Inria / Raweb 2004
Project-Team: CALLIGRAMME

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Project-Team : calligramme

Section: Scientific Foundations


Introduction

Project-team Calligramme's research is conducted at the juncture of mathematical logic and computer science. The scientific domains that base our investigations are proof theory and the $ \lambda$-calculus, more specifically linear logic. This latter theory, the brainchild of J.-Y. Girard [41] results from a finer analysis of the part played by structural rules in Gentzen's sequent calculus [40]. These rules, traditionally considered as secondary, specify that the sequences of formulas that appear in sequents can be treated as (multi) sets. In the case of intuitionistic logic, there are three of them:

Im1 ${\mtable{...}{\mtext (Weakening)}~\mtable{...}{\mtext (Contraction)}~\mtable{...}{\mtext (Exchange)}}$

These rules have important logical weight: the weakening rule embodies the fact that some hypotheses may be dropped during a derivation; in a similar fashion the contraction rule specifies that any hypothesis can be used an unlimited number of times; as for the exchange rule it stipulates that no order of priority holds between hypotheses. Thus, the presence of the structural rules in the ordinary sequent calculus strongly conditions the properties of the logic that results. For example, in the Gentzen-style formulations of classical or intuitionistic logic, the contraction rule by itself entails the undecidability of the predicate calculus. In the same manner, the use of the weakening and contraction rules in the right half of the sequent in classical logic is responsible for the latter's non-constructive aspects.

According to this analysis, linear logic can be understood as a system that conciliates the constructivist aspect of intuitionistic logic and the symmetry of classical logic. As in intuitionistic logic the constructive character comes from the banning of the weakening and contraction rules in the right part of the sequent. But simultaneously, in order to preserve symmetry in the system, the same rules are also rejected in the other half.

Table 1.
Propositional linear logic
Rudimentary linear logic
NegationMultiplicativesAdditivesExponentials
NegationIm2 $A^\#8869 $
ConjunctionA$ \otimes$BA&B
DisjunctionIm3 ${A\#9739 B}$A$ \oplus$B
ImplicationIm4 ${A\#8888 B}$
Constants1, $ \bottom$$ \top$, 0
Modalities!A, ?A

The resulting system, called rudimentary linear logic, presents many interesting properties. It is endowed with four logical connectors (two conjunctions and two disjunctions) and the four constants that are their corresponding units. It is completely symmetrical, although constructive, and equipped with an involutive negation. As a consequence, rules similar to De Morgan's law hold in it.

In rudimentary linear logic, any hypothesis must be used once and only once during a derivation. This property, that allows linar logic to be considered as a resource calculus, is due, as we have seen, to the rejection of structural rules. But their total absence also implies that rudimentary linear logic is a much weaker system than intuitionistic or classical logic. Therefore, in order to restore its strength it is necessary to augment the system with operators that recover the logical power of the weakening and contraction rules. This is done via two modalities that give tightly controlled access to the structural rules. Thus, linear logic does not question the usefulness of the structural rules, but instead, emphasizes their logical importance. In fact, it rejects them as epitheoretical rules [37] to incorporate them as logical rules that are embodied in new connectors. This original idea is what gives linear logic all its subtlety and power.

The finer decomposition that linear logic brings to traditional logic has another consequence: the Exchange rule, which so far has been left as is, is now in a quite different position, being the only one of the traditional structural rules that is left. A natural extension of Girard's original program is to investigate its meaning, in other words, to see what happens to the rest of the logic when Exchange is tampered with. Two standard algebraic laws are contained in it: commutativity and associativity. Relaxing these rules entails looking for non-commutative, and non-associative, variants of linear logic; there are now several examples of these. The natural outcome of this proliferation is a questioning of the nature of the structure that binds formulas together in a sequent: what is the natural general replacement of the notion of (multi) set, as applied to logic? Such questions are important for Calligramme and are addressed, for example, in [6].

The activities of project-team Calligramme are organized around three research actions:

The first one of these is essentially theoretical, the other two, presenting both a theoretical and an applied character, are our privileged fields of application.


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