∀x ∈ [0,1]: EyIneq(c, x) Given c ∈ [0,2], does every x ∈ [0,1] make EyIneq(c, x) hold? EyIneq(c ∈ [0,2], x ∈ [0,1]) := ∃y ∈ [0,1]: Ineq(c, x, y) Given c ∈ [0,2] and x ∈ [0,1], is there y ∈ [0,1] that makes Ineq(c, x, y) hold? Ineq(c ∈ [0,2], x ∈ [0,1], y ∈ [0,1]) := x + y > c Given c ∈ [0,2], x ∈ [0,1] and y ∈ [0,1], does the inequality x+y > c hold?

28. 3 Toy LabsCompact Notation AxEyIneq(c ∈ [0,2]) := ∀x ∈ [0,1]:EyIneq: ∃y ∈ [0,1]:Ineq: x + y > c Given c ∈ [0,2], does every x ∈ [0,1] has a y ∈ [0,1] such that x+y>c .

29. Developing Avatars Implement an interface derived from the logical statement.

30. An Avatar for the “Toy” Lab interface AxEyIneqAvatar { /** Given c ∈ [0,2] and x ∈ [0,1], which y ∈ [0,1] makes Ineq(c, x, y) hold? */ double move (EyIneq dummy, double c, double x); /** Given c ∈ [0,2], which x ∈ [0,1] makes EyIneq(c, x) NOT hold? */ double move (AxEyIneq dummy, double c); /** Given c ∈ [0,2], do all x ∈ [0,1] make EyIneq(c, x) hold? */ Position position (AxEyIneq dummy, double c); }

31. An Avatar for the “Toy” Lab AxEyIneq(c ∈ [0,2]) := ∀x ∈ [0,1]:EyIneq: ∃y ∈ [0,1]:Ineq: x + y > c EyIneq: ∃y ∈ [0,1]:Ineq: x + y > c /** Given c ∈ [0,2] and x ∈ [0,1], which y ∈ [0,1] makes Ineq(c, x, y) hold? */ double move (EyIneq dummy, double c, double x){ return 1; } Best strategy: y as big as possible

32. An Avatar for the “Toy” Lab AxEyIneq(c ∈ [0,2]) := ∀x ∈ [0,1]:EyIneq: ∃y ∈ [0,1]:Ineq: x + y > c ∀x ∈ [0,1]:EyIneq /** Given c ∈ [0,2], which x ∈ [0,1] makes EyIneq(c, x) NOT hold? */ double move (AxEyIneq dummy, double c){ return 0; } Best strategy: x as small as possible

33. An Avatar for the “Toy” Lab AxEyIneq(c ∈ [0,2]) := ∀x ∈ [0,1]:EyIneq: ∃y ∈ [0,1]:Ineq: x + y > c ∀x ∈ [0,1]:EyIneq /** Given c ∈ [0,2], do all x ∈ [0,1] make EyIneq(c, x) hold? */ Position position (AxEyIneq dummy, double c){ return c<1?Position.VERIFIER:Position.FALSIFIER; } AxEyIneq(c) is true for c ∈ [0,1) and false for c ∈ [1,2]

34. How Are Avatars Used?

36. How to contribute to FSW?(as a user)

37. Coding Labs Develop a set of nested Java classes

38. Coding the “Toy” Lab AxEyIneq(c) := publicclass AxEyIneq { @Paramdoublec; }

39. Coding the “Toy” Lab AxEyIneq(c) := ∀x publicclass AxEyIneq implements IForall{ @Paramdoublec; doublex; }

40. Coding the “Toy” Lab AxEyIneq(c) := ∀x :EyIneq: ∃y publicclass AxEyIneq implements IForall{ @Paramdoublec; doublex; publicclass EyIneq implements IExists{ doubley; } }

41. Coding the “Toy” Lab AxEyIneq(c) := ∀x :EyIneq: ∃y : Ineq: x + y > c publicclass AxEyIneq implements IForall{ @Paramdoublec; doublex; publicclass EyIneq implements IExists{ doubley; publicclass Ineq implements IPred{ publicboolean execute() { returnx+y>c; } } } }

42. Coding the “Toy” Lab AxEyIneq(c) := ∀x ∈ [0,1]:EyIneq: ∃y : Ineq: x + y > c ∈ [0,1] publicclass AxEyIneq implements IForall{ @Paramdoublec; doublex; publicboolean checkecProvidedValue() { return 0<=x && x<=1; } publicclass EyIneq implements IExists{ doubley; publicclass Ineq implements IPred{ publicboolean execute() { returnx+y>c; } } } }

43. Coding the “Toy” Lab AxEyIneq(c ∈ [0,2]) := ∀x ∈ [0,1]:EyIneq: ∃y ∈ [0,1]:Ineq: x + y > c ∈ [0,2] publicclass AxEyIneq implements IForall{ @Paramdoublec; doublex; publicclass EyIneq implements IExists{ doubley; publicclass Ineq implements IPred{ publicboolean execute() { returnx+y>c; } } publicboolean checkecProvidedValue() { return 0<=y && y<=1; } } publicboolean checkecProvidedValue() { return 0<=x && x<=1; } public AxEyIneq(double c) { if( c<0 || c>3 ){ thrownew RuntimeException(); } this.c = c; } }