Propositional Calculus – Methods of Proof Predicate Calculus

1. Propositional Calculus – Methods of ProofPredicate Calculus Math Foundations of Computer Science

2. Propositional Calculus – Methods of Proof • Logic is useful in design theory • Also useful in reasoning about mathematical statements: • Case Analysis • Proof of the contrapositive • Proof by contradiction • Proof by reduction to truth Math Foundations of Computer Science

3. Law of Excluded Middle • Handy for Case Analysis: • Can now prove

4. Dual of the Excluded Middle • A proposition and its negative can’t be simultaneously true • Does this jive with the real world? • Do we have contradictions in mathematics? • Handy for proofs by Contradiction

5. Contrapositive • Example: Prove “If x is greater than 2 and prime, then x is odd” • The contrapositive is: “If x is even, then x <= 2 or x is composite” • Propositional logic fails at this point; we need to talk about the meaning of our terms

6. Contradiction • Rather than proving E, we assume NOT E, and look for a contradiction • Example (from previous slide) • We want to prove ab → c (cont.)

7. Contradiction - example • So, as assume a b and NOT c • Derive a contradiction

8. Equivalence by Truth (p ≡ 1) ≡ p • Use the tautologies to reduce the expression to 1 • Probably most like the examples we’ve been looking at

9. Deduction • The use of logic in sequences of statements that constitute a complete proof • Start with certain hypotheses (“givens”) • Apply a sequence of inference rules • Results in a conclusion • Most familiar to you, from geometry

10. Deduction • Given expressions E1, E2, …, Ek as hypotheses, we wish to draw conclusion E, another logical expression • Generally, none of these is a tautology • Show that E1∩E2∩ … ∩Ek → E is a tautology

11. Deduction – guidelines • Any tautology may be a line in a proof • modus ponens – if E and E→F are lines, then F may be added as a line • If E and F are lines in a proof, then we may add the line E ∩F • If E and E≡F are lines, the F can be added

12. Resolution – a handy inference rule • Based on this tautology: • Just another inference rule • But a common one • Often used in a deductive proof

13. Resolution • Applied to clauses (your hypotheses) (as in deduction) • Convert hypotheses into clauses (conjunctive normal form) • Write each clause as a line • Use resolution to write other lines

14. Simplifying clauses • Consider a clause as a set of literals • Given commutativity, associativity and idempotence of OR • Remove duplicate literals:

15. Simplifying clauses • Eliminate clauses that have contradictory literals • , by the annihilator law of OR • These clauses are equivalent to 1, and are not needed in a proof

16. Resolution - example • Given these 2 clauses: • Rearrange terms, and apply resolution • Remove duplicates

17. Put Expressions into Conjunctive Normal Form • Get rid of all operators but NOT, AND, and OR NAND and NOR are easily replaced with ANDandOR followed by a NOT • Apply DeMorgan’s laws to push negations down as far as they will go

18. Put Expressions into Conjunctive Normal Form • Apply distributive law for OR over AND • Push the ORs as low as they’ll go • E.g. • Replace the implication • Push that outer NOT down:

19. CNF – Example (cont.) • Push the first OR below the first AND • Regroup • Push that OR down over the inner AND • And now you have an expression in CNF

20. Resolution Proofs by Contradiction • A more common use • Start with both the hypotheses, and the negation of the conclusion • Try to drive a clause w/no literals • This clause has value 0