Mathematics for Computer Science Proving Validity 6.042J/18.062J Instead of truth tables, The Logic of can try to prove valid Propositions formulas symbolically using axioms and deduction rules propositional logic.1 propositional logic.2 February 14, 2014 February 14, 2014 Albert R Meyer Albert R Meyer Proving Validity Algebra for Equivalence The text describes a for example, bunch of algebraic rules to the distributive law prove that propositional P AND (Q OR R) ≡ formulas are equivalent (P AND Q) OR (P AND R) propositional logic.3 propositional logic.4 February 14, 2014 February 14, 2014 Albert R Meyer Albert R Meyer 1
Algebra for Equivalence Algebra for Equivalence for example, The set of rules for in ≡ DeMorgan’s law the text are complete: if two formulas are , ≡ NOT (P AND Q) ≡ these rules can prove it. NOT (P) OR NOT (Q) propositional logic.5 propositional logic.6 February 14, 2014 February 14, 2014 Albert R Meyer Albert R Meyer A Proof System A Proof System Another approach is to Lukasiewicz’ proof system is a particularly elegant example of start with some valid this idea. formulas (axioms) and deduce more valid formulas using proof rules propositional logic.7 propositional logic.8 February 14, 2014 February 14, 2014 Albert R Meyer Albert R Meyer 2
A Proof System Lukasiewicz’ Proof System Axioms: Lukasiewicz’ proof system is a particularly elegant example of 1) ( ¬ P → P) → P this idea. It covers formulas 2) P → ( ¬ P → Q) whose only logical operators are 3) (P → Q) → ((Q → R) → (P → R)) IMPLIES ( → ) and NOT. The only rule: modus ponens propositional logic.9 propositional logic.10 February 14, 2014 February 14, 2014 Albert R Meyer Albert R Meyer Lukasiewicz’ Proof System Lukasiewicz’ Proof System Prove formulas by starting with Prove formulas by starting with axioms and repeatedly applying axioms and repeatedly applying the inference rule. the inference rule. To illustrate the proof system For example, to prove: we’ll do an example, which you P P → may safely skip. propositional logic.12 propositional logic.13 February 14, 2014 February 14, 2014 Albert R Meyer Albert R Meyer 3
A Lukasiewicz’ Proof A Lukasiewicz’ Proof 3 rd axiom: 3 rd axiom: (P → Q ) → (P → Q ) → (( Q → R) → (P → R)) (( Q → P) → (P → P)) replace R by P replace Q by ( P P) → propositional logic.14 propositional logic.15 February 14, 2014 February 14, 2014 Albert R Meyer Albert R Meyer A Lukasiewicz’ Proof A Lukasiewicz’ Proof so apply modus ponens: 3 rd axiom: Axiom 2) Axiom 2) (P → (P → P) ) → (P → (P → P) ) → (((P → P) → P) → (P → P)) (((P → P) → P) → (P → P)) propositional logic.16 propositional logic.17 February 14, 2014 February 14, 2014 Albert R Meyer Albert R Meyer 4
A Lukasiewicz’ Proof A Lukasiewicz’ Proof so apply modus ponens: so apply modus ponens: Axiom 1) (((P → P) → P) → (P → P)) (P → P) QED propositional logic.18 propositional logic.19 February 14, 2014 February 14, 2014 Albert R Meyer Albert R Meyer z is S ound Lukasiewicz is Complete Luka Luka siew si i e cz w ’ ic Proof System Conversely, every valid The 3 Axioms are all valid ( NOT , → )-formula is provable: (verify by truth table). system is “ complete ” We know modus ponens is Not hard to verify but would take s S ound. o every provable a full lecture; we omit it. formula is also valid. propositional logic.20 propositional logic.21 February 14, 2014 February 14, 2014 Albert R Meyer Albert R Meyer 5
validity checking still inefficient Algebraic & deduction proofs in general are no better than truth tables. No efficient method for verifying validity is known. propositional logic.22 February 14, 2014 Albert R Meyer 6
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