Normal forms in Software Use barcode 3 of 9 in Software Normal forms

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1.5 Normal forms generate, create code 39 extended none for software projects Code 128 Code Set C De nition 1.44 Given a formul 3 of 9 barcode for None a in propositional logic, we say that is satis able if it has a valuation in which is evaluates to T. For example, the formula p q p is satis able since it computes T if we assign T to p.

Clearly, p q p is not valid. Thus, satis ability is a weaker concept since every valid formula is by de nition also satis able but not vice versa. However, these two notions are just mirror images of each other, the mirror being negation.

Proposition 1.45 Let be a formula of propositional logic. Then is satis able i is not valid.

Proof: First, assume that is satis able. By de nition, there exists a valuation of in which evaluates to T; but that means that evaluates to F for that same valuation. Thus, cannot be valid.

Second, assume that is not valid. Then there must be a valuation of in which evaluates to F. Thus, evaluates to T and is therefore satis able.

(Note that the valuations of are exactly the valuations of .) 2 This result is extremely useful since it essentially says that we need provide a decision procedure for only one of these concepts. For example, let s say that we have a procedure P for deciding whether any is valid.

We obtain a decision procedure for satis ability simply by asking P whether is valid. If it is, is not satis able; otherwise is satis able. Similarly, we may transform any decision procedure for satis ability into one for validity.

We will encounter both kinds of procedures in this text. There is one scenario in which computing an equivalent formula in CNF is really easy; namely, when someone else has already done the work of writing down a full truth table for . For example, take the truth table of (p q) (q p) in Figure 1.

8 (page 40). For each line where (p q) (q p) computes F we now construct a disjunction of literals. Since there is only one such line, we have only one conjunct 1 .

That conjunct is now obtained by a disjunction of literals, where we include literals p and q. Note that the literals are just the syntactic opposites of the truth values in that line: here p is T and q is F. The resulting formula in CNF is thus p q which is readily seen to be in CNF and to be equivalent to (p q) (q p).

Why does this always work for any formula Well, the constructed formula will be false i at least one of its conjuncts i will be false. This means that all the disjuncts in such a i must be F. Using the de Morgan.

1 Propositional logic rule 1 2 n Software barcode 3/9 ( 1 2 n ), we infer that the conjunction of the syntactic opposites of those literals must be true. Thus, and the constructed formula have the same truth table. Consider another example, in which is given by the truth table: p T T T T F F F F q T T F F T T F F r T F T F T F T F T F T T F F F T.

Note that this table is reall y just a speci cation of ; it does not tell us what looks like syntactically, but it does tells us how it ought to behave. Since this truth table has four entries which compute F, we construct four conjuncts i (1 i 4). We read the i o that table by listing the disjunction of all atoms, where we negate those atoms which are true in those lines: 1 = p q r (line 2) 3 = p q r.

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