Citations: Archive for Mathematical Logic - Danos, Regnier, of (ResearchIndex)

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V. Danos and L. Regnier. The structure of multiplicatives. Archive for Mathematical Logic, 28:181--203, 1989.

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On Semantic and Type-Theoretic - Aspects Of Polynomial-Time (Correct)

....ane fragments (diagonal arrows) The path we decided to take is mapped out with dashed arrows. 54 IMLL2 IMLLL2 IMLLL To our knowledge, this is the rst attempt to extend essential nets to these fragments. Proof nets for rst order MLL which extend the Danos R egnier (DR) criterion [34] have already been proposed in [40] from which it is an easy step to derive a correctness criterion for MLL2 (i.e. second order classical MLL) Although IMLL2 is a sublogic of MLL2, essential nets are quite a di erent kind of structure from proof nets; knowing the corresponding DR criterion is ....

V. Danos and L. Regnier. The structures of multiplicatives. Archive for Mathematical Logic, 28:181-203, 1989.

Polarity Items in Resource Logics. A comparison - Bernardi (2000) (2 citations) (Correct)

....power of the structural component allowing to permute and change the order of the composed structures. Following the rst solution we obtain Multimodal (non associative) Lambek Calculus (MMNL) Moortgat,1997] whereas adding permutation and associativity we reach Multiplicative Linear Logic (MLL) [Danos and Regnier,1989]. The aim of this paper is to show the advantages of having unary operators at our disposal when reasoning with linguistic structures. In particular, we will look at expressions, known as polarity items, which are sensitive to the monotonicity properties of the context they occur in ....

V. Danos and L. Regnier. The structure of multiplicatives. Archive for Mathematical Logic, 28:181-203, 1989.

Proof Nets with Explicit Negation for Multiplicative Linear Logic - Puite (1998) (Correct)

....we call X Y provable. A substructure of a proof structure (in particular of a proof net) S is called a subnet of S iff, regarded as a proof structure, it is a proof net. Upsilon Alternatively we can define the notion of proof net by means of a generalization of the long trip condition ( Gi87] [DR89]; see also [Tr92] or by means of a homological characterization (cf. Me94] The only thing we have to do is to treat our thin links as if they were R links, and our thick links as if they were R Omega links, while the X links may be treated as Ax or Cut links. Example 3.2 (Proof nets) ....

V. Danos and L. Regnier. The structure of multiplicatives. Archive for Mathematical Logic, 28:181--203, 1989.

Games and Full Completeness for Multiplicative Linear Logic - Samson Abramsky And (1994) (132 citations) (Correct)

....does not. Then, we prove a Full Completeness Theorem for this semantics, with respect to MLL MIX (Multiplicative Linear Logic plus the Mix Rule) Recall that the MIX rule [Gir87] has the form Gamma; Delta There is a notion of proof net for this logic: this uses the Danos Regnier criterion [DR89], simply omitting the connectedness part. Thus, a proof structure will be a valid proof net for MLL MIX just if, for every switching, the corresponding graph is acyclic. This criterion was studied by Fleury and Retor e [FR90] used by Blute in his work on coherence theorems [Blu92] and adapted ....

....that 1; 1 and ; are derivable from MIX and MIX0. But with 1 = clearly any sequent will be equivalent to one in which the units do not occur. Thus, we prefer to omit the units from our system. 2. 2 Proof nets for MLL MIX Proof structures can be defined for MLL MIX just as for MLL [Gir87, DR89]. Alternatively, since we only allow atomic instances of identity axioms, we can define a proof structure to be a pair ( Gamma; OE) where Gamma is a sequent and OE is a fixpoint free involution on the set of occurrences of literals in Gamma, such that, if o is an occurrence of l, OE(o) is an ....

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V. Danos and L. Regnier. The structure of multiplicatives. Archive for Mathematical Logic, 28:181--203, 1989.

Evolving Games and Essential Nets for Ane Polymorphism - Murawski Ong Oxford (3 citations) (Correct)

....correctness criterion is motivated by the evolving game model and corresponds to the rule that the game imported by P is determined uniformly by those imported by O earlier. To our knowledge, the results in this section are also new. Proof nets for rst order MLL which extend the DR criterion [5] for MLL have already been proposed in [6] from which it is an easy step to derive a correctness criterion for MLL2 (i.e. second order classical MLL) Although IMLL2 is a sublogic of MLL2, essential nets are quite a di erent kind of structure from proof nets; knowing the corresponding ....

Danos, V., Regnier, L.: The Structures of Multiplicatives. Archive for Mathematical Logic, Vol. 28, 1989, 181-203

The Additive Multiboxes - de Falco (2000) (Correct)

....2 Preliminaries: Proof nets of MALL and their normalization This section is devoted to set the stage. Proof nets of LL were introduced by Jean Yves Girard in [Gir87] Several simpli cations have been discovered 4 since that rst work, mainly by Vincent Danos and Laurent Regnier (see for example [DR89]) Despite the extensive use of the notion of proof net, after the paper [Gir87] the rst work dealing with a notion of proof net for full second order LL (without constants) is [TdF01] see also [TdF00b] at least as far as we know. We present here the restriction to the fragment MALL of the ....

....of MALLS is associated a (unique) proof net. What is less trivial is that the converse also holds: 2.7. Theorem. If R is a proof net, then there exists a proof of MALLS s.t. R is the proof net associated with . The proof is then called a sequentialization of R. Proof: Straightforward from [DR89]. 2.8. Remark. The proof net associated with a proof of MALLS which does not make use of the rule Hyp does not contain any link H, and conversely. In the sequel, we will restrict to these notions of proof and proof net, and will speak of proof nets with hypothesis (resp. proof with hypothesis ....

