Class 5

Review

Transcript

1. cs2102: Discrete Mathematics Class 6: Quantifiers, Program Correctness, Equivalence David Evans, Mohammad Mahmoody University of Virginia
2. Plan Finish Well-Ordering with Quantifiers Program Correctness (PS1) Validity, Satisfiability, Equivalence Negating Quantifiers Converting Formulas to DNF/CNF/3CNF SAT Solving: why satisfiability matters
3. Well-Ordering Principle Every nonempty set of non-negative integers has a smallest element. ∀ ∈ pow ℕ − ∅ .
4. Well-Ordering Principle ∀ ∈ pow ℕ − ∅ . ∃ ∈ . ∀ ∈ − . <
5. Problem Set 1
6. Disambiguating the English “Proofs also play a growing role in computer science; they are used to certify that software and hardware will always behave correctly, something that no amount of testing can do.” “Proofs can certify that a computing system will always behave correctly, something that no amount of testing can do.”
7. Certifying Computing Systems “Proofs can certify that a computing system will always behave correctly, something that no amount of testing can do.”
8. Certifying Computing Systems “Proofs can certify that a computing system will always behave correctly, something that no amount of testing can do.” ∀ ∈ . ¬(Test ⟹ Correct()) What does it mean to test a computing system?
10. Certifying Computing Systems “Proofs can certify that a computing system will always behave correctly, something that no amount of testing can do.” ∀ ∈ . ¬(Test ⟹ Correct()) Test s = ∀ ∈ (). ℎ , ∈ ℎ(, ) Correct s = ∀ ∈ (). ℎ , ∈ ℎ(, )
11. ∀ ∈ . ¬(Test ⟹ Correct()) Test s = ∀ ∈ (). ℎ , ∈ ℎ(, ) Correct s = ∀ ∈ (). ℎ , ∈ ℎ(, ) When can testing certify a computing system is correct?
12. TicTacToe(boardstate) – Acceptable behavior is to always pick a move that is legal (when one exists) and leads to best possible outcome.
13. Proofs about Computing Systems “Proofs can certify that a computing system will always behave correctly, something that no amount of testing can do.” ∀ ∈ . ∃ ∈ . ⟹ Correct() ∃ ∈ . ∃ ∈ . ⟹ Correct()
14. Proving Programs Correct def max(a, b): “Returns maximum of a and b” How should we define AcceptableBehaviors(max, x)?
15. Proving Programs Correct def max(a, b): “Returns maximum of a and b” AcceptableBehaviors(max, x = (a, b)): result = max , no other state modified result ∈ , ∧ result ≥ ∧ result ≥ .
16. Proving Programs Correct AcceptableBehaviors(max, x = (a, b)): result = max , no other state modified result ∈ , ∧ result ≥ ∧ result ≥ . def max(a, b): if a > b: result = a else: result = b return result Coq
17. Real Computing Systems def max(a, b): if a > b: result = a … Idealized Computing Model ∀, ∈ ℕ . result = max a, b , result ∈ , ∧ result ≥ ∧ result ≥ .
18. Real Computing Systems def max(a, b): if a > b: result = a … Idealized Computing Systems ∀, ∈ ℕ . result = max a, b , result ∈ , ∧ result ≥ ∧ result ≥ .
20. Review • Validity, Satisfiability, Equivalence • Negating Quantifiers • Converting Formulas to DNF/CNF/3CNF
21. Valid Formulas A formula is valid if there is no way to make it false. → ∧ → ⇒ ( → )
22. Satisfiable Formulas A formula is satisfiable if there is some way to make it true. How to say something *is* valid using quantifiers? → ∧ → ⇒ ( → )
23. ∀ , , ∈ , , → ∧ → ↔ ( → )= Two Equivalent Formulas Two formulas are equivalent, if for all true/false assignment to the variables they evaluate to equal values. How to say it using quantifiers? → ∧ → ≡ ( → )
24. Negating Universal Quantifiers • What is the negation of ∀ ∈ , () ?
25. Negating Existential Quantifiers • What is the negation of ∃ ∈ , () ? All integers
26. Proof by Contradiction vs. Counter Example • Suppose we want to prove ∀ ∈ , () is True. How prove it by contradiction? • Suppose we want to show that ∀ ∈ , () is False. What should we do?
27. Review of DNF, CNF and 3CNF
28. DNF: Disjunctive Normal Form • Disjunction (OR) of some number of conjunctive clauses. • Conjunctive clause: AND of distinct literals • Literal: a variable or its negation • Disjunction of 0 number of clauses is also a DNF, but what is it? 1 ∧ 2 ∧ ¬3 ∨ 1 ∧ ¬2 ∧ 3 ∨ ¬1 ∧ 2 ∧ ¬3 F F
29. Converting to DNF ⊕ T T F T F T F T T F F F
30. T T T T F T F T T F F T T T F T F F F T F F F F ( ∧ ) ∨ ( ∧ ¬) ∨ (¬ ∧ ) ∨ (¬ ∧ ¬)
31. CNF: Conjunctive Normal Form • Conjunction (AND) of some number of (disjunctive) clauses. • Clause: OR of distinct literals • Literal: a variable or its negation • Disjunction of 0 number of clauses is also a CNF, but what is it?
32. Converting to CNF ⊕ T T F T F T F T T F F F
33. T T T T F T F T T F F T T T F T F F F T F F F F (¬ ∨ ¬) ∧ (¬ ∨ ) ∧ ( ∨ ¬) ∧ ( ∨ )
34. Any logical formula → truth table → CNF or DNF Universality of CNF/DNF
35. Converting to 3CNF Suppose we convert CNF = ( ∨ ∨ ∨ ) Into 3CNF = ∨ ∨ ∧ (¬ ∨ ∨ ) In this case, it is easy to see that is satisfiable if and only if is satisfiable. Using quantifiers: ∃ , , , . ↔ ∃ , , , .
36. Charge • PS2 Due Friday (6:29pm) • Next week: (Mathematical) Data Types – Sets, Functions – Read MCS Chapter 4