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