Instrumento de aplicación para el estudio de caso

(Berger et al., 2002) identifies the atoms containing some special predi- cate symbols, called L-critical, within a class of formulas that are not def- inite/goal formulas. In what follows, we prove that double negating these atoms is sufficient in order to turn these formulas into definite/goal ones.

The notion of “positive subformulas” is as given by Definition 2.8. Definition 8.1 (L-critical predicate symbols). We consider Π0₂-formulas:

∀xH1 →...→ ∀xHn→ ∀y(G~ → ⊥)→ ⊥, (8.1)

withHi, i∈ {1, ..., n} and G quantifier-free and takeL to be

L:={H1, ..., Hn, ~G→ ⊥}. (8.2)

Let Ci be quantifier-free and Ri predicate symbols occurring in L. Then

L-critical predicate symbols are specified by

• ⊥ is L-critical

• if (C~1 →R1) →...→(C~m →Rm) →R is a positive subformula ofL

and if some Ri is L-critical, then R isL-critical (∗).

An atom formed with L-critical predicate symbols is called an L-critical atom.

Remark 8.1. The set of L-critical predicate symbols is the least set of pred- icate symbols containing⊥ and closed under (∗).

Let us illustrate the intended meaning of Definition 8.1 by some simple and intuitive examples.

Example 11. We consider a fixed set L and A0_i, Ai, B, Bi, i= 1,2 atomic

formulas different from ⊥, such that the formulas C1, ..., C4 are positive

subformulas ofL. In the following formulas the conclusionB is an L-critical atom:

• C1 := (A~ → ⊥)→B • C2 :=A~0 →(A~ → ⊥)→B • C3 := (A~ → ⊥)→A~0→B.

C2 andC3 indicate that the order of premise does not play any role. Notice

that the premise needs to contain a negation in order for the conclusion to be L-critical. However, in

• C4 := ((A~ → ⊥)→A~0)→B.

A0 is not L-critical, since it occurs in a negative formula of L, B is not

L-critical either.

Definition 8.1 identifies the situations in which the premise of an impli- cation is relevant, whereas the conclusion is an atom different from ⊥, so is an irrelevant formula. Such formulas are also not definite. We illustrate this by the following example:

Example 12. We consider C1, C2, C3 as in Example 11.

• C1 is not a definite formula, because it is neither relevant, nor is ¬˜A~

in I_G.

• Since C1 is not a definite formula, neither is C2 =A~0 →C1.

• C3 is also not definite, since it is not in RD and since A~ → ⊥ 6∈ IG.

• Since C3 is neither in RD, nor in D, the formula H := C → C3 is

also not definite, for any formula C.

• C3 is however an irrelevant goal formula, since its premise is in RD

and its conclusion is clearly irrelevant. Thus C3 → C is a definite

formula, when C is definite.

C4 is a definite formula, since its premise is an irrelevant goal formula. (Berger et al., 2002) proposes to repair some of the situations in which we need the formulas to be definite/goal. In case they are not as such, but contain L-critical atoms, it is possible to transform them into definite/goal formulas, by double negating the atoms formed with L-critical predicate symbols different from⊥. By carrying this out recursively over the formulas, we obtain definite/goal formulas, as shown in what follows.

Note: By definition all atomic formulas belong to both Dand G. In order to understand why the L-critical predicate symbols were defined only for positive subformulas of L, we formulate the following consequence of Definition 3.5, using Definition 2.8 for the notions of positive/negative subformulas:

Lemma 8.1. ForDandG the classes of definite, respectively goal formulas, we have that:

(1) each positive subformula of a D-formula is definite. (2) each negative subformula of a D-formula is goal. (3) each positive subformula of a G-formula is goal. (4) each negative subformula of a G-formula is definite.

Proof. We show each claim by case distinction on the subformulas and treat (1) and (2) simultaneously. The proof for (3) and (4) is very similar, so we leave it as an exercise.

LetA be a definite formula.

(1) We show that its positive subformulas are definite.

CaseP. P is a positive subformula of itself and definite.

Case ∀_xB(x). By Definition 3.5, B(x) is also a positive subformula of A and thus by the induction hypothesis it is definite. Then, by Definition 2.8 ∀xB(x) is definite.

Case B → C. If B → C is a positive subformula of A, then B is a negative subformula and C a positive subformula of B → C. Using the induction hypothesis, we have from (2) that B ∈ G and from (1) that C ∈ D. Thus, by Definition 3.5, B → C is a definite formula.

(2) We show that the negative subformulas of A are inG. If the negative subformula of A is

P, then this is a negative subformula of itself and in G.

∀xB(x), then by Definition 2.8 B(x) is also a negative subformula

of A and thus by the induction hypothesis it is in G. Then, by Definition 3.5 ∀_xB(x) is also inG.

B→C, thenBis a positive subformula andCa negative subformula of A. Using the induction hypothesis, we have A ∈ D (by (1)) and B ∈ G (by (2)). By Definition 3.5,A→B is therefore a goal formula.

Example 13. (Intuitive view of Lemma 8.1)

Consider the formulaD:= (A→B)→C to be a definite formula. By Definition 2.8, A is a positive subformula, B a negative subformula and C a positive subformula of D. The notions of positive and negative subformula are thus alternating.

Likewise, for D to be definite, by Definition 3.5, A needs to be a defi- nite, B a goal (such that A→B is in G) and C a definite formula. Hence, definite/goal subformulas are alternating in the same manner as the posi- tive/negative subformulas.

Thus, Lemma 8.1 can be viewed as mapping these notions - positive/negative formulas to definite/goal formulas.