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Logic March 28, 2005 Propositional Logic Information definition: a - PowerPoint PPT Presentation

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Logic March 28, 2005 Propositional Logic Information definition: a proposition is a statement of fact It is raining (english) Raining Connectives: operators on propositions And, or, not, implies, if and only if

  1. Logic March 28, 2005

  2. Propositional Logic Information definition: a proposition is a statement of fact � “It is raining” (english) Raining Connectives: operators on propositions � And, or, not, implies, if and only if ∧ ∨ ¬ → ↔ , , , ,

  3. Syntax Symbols: p, q, r, s, t (variables) Constants: T, F Functions: f,g,h (n-ary) and connectives ∧ ∨ ¬ → ↔ , , , , Relations: R, S (n-ary) Parentheses: ),( Equality ≡

  4. Examples → p q ∧ ¬ ∨ ( p p ) r ∨ ∧ ≡ ))

  5. Formulas and Terms Rules: � All symbols are formulas � All constants are formulas � If t 0 ,t 1 are formulas then so are ∧ ∨ → ↔ ¬ t t , t t , t t , t t , t , ( t ) 0 1 0 1 0 1 0 1 0 0 � If t 0 ,t 1 are formulas then so are ≡ Rt 0 , t t t 1 0 1 � Formulas composed from symbols, constants and functions are called terms

  6. Semantics ¬ ∧ ∨ → ↔ p q p p q p q p q p q T T F T T T T T F F F T F F F T T F T T F F F T F F T T

  7. Semantics II Semantics of any formula is given by an evaluation function Φ from formulas to {T,F} To define the semantics, it suffices to define evaluation of symbols and functions (and use the previous slide)

  8. Examples II ∨ ¬ p p A tautology is always true ∧ ¬ p p A contradiction is always false One way to derive truth of a formula is to use a truth table. → ≡ ¬ ∨ ( p q ) ( p q )

  9. Laws of Propositional Logic Commutativity Associativity Distributivity DeMorgan

  10. Rules of Inference Modus Ponens Modus Tollens Syllogism Disjunctive Syllogism Specialization Conjuction

  11. Theories A Theory in propositional logic is a set of constants, functions, relations and axioms. Example: (theory of ordered integers) Constants: non-negative integers � Function: +, Relation: < � Axioms: � ¬ x < ( x ) < → < + 0 x y x y < → ¬ < ( x y ) ( y x )

  12. Why? Why do computer scientists care? Because theories are specifications of a collection of structures To reason about code correctness To enable code transformations � Must preserve invariants

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