Vincent Danos and Laurent Regnier. The structure of multiplicatives. Archive for Mathematical Logic, 28:181-203, 1989.

Interface in Linear Logic: A Finite System of Generators - Bechet (Correct)

....blocked forever by an other one in a cycle) To check if a net is correct, a set of switching positions is associated to each king of cell. A net is said to be correct if for every switching of the cells, no cycle appears. To characterize the connection property of a bit of net (called a module [DR89]) with other nets, its interface may be computed. Now, two modules can be connected together if there interfaces are orthogonal. More precisely, an interface is a set of partitions and a partition is an equivalence relation between the external ports of a module. Two partitions are said to be ....

V. Danos and L. Regnier. The structure of multiplicatives. Archive for Mathematical Logic, 28:181--203, 1989.

Constructing Different Phonological Bracketings From a Proof-Net - Bechet, de Groote (1996) (2 citations) (Correct)

.... some correctness criterion that allows them to be discriminated from the other proof structures (therefore the notion of proof net may be dened without making any explicite reference to the sequent calculus) In order to introduce such a correctness criterion, which is due to Danos and Regnier [4], we must dene the notion of switching. A switching of a proof structure is a selection for every link between the left or the right position. The graph underlying such a switching is obtained by replacing each link by a single edge as follows: 5 J J J Omega Omega Omega left ....

....each of these axioms by a path axiom cut axiom. 2. If n = 1 then stop. the interpolation problem is already solved) 3. Otherwise, nd two adjacent cuts that obey the property described in Lemma 7.2. This corresponds to nding, in a module, two conclusions that are connected for any switching [4]. This last property is reminiscent of the notion of empire and may be checked using Girard s closure conditions on empires [8] 4. Replace the two adjacent cuts by a single cut as specied in Lemma 7.2. 5. Decrease n by 1 and go to step 2. 14 This algorithm, which works on proof nets, does not ....

V. Danos and L. Regnier. The structure of multiplicatives. Archive for Mathematical Logic, 28:181203, 1989.

On Double Categories and Multiplicative Linear Logic - Melliès (1999) (Correct)

....1994) In this article, we prefer the more elegant solution o ered by the concept of module de ned hereafter. parsing net. A parsing box is a link with a variable number of non distinguished ports: A parsing net is a net possibly containing parsing boxes. Well formed parsing nets. (Danos and Regnier 1989) presents a geometrical characterisation of well formed nets inspired by the sequentialisation theorem of (Girard 1987) a net is well formed if and only if every switching de nes a tree ( connected acyclic z To explain this phenomenon in the calculus, let us say that a term w:yP where ....

V. Danos, L. Regnier, The structure of multiplicatives. Archive for Mathematical Logic, 28:181203, 1989.

Evolving Games and Essential Nets for Affine Polymorphism - Murawski, Ong (1 citation) (Correct)

....correctness criterion is motivated by the evolving game model and corresponds to the rule that the game imported by P is determined uniformly by those imported by O earlier. To our knowledge, the results in this section are also new. Proof nets for rst order MLL which extend the DR criterion [5] for MLL have already been proposed in [6] from which it is an easy step to derive a correctness criterion for MLL2 (i.e. second order classical MLL) Although IMLL2 is a sublogic of MLL2, essential nets are quite a di erent kind of structure from proof nets; knowing the corresponding ....

V. Danos and L. Regnier. The structures of multiplicatives. Archive for Mathematical Logic, 28:181-203, 1989.

Digital Equipment Corporation 1991 - This Work May (Correct)

....a sequential deduction. In order to single out which deduction nets really correspond to sequential deductions, one needs a global criterion. In his seminal paper, Girard gave such a criterion for proof nets, the long trip condition [7] Later, Danos and Regnier proposed a different criterion [5]. We now present the Danos Regnier criterion for soundness of a deduction net. This criterion is equivalent to Girard s original trip conditions criterion, but it is somewhat more manageable. It is convenient to consider that there are two kinds of edges: 1) Edges connecting the exit of A and B ....

.... net P, a switch graph associated with P is any subgraph of P obtained by deleting exactly one of the two soft edges associated with every par node in P (and keeping the other soft edge) The Danos Regnier criterion for soundness of a deduction net is stated as follows (see Danos and Regnier [5], and Danos [4] Definition 23 A deduction net P satisfies the Danos Regnier criterion, or is sound, iff every switch graph associated with P is a tree. Research Report No. 9 May 1991 36 Jean Gallier For example, the following is a sound proof net, since both switch graphs are trees: A B ....

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V. Danos and L. Regnier. The structure of multiplicatives. Archive for Mathematical Logic, 28:181--203 (1989).

Polarized Proof-Nets: Proof-Nets for LC (Extended Abstract) - Laurent (Correct)

....in the study of proof nets is the problem of correctness criterions that is the problem to know whether a proof structure is a proof. More technically, can you inductively deconstruct a proof structure There exist different correctness criterions for multiplicative proof structures like [3] or [2] which lead to the criterion of [7] for the full case. We present here this general criterion. Definition 8 (Sequentialization of a proof structure) The relation L sequentializes R into E is defined for each possible L. R is a proof structure, E is a set of proof structures and L is a ....

Vincent Danos and Laurent Regnier. The structure of multiplicatives. Archive for Mathematical Logic, 28:181--203, 1989.

Concurrent Construction of Proof-Nets - Andreoli, Mazare (2003) (Correct)