Naturally growing trees

In general, if our task is to establish a derivability relation  C, with  a list of assumptions, we can keep proof search local by a suitable notation for derivations. If the conclusion under the assumptions is a conjunction 64

Pro of s e a rc h 65

A & B , it is sufficient to establish   A and   B . If it is an implication A ⊃ B , it is sufficient to establish A,   B . If it is a disjunc tion A ∨ B , it is sufficient to establish one of   A or   B .

Instead of analysing the conclusion, we can wor k on the assumption side.

If one assumption is a conjunc tion A & B and the der ivabilit y re lation to be established A & B,   C , it is sufficient to establish A, B,   C . If sequent calculus for ro ot-first pro of search. Instead of w r iting I and E for int ro duc tion and e limination, we w r ite R and L . T he for mer g ive a s a conclusion a sequent w ith the pr incipal for mula, i.e., the for mula w ith the connec t ive of the r ule, at r ig ht of the der ivabilit y relation, the latter w ith the pr incipal for mula at left. T he components of the pr incipal for mula in the premisses are the active formulas of a rule.

Table 4.1 Rules for root-first proof search

  A   B

These rules are closely related to the rules of natural deduction of Sec t ion 3.4. We say that we const r uc t der ivations w ith them, and that a sequent  C is derivable if it is the last line of a derivation constructed by the rules. This derivability is of a higher level as compared to derivability in the sense of the derivability relation for which the symbol is used. The terminology of sequents avoids the double meaning of derivability. The last sequent in a derivation is the endsequent of the derivation.

The R-rules for conjunction and implication are invertible: Whenever a sequent of the form   A & B is derivable, the sequents   A and

  B are derivable. Note that the last rule in the derivation of   A & B need not be R&. By invertibility, however,   A & B can be concluded by R&. Similarly, whenever a sequent  A ⊃ B is derivable, the sequent A,   B is derivable. Rules R∨1 and R∨2 are not invertible. On the antecedent side, rules L & and L∨ are invertible. With rule L⊃, if a sequent

of the form A⊃ B,   C is derivable, the sequent B,   C that matches the second premiss of the rule is derivable, but the sequent A⊃ B,   A need not be derivable.

It may seem strange that the principal formula of rule L⊃ is repeated in the first premiss. The repetition can be justified, though not motivated, as follows: If in an endsequent it is allowed to assume A⊃ B, the same assumption must be allowed anywhere.

To show that a sequent is derivable, we start decomposing formulas in it in the root-first direction. It is best to decompose first formulas that correspond to invertible rules, first those that have one premiss. Successive decomposition amounts to the root-first construction of branches in a derivation tree. A branch ends when we reach an initial sequent, either one that has the same formula as an assumption and conclusion, or one that has

⊥ as an assumption.

Table 4.2 Initial sequents A,   A ⊥,   C

A sequent of the form A,   A corresponds to the making of an assump-tion in derivaassump-tions of Gentzen’s tree form. A sequent of the form⊥,   C corresponds to rule⊥E . That is why it is sometimes written as a limiting case of a rule when the number of premisses is 0.

Some examples will show how the calculus of sequents works. It must be kept in mind that the derivations are best read the way they are constructed, i.e., starting from the root.

a. ( A⊃ B) & (B ⊃ C)  A ⊃ C

A⊃ B, B ⊃ C, A  A B, B ⊃ C, A  B C, B, A  C

B, B ⊃ C, A  C ^L⊃

A⊃ B, B ⊃ C, A  C ^L⊃

( A⊃ B) & (B ⊃ C), A  C ^{L &}

( A⊃ B) & (B ⊃ C)  A ⊃ C ^R⊃ b.¬¬(A ∨ ¬A)

¬(A ∨ ¬A), A  A

¬(A ∨ ¬A), A  A ∨ ¬A^R∨1 ⊥, A  ⊥

¬(A ∨ ¬A), A  ⊥ ^L⊃

¬(A ∨ ¬A)  ¬A ^R⊃

¬(A ∨ ¬A)  A ∨ ¬A^R∨2 ⊥  ⊥

¬(A ∨ ¬A)  ⊥ ^L⊃

 ¬¬(A ∨ ¬A) ^R⊃

Proof search 67

This second example goes through smoothly, but the same cannot be said of the corresponding derivation in Gentzen’s natural deduction, as in exercise 3(e) of the previous chapter.

The first, i.e. downmost, rule in example (b) has to be R⊃, the second L⊃. At this point, we notice that if the principal formula ¬(A ∨ ¬A) had not been repeated in the left premiss, that premiss would be  A ∨ ¬A which is underivable in intuitionistic logic. There were no choices up to that point, so that without repetition the calculus would not derive what it should derive. The first choice in the proof search was when we came to the said left premiss. Had we tried again rule L⊃, the part of derivation would have become:

¬(A ∨ ¬A)  A ∨ ¬A ⊥  A ∨ ¬A

¬(A ∨ ¬A)  A ∨ ¬A ^L⊃

The left premiss is identical to the conclusion, so nothing was gained by the step, but only a loop produced. Therefore:

No derivation branch must contain the same sequent twice.

The rules of the calculus of sequents are local, and therefore loops can be eliminated by deleting the part of derivation between the two occurrences.

In this case, the second premiss⊥  A ∨ ¬A and its (degenerate) derivation would have fallen off the derivation tree.

With loops forbidden, the only choice for the third rule root first is R∨1

or R∨2. Then, if we try R∨1, the premiss is¬(A ∨ ¬A)  A and the only applicable rule, namely L ⊃, will again give a loop:

¬(A ∨ ¬A)  A ∨ ¬A ⊥  A

¬(A ∨ ¬A)  A ^L⊃

¬(A ∨ ¬A)  A ∨ ¬A^R∨1

The identical sequents are separated by two steps. The loop is deleted by continuing from the left premiss of L⊃ in the way the original derivation continues from the conclusion of R∨1. A whole derivation branch falls off, as above.

With loops forbidden, the only rule applicable above the two last ones in the derivation is R∨2. If to the premiss¬(A ∨ ¬A)  ¬A rule L⊃ is applied, a loop is again produced. Therefore the rule to be applied is R⊃

and the premiss is¬(A ∨ ¬A), A  A ∨ ¬A. At this point, it is seen that rule R∨1gives an initial sequent as a premiss.

If instead of R∨1 as an upp er most r ule we t r y R∨2 , we get the sequent

¬( A ∨ ¬A), A  ¬A to w h i c h R⊃ has to b e applied lest a loop b e pro-duce d. The result is ¬( A ∨ ¬A), A, A  ⊥. Now rule L⊃ gives as the first premiss¬( A ∨ ¬A), A, A  A ∨ ¬A, and it is the only applicable r ule. We get a sequent that is just like an earlier one, except for the duplicat ion of the for mula A in the antecedent par t. O bv iously nothing is g ained by the dupli-cation, but we note that proof search can fail for the reason that the search can go on pro ducing multiplications fore ver. To avoid the phenomenon, we show that no r ule instances are needed in which a sequent is pro duce d that is exac tly like some pre v ious sequent in the same br anch, save for possible duplications.