Propositional Calculus – Methods of Proof Predicate 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

Law of Excluded Middle   Handy for Case Analysis:  Can now prove

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

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.) ax > 2 bx is prime cx is odd

Contradiction - example  So, as assume a b and NOT c  Derive a contradiction

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

Deduction  Given expressions E 1, E 2, …, E k as hypotheses, we wish to draw conclusion E, another logical expression  Generally, none of these is a tautology  Show that E 1 ∩ E 2 ∩ … ∩ E k → E is a tautology

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

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

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

Simplifying clauses  Consider a clause as a set of literals  Given commutativity, associativity and idempotence of OR  Remove duplicate literals:

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

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

Put Expressions into Conjunctive Normal Form 1.Get rid of all operators but NOT, AND, and OR  NAND and NOR are easily replaced with AND and OR followed by a NOT 2.Apply DeMorgan’s laws to push negations down as far as they will go

Put Expressions into Conjunctive Normal Form 3.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:

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

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