V. Danos and L. Regnier. The structure of multiplicatives. Arch. Math. Logic, 28:181--203, 1989.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....structures to those of Girard. The second requirement for a set of linkings # to be a proof net is pointwise MLL correctness : P2) Every linking of # induces an MLL proof net. In other words, for each linking # #, the MLL proof structure induced by # is an MLL proof net, in the usual sense [Gir87, DR89]. To be self contained, we characterise (P2) explicitly below. Henceforth view a sequent