Calculation of the séquents

In 1935 Gentzen proposed the natural Déduction, a formalism to describe the evidence of the calculation of the predicates, whose idea was to stick to more close with the way in which the mathematicians reason. It then tried to use the natural deduction to produce a syntactic proof of the coherence of the Arithmétique, but the technical difficulties led it to reformulate the formalism in a version more symétrique : the calculation of the séquents . It is within this framework that it showed what was to become one of the principal theorems of the Théorie of the demonstration : the theorem D ''' elimination of the cuts ''. Dag Prawitz showed into 1965 that this theorem could be transported to the natural deduction

Definition

One gives here a version slightly modernized compared to that of Gentzen of the calculation of the séquents LK which formalizes the traditional Logique. One will see low than on the same principle one can define LJ, a calculation of the séquents for the Logique intuitionalist, L for the linear Logique… the definition below supposes a minimum of familiarity with the Calcul of the predicates.

Terminological note

The term calculation of the séquents is a translation of English sequent calculus , itself inherited German Sequenzenkalkül . The literal translation of French German would rather give calculation of the sequences and one finds certain authors using this terminology. However the use more the current imposed the use of the neologism séquent .

Séquent

The basic object of calculation is the séquent , which is a couple of finished lists (possibly empty) of formulas. The séquents are usually notés :

A_1, \ dowries, A_n \ vdash B_1, \ dowries, B_p

where the

A_i

are the assumptions and the

B_j

are the conclusions séquent. One sees here appearing the major innovation of the calculation of the séquents : perfect symmetry between assumptions and conclusions. Another point important to note is that the same formula can appear several times on the left and/or on the right in one séquent, for example

A_1

can be the same formula as

A_n

; it is said whereas this formula has several occurrences in séquent.

Interpretation of one séquent

The notation séquent can be included/understood like a syntactic easy way to indicate formulas particulières ; the comma on the left is interpreted like a conjunction, the comma on the right like a disjunction and the symbol $\ vdash$ like a Implication, so that séquent it above can include itself/understand like a notation for the formule :

A_1 \ and \ dowries \ and A_n \ rightarrow B_1 \ gold \ dowries \ gold B_p

Rules

The rules of the calculation of the séquents specify how starting from a certain number (possibly no one) of séquents premises , one can derive new séquent conclusion. In each rule the Greek letters Gamma, Delta, etc indicate continuations of formulas, one makes appear the séquents premises above, separated by a horizontal feature from séquent conclusion. They are divided into three groupes : the group identity , the structural group and groups it logical .

; group identity:

A discussion on the rule of cut appears low. Let us notice that the rule axiom is the only one of all the calculation which does not have séquent premise.

For the groups structural and logical, the rules come per pair according to the side, left or right, of séquent where they act.

; structural group:

In the rules of exchange the notation $\ sigma (\ Gamma)$ indicates a permutation of (occurrences of) the formulas appearing in $\ Gamma$ . The rules of exchanges are responsible for the commutation logic.

The rules of contraction and weakening express the shape of idempotence of the operators “ $\ scriptstyle \ lor$ ” and “ $\ scriptstyle \ land$ ” of logic : a formula $A$ is equivalent to the formulas $\ scriptstyle has \ lor has$ and $\ scriptstyle has \ A$ Land.

; logical group:

The rules of quantifiers are subjected to restrictions : in the rules of $\ forall$ -droite and $\ exists$ -gauche, one asks that the variable $x$ not appear in any the formulas of $\ Gamma$ and $\ Delta$ . The notation $A$ indicates the formula $A$ in which the variable $x$ is replaced by a $t$ term.

Demonstrations

A demonstration of LK is a tree of séquents built by means of the rules above so that each séquent is conclusion of exactly a règle ; the demonstration is finished if one arrives so that all the sheets of the tree are rules axiom. Séquent root of the tree is the conclusion of the demonstration. Here an example of demonstration of the principle of contraposition of the implication ; to simplify one omitted the uses of rules of exchange and represented in fat them (occurrences of) formulas in the séquents premises to which each règle relates;:

Discussion

The calculation of the séquents and other formalisms logical

The calculation of the séquents, contrary to the Systems in Hilbert or the natural Deduction, is not a formalism very intuitif ; the matter of Gentzen was to solve certain engineering problems encountered in natural deduction during the demonstration of the coherence of the arithmetic one. This is why the calculation of the séquents must be thought like a formalism for to rather reason on the formal evidences than for to write formal evidences, it to what the natural deduction is adapted. One can however easily show that any provable formula in one of the systems is also in the other.

The interest of the calculation of the séquents is to make explicit a great number of properties of the logique :

the duality assumption/conclusion expressed by perfect symmetry enters the right-hand side and the left the séquents ;
the symmetry of the traditional logic expressed by the partitioning of the rules in left and droite ;
properties structural (commutation, idempotence) expressed by the rules of the structural group.

The rule of cut

The rule of cut is a generalization of the modus-ponens . To see it is necessary to consider the particular case of the rule où :

$\ Gamma$ , $\ Delta$ and $\ Gamma'$ are empty sequences of formules ;
$\ Delta'$ contains a single formula $B$ .

In this particular case both séquents premise of the rule become respectively $\ vdash A$ and $A \ vdash B$ while séquent it conclusion becomes $\ vdash B$ .

The rule of cut plays a very particular part in the calculation of the séquents for two raisons :

it is essential to formalize the evidence mathématiques ; indeed the experiment shows that, except for the easy demonstrations of tautologies, the least proof of mathematics almost exclusively uses the rule of coupure ;
more technically, it is the only rule to violate the property of the subformula ; if one observes the rules of the calculation of the séquents one sees that for each rule, except the cut, the formulas appearing in the séquents premise are still present in séquent conclusion. In other words, if a demonstration of the calculation of the séquents does not use the rule of cut, then all the formulas appearing inside are subformulas of séquent conclusion, i.e. séquent it conclusion of the demonstration is most complicated of all the demonstration.

The property of the subformula was very important for Gentzen because it is it which makes it possible to show the coherence of the calcul : indeed simplest possible is séquent it vacuum which séquent it contains any formula on the left neither, nor on the right. However this séquent expresses a contradiction. To see that calculation is coherent it is thus enough to see that séquent it vacuum is not prouvable ; but a demonstration without cut of séquent vacuum should contain only simpler séquents, and there is none of it. Thus any demonstration of séquent vacuum must use the rule of cut. And here is how Gentzen was led to show its famous theorem D ''' elimination of the cuts '' which precisely stipulates that of any demonstration of one séquent, one can extract a demonstration without cut from same séquent, which shows in corollary that there does not exist any demonstration of séquent vacuum. It is by adapting this demonstration of elimination of the cuts to the case of arithmetic that Gentzen produced its proof of coherence of the arithmetic one.

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