Class 5: CNF, Quantifiers, Proofs about Software

cs2102: Discrete Mathematics
University of Virginia, Fall 2016

See course site for notes:
https://uvacs2102.github.io

Transcript

1. cs2102: Discrete Mathematics Class 5: Satisfiability David Evans University of Virginia
2. Plan Review: Logic Formulas, Valid, Satisfiable Conjunctive Normal Form, 3CNF PS1: Problems 1 and 2 This week’s goal: understand the SAT problem (3.5), and why it is so significant. (graded PS1s should be returned by tomorrow)
3. A formula is valid if there is no way to make it false. A formula is satisfiable if there is some way to make it true. If NOT(X) is valid, what does that mean about X?
4. Conjunctive Normal Form
5. Conjunctive Normal Form clause literal variable or its negation
6. Conjunctive Normal Form Can we write any logical formula in CNF? A ∧ B
7. Conjunctive Normal Form Can we write any logical formula in CNF? A ∧ B A ⇒ B
8. Logical Formula to Truth Table Can we convert any logical formula to a truth table?
9. Logical Formula to Truth Table How big is the table? a1 a2 a3 … an Result T T T … T T T T … F … … … … … … … … F F F … T F F F … F
10. Truth Table to Disjunctive Normal Form Disjunctive normal form: OR of clauses, each clause is AND of literals. a1 a2 a3 … an Result T T T … T T T T … F … … … … … … … … F F F … T F F F … F
11. Truth Table to Disjunctive Normal Form a1 a2 a3 … an Result T T T … T T T T … F … … … … … … … … F F F … T F F F … F clauses = Set() # empty is False for each row in table: if result(row): clauses.add(row as conjunt clause)
12. Truth Table to Conjunctive Form De Morgan’s Law
13. Truth Table to Conjunctive Normal Form a1 a2 a3 … an Result T T T … T T T T … F … … … … … … … … F F F … T F F F … F clauses = Set() for each row in table: if not result(row): clauses.add(row as disjunctive clause)
14. 3CNF Definition. A logical formula that is written as a conjunction of clauses, where each clause is a disjunction of exactly three literals, and each literal is either a variable or a negation of a variable, is in three- conjunctive normal form (3CNF).
15. 3CNF Definition. A logical formula that is written as a conjunction of clauses, where each clause is a disjunction of exactly three literals, and each literal is either a variable or a negation of a variable, is in three- conjunctive normal form (3CNF). (A ∨ B ∨ ¬C) ∧ (A ∨ B ∨ C) ∧ (¬A ∨ ¬B ∨ C)
16. Definition. A logical formula that is written as a conjunction of clauses, where each clause is a disjunction of exactly three literals, and each literal is either a variable or a negation of a variable, is in three- conjunctive normal form (3CNF). (A ∨ B) Convert clause with two literals to 3CNF:
17. Definition. A logical formula that is written as a conjunction of clauses, where each clause is a disjunction of exactly three literals, and each literal is either a variable or a negation of a variable, is in three- conjunctive normal form (3CNF). (A) Convert clause with one literal to 3CNF:
18. Definition. A logical formula that is written as a conjunction of clauses, where each clause is a disjunction of exactly three literals, and each literal is either a variable or a negation of a variable, is in three- conjunctive normal form (3CNF). (A ∨ B ∨ C ∨ D) Convert clause with four literals to 3CNF:
19. (A ∨ B ∨ C ∨ D ∨ E) Convert clause with five (or more) literals to 3CNF:
20. Any logical formula → truth table → CNF → 3CNF Universality of 3CNF
21. Can any logical formula be converted to 2CNF? (Slack break…) Definition. A logical formula that is written as a conjunction of clauses, where each clause is a disjunction of exactly two literals, and each literal is either a variable or a negation of a variable, is in two- conjunctive normal form (2CNF).
22. (A ∨ B ∨ C ∨ D) Convert clause with four literals to 2CNF?
23. Problem Set 1
24. 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.”
25. Logical Formula “Proofs can certify that a computing system will always behave correctly, something that no amount of testing can do.”
26. Logical Quantifiers ∀ ∈ . () ∃ ∈ . ()
27. Nesting Quantifiers ∀ ∈ ℕ. ∃ ∈ ℕ. = + 1
28. Nesting Quantifiers ∀ ∈ ℕ − {0}. ∃ ∈ ℕ. = + 1
29. Well-Ordering Principle Every nonempty set of non-negative integers has a smallest element. Can we express this using a quantified logical formula?
30. Well-Ordering Principle Every nonempty set of non-negative integers has a smallest element. ∀ ∈ pow ℕ − ∅ .
31. Well-Ordering Principle ∀ ∈ pow ℕ − ∅ . ∃ ∈ . ∀ ∈ − . <
32. Certifying Computing Systems “Proofs can certify that a computing system will always behave correctly, something that no amount of testing can do.”
33. 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?
35. 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(, )
36. ∀ ∈ . ¬(Test ⟹ Correct()) Test s = ∀ ∈ (). ℎ , ∈ ℎ(, ) Correct s = ∀ ∈ (). ℎ , ∈ ℎ(, ) When can testing certify a computing system?
37. TicTacToe(boardstate) – Acceptable behavior is to always pick a move that is legal (when one exists) and leads to best possible outcome.
38. 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()
39. Proving Programs Correct def max(a, b): “Returns maximum of a and b” How should we define AcceptableBehaviors(max, x)?
40. 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 ≥ .
41. 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
42. Hoare Logic Sir Tony Hoare { P } S { Q }
43. def max(a, b): if a > b: result = a else: result = b return result AcceptableBehaviors(max, x = (a, b)): result ≡ max , no other state modified result ∈ , ∧ result ≥ ∧ result ≥ . Hoare triple for if statement: ∧ 1 , ∧ ¬ 2 if then 1 else 2
44. def max(a, b): if a > b: result = a else: result = b return result AcceptableBehaviors(max, x = (a, b)): result ≡ max , no other state modified result ∈ , ∧ result ≥ ∧ result ≥ . Hoare triple for if statement: ∧ 1 , ∧ ¬ 2 if then 1 else 2 P = True = result ∈ , ∧ result ≥ ∧ result ≥
45. def max(a, b): if a > b: result = a else: result = b return result Hoare triple for if statement: ∧ 1 , ∧ ¬ 2 if then 1 else 2 P = True = result ∈ , ∧ result ≥ ∧ result ≥ ∧ a > b result = a
46. def max(a, b): if a > b: result = a else: result = b return result AcceptableBehaviors(max, x = (a, b)): result ≡ max , no other state modified result ∈ , ∧ result ≥ ∧ result ≥ . ∃ ∈ . ⟹ Correct()
47. def max(a, b): if a > b: result = a … Idealized Computing Systems ∧ a > b result = a result ∈ , , result ≥
48. Real Computing Systems def max(a, b): if a > b: result = a … Idealized Computing Systems ∧ a > b result = a result ∈ , , result ≥
49. Charge • Add date is today: make sure you are enrolled (or bring me a course action to sign now!) • Be paranoid: assumptions about computing systems are not true in practice • Thursday: how hard is it to determine if a 3CNF can be satisfied (and why we care) Due Friday (6:29pm): PS2