Class 21

Review

Transcript

1. cs2102: Discrete Mathematics Class 21: Review David Evans, Mohammad Mahmoody University of Virginia
2. Plan Today: review! • State Machine and Invariance  Program correctness and Termination • Recursive Defenitions  Structural Induction • Infinite Sets and Cardinalities
3. State Machine and Invariants • State Machine: M = , ⊆ × , 0 ∈ • is a preserved invariant if ∀ ∈ . ∧ ( → ) ∈ ⟹ () • Invariant Principle: If is preserved invariant and (0 ) is true  it is true for all reachable states.
4. State Machine and Invariants
5. State Machine and Invariants
6. State Machine and Invariants
7. Program Correctness • To prove a program P outputs “correctly” : 1. Model it as a state machine M 2. Show it eventually terminates 3. Show partial correctness (if it ends  it is correct) a) Find a “good” preserved invariant P for M b) Show ()=true for final state q implies correct output c) Show 0 =true for start state
8. Program Correctness • State Machine :
9. Program Correctness • Why does it terminate?
10. Program Correctness • What is a good invariant here?
11. Program Correctness • Proving partial correctness
12. Recursive Data Types • To define recursive data type – Base case(s) that specify some elements are in – Constructor case(s) that specify how to construct new data element ∈ from previously constructed 1 , 2 , … ∈ • To define (new) operators on we again use base and constructor cases!
13. Recursive Data Types
14. Recursive Data Types
15. Structural Induction Goal: To prove for all elements in data type 1. Prove for all base cases ∈ 2. Prove 1 ∧ (2 ) ⇒ for all constructable from 1 , 2 15
16. Using Structural Induction
17. Infinite Sets and Cardinalities A set is countably infinite iff there is a bijection between and ℕ. is infinite, if there is no bijection between and any ℕ . Equivalent : ℕ ≤ || : there is a surjective function from to ℕ is Countable : ≤ |ℕ| : there is a surjective function from ℕ to . A set is uncountable iff it is not countable.
18. Infinite Sets and Cardinalities • Set of all English words… is finite. • Set of all words using English letters… is an infinite set. It includes all the finite sequences of a,b, which by interpreting = 0, = 1 we get all the natural numbers written in binary (and a lot more). So there is a surjective function from this set to ℕ.
19. Infinite Sets and Cardinalities • We know (ℕ) is uncountable (Cantor’s theorem) and that ℕ = 0,1 = |ℝ| • To show is uncountable, prove that ≥ || for an uncountable set. • Example: 0,2 , 1,2 , .