# Final-Example-Sol - Name: Perm #: Final Exam - Example CS...

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2Inference Rules for Propositional LogicModus Ponens (a.k.a. Implication-Elimination)αβ,αβAnd-Eliminationα1α2αnαiAnd-Introductionα1,α2,… ,αnα1α2αnOr-Introductionα1α1α2αnDouble-Negation Elimination¬¬ααUnit Resolutionαβ,¬βαResolutionαβ,¬βγαγAdditional Inference Rules for First-Order LogicUniversal EliminationExistential EliminationExistential Introduction)},/({ααgvSUBSTv)},/({ααkvSUBSTv)},/({ααvgSUBSTvRules, equations, etc.
3Standard Logical EquivalencesFor atomic sentencespi, pi' ,andq ,where there is a substitutionθ,such thatSUBST(θ,pi') = SUBST(θ,pi) for alli, thenFor literals piand qi, where UNIFY(pj,¬qk) =θ),()(,,,,2121qSUBSTqppppppnnθGeneralized Modus Ponens),(,1111kjnmnkmjqandpexceptqqppSUBSTqqqpppθGeneralized Resolution
4Procedure to convert sentences to Conjunctive Normal Form:Eliminate equivalence: Replace PQ with (PQ) and (QP)Eliminate implications: Replace (PQ) with (¬PQ)Move¬inwards:¬∀,¬∃,¬¬,¬(PQ),¬(PQ)Standardize variables apart:x Px Qbecomesx1Px2QMove quantifiers left in order:x Py Qbecomesxy PQEliminateby “Skolemization” – whenis on the outside, do existentialelimination; otherwise use a “skolem function” H(xi) to enclose the universallyquantified variablesDrop universal quantifiersDistributeover, e.g.: (PQ)Rbecomes(PR)(QR)Flatten nesting: (PQ)Rbecomes PQR
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