Class 14

Invariance Principle

Transcript

1. Class 14: Invariant Principle cs2102: Discrete Mathematics David Evans & Mohammad Mahmoody University of Virginia
2. Plan State Machines Review Invariant Principle Read this week: MCS Chapter 6
3. State Machine = (, ⊆ × , 0 ∈ )
4. = , = → , → 0 = Binary Relation View State Machine View
5. Bishop Moves = = 0 =
6. Reachable States A state is reachable if it appears in some execution. The execution of a state machine, = (, ⊆ × , 0 ∈ ) is a (possibly infinite) sequence of states, (0 , 1 , … , ) that: 1. 0 = 0 (it begins with the start state) 2. ∀ ∈ 0, 1, … , − 1 . → +1 ∈ (if and are consecutive states in the sequence, there is an edge → ∈ .
7. Bishop Moves = , , ∈ ℕ } 0 = (0, 2) What states are reachable? = , → ′, ′ , , ′, ′ ∈ ℕ ∧ ∃ ∈ ℕ+ : ′ = ± and ′ = ± }
8. Preserved Invariant is a preserved invariant of = (, ⊆ × , 0 ∈ ) if ∀ ∈ . ∧ ( → ) ∈ ⟹ ()
9. = , ≔ ( > 3) Is a preserved invariant? = , , ∈ ℕ } 0 = (0, 2) = , → ′, ′ , , ′, ′ ∈ ℕ ∧ ∃ ∈ ℕ+ : ′ = + and ′ = + } (up-right only bishops)
10. Is a preserved invariant? = , , ∈ ℕ } = , ∶= ( = ) = , → ′, ′ , , ′, ′ ∈ ℕ ∧ ∃ ∈ ℕ+ : ′ = + and ′ = + } (up-right only bishops)
11. Is a preserved invariant? = , , ∈ ℕ } = , ∶= ( is even) = , → ′, ′ , , ′, ′ ∈ ℕ ∧ ∃ ∈ ℕ+ : ′ = + and ′ = + } (up-right only bishops)
12. = , , ∈ ℕ } = , → ′, ′ , , ′, ′ ∈ ℕ ∧ ∃ ∈ ℕ+ : ′ = ± and ′ = ± } regular bishops What are preserved invariants for the Bishop machine? Any questions about definitions: state machine, execution, reachable, preserved invariant
14. = , , ∈ ℕ } = , → ′, ′ , , ′, ′ ∈ ℕ ∧ ∃ ∈ ℕ+ : ′ = ± and ′ = ± } regular bishops What are preserved invariants for the Bishop machine?
15. = , , ∈ ℕ } ∶= ’s color = , → ′, ′ , , ′, ′ ∈ ℕ ∧ ∃ ∈ ℕ+ : ′ = ± and ′ = ± } regular bishops
16. = , , ∈ ℕ } = , → ′, ′ , , ′, ′ ∈ ℕ ∧ ∃ ∈ ℕ+ : ′ = ± and ′ = ± } = (, ) ∶= + is even regular bishops
17. Invariant Principle If a preserved invariant is true for the start state, it is true for all reachable states.
18. = , , ∈ ℕ } = , → ′, ′ , , ′, ′ ∈ ℕ ∧ ∃ ∈ ℕ+ : ′ = ± and ′ = ± } 0 = (0, 2) Prove never enters a state corresponding to a white square.
19. = , , ∈ ℕ } = , → ′, ′ , , ′, ′ ∈ ℕ ∧ ∃ ∈ ℕ+ : ′ = ± and ′ = ± } 0 = (0, 2) Prove never enters a state where + is odd.
20. Fast (and Slow) Exponentiation
23. Exponentiation in RSA
24. Slow Exponentiation How many multiplications?
25. Modeling Slow Exponentiation ∷= ∷= 0 ∷=
26. Correctness of Slow Exponentiation ∷= ℕ × ℕ ∷= { , ⟶ , − 1 | ∀ ∈ ℕ, ∈ ℕ+} 0 ∷= (1, ) What preserved invariant would be useful? If we prove this to be invariant, at the end, 0 = means = . We will see more subtle aspects of this argument next time..
27. Charge • PS6 will be posted by tomorrow, due next Friday • Read rest of Chapter 6