Class 14: Invariant Principle cs2102: Discrete Mathematics David Evans & Mohammad Mahmoody University of Virginia 

Plan State Machines Review Invariant Principle Read this week: MCS Chapter 6 

State Machine = (, ⊆ × , 0 ∈ ) 

= , = → , → 0 = Binary Relation View State Machine View 

Bishop Moves = = 0 = 

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 → ∈ . 

Bishop Moves = , , ∈ ℕ } 0 = (0, 2) What states are reachable? = , → ′, ′ , , ′, ′ ∈ ℕ ∧ ∃ ∈ ℕ+ : ′ = ± and ′ = ± } 

Preserved Invariant is a preserved invariant of = (, ⊆ × , 0 ∈ ) if ∀ ∈ . ∧ ( → ) ∈ ⟹ () 

= , ≔ ( > 3) Is a preserved invariant? = , , ∈ ℕ } 0 = (0, 2) = , → ′, ′ , , ′, ′ ∈ ℕ ∧ ∃ ∈ ℕ+ : ′ = + and ′ = + } (up-right only bishops) 

Is a preserved invariant? = , , ∈ ℕ } = , ∶= ( = ) = , → ′, ′ , , ′, ′ ∈ ℕ ∧ ∃ ∈ ℕ+ : ′ = + and ′ = + } (up-right only bishops) 

Is a preserved invariant? = , , ∈ ℕ } = , ∶= ( is even) = , → ′, ′ , , ′, ′ ∈ ℕ ∧ ∃ ∈ ℕ+ : ′ = + and ′ = + } (up-right only bishops) 

= , , ∈ ℕ } = , → ′, ′ , , ′, ′ ∈ ℕ ∧ ∃ ∈ ℕ+ : ′ = ± and ′ = ± } regular bishops What are preserved invariants for the Bishop machine? Any questions about definitions: state machine, execution, reachable, preserved invariant 

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= , , ∈ ℕ } = , → ′, ′ , , ′, ′ ∈ ℕ ∧ ∃ ∈ ℕ+ : ′ = ± and ′ = ± } regular bishops What are preserved invariants for the Bishop machine? 

= , , ∈ ℕ } ∶= ’s color = , → ′, ′ , , ′, ′ ∈ ℕ ∧ ∃ ∈ ℕ+ : ′ = ± and ′ = ± } regular bishops 

= , , ∈ ℕ } = , → ′, ′ , , ′, ′ ∈ ℕ ∧ ∃ ∈ ℕ+ : ′ = ± and ′ = ± } = (, ) ∶= + is even regular bishops 

Invariant Principle If a preserved invariant is true for the start state, it is true for all reachable states. 

= , , ∈ ℕ } = , → ′, ′ , , ′, ′ ∈ ℕ ∧ ∃ ∈ ℕ+ : ′ = ± and ′ = ± } 0 = (0, 2) Prove never enters a state corresponding to a white square. 

= , , ∈ ℕ } = , → ′, ′ , , ′, ′ ∈ ℕ ∧ ∃ ∈ ℕ+ : ′ = ± and ′ = ± } 0 = (0, 2) Prove never enters a state where + is odd. 

Fast (and Slow) Exponentiation 

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Exponentiation in RSA 

Slow Exponentiation How many multiplications? 

Modeling Slow Exponentiation ∷= ∷= 0 ∷= 

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.. 

Charge • PS6 will be posted by tomorrow, due next Friday • Read rest of Chapter 6