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Converting Logical Assertions to Conjunctive Normal Form: A Detailed Approach

This guide explains the process of converting logical assertions into conjunctive normal form (CNF) using First-Order Predicate Logic (FOPL). Starting from statements about Romans who know Marcus, it simplifies expressions through a series of steps. Implications are eliminated, quantifiers standardized, and existential quantifiers Skolemized. Finally, the process culminates in rewriting the logical expressions as a conjunction of disjuncts, detailing each step for clarity. This structured method showcases how to represent logic clearly and concisely.

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Converting Logical Assertions to Conjunctive Normal Form: A Detailed Approach

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  1. Conversion to Conjunctive Normal Form

  2. Assertion All Romans who know Marcus either hate Caesar or think than anyone who hates anyone is crazy. All Romans wish they were Greeks.

  3. Expressed in FOPL

  4. For the transformation, let’s simplify Let P = roman(X) Q = know(X, marcus) R = hate(X, caesar) S = hate(Y,Z) T = thinkcrazy(X,Y) V = wish_greek(X)

  5. Giving

  6. Step 1: Eliminate implication using the identity So becomes

  7. Applied to the Original Expression Now eliminate the second =>

  8. Applied to the original expression

  9. Eliminating the Third ImplicationGives

  10. Step 2: Invoke deMorgan

  11. Step 3: Standardize the quantifiers so that each binds a unique variable For Example Given: We write:

  12. Step 4: Move all quantifiers to the left without changing their order Step 3 makes this legal

  13. Step 5: Eliminate Existential Quantifiers: Skolemization • Type 1 Given Tells us that there is an individual assignment to X drawn from its domain under which school(X) is satisfied. .

  14. So, invent a function that goes into the domain of X and picks out just that item that satisfies school. • Call it pick • The original expression is transformed to: school(pick()) • Where • Pick is a function with no arguments • That returns the value from the domain of X that satisfies School • We might not know how to get the value. • But we give a name to the method that we know exists

  15. Type 2 Suppose we have Where P,Q are elements of the set of integers In English, given an integer P, there is another integer Q, such that Q > P We can’t invent a single function, because Q depends on P Instead, invent a function whose single argument is the universally quantified variable.

  16. get(P) returns an integer > P Becomes

  17. Our example has one instance of an existentially quantified variable within the scope of a universally quantified variable Becomes

  18. Giving

  19. Step 6: Drop the remaining quantifiers • Legal since everything is universally quantified

  20. Step 7: Rewrite the expression as a conjunct of disjuncts using dist. And assoc. laws This Gives:

  21. Step 8: Rewrite each conjunct as a separate clause that are implicitly anded

  22. Step 9: Rename variables in clauses so that no two clauses use the same variable name This is already the case The Expression is now in conjunctive normal form.

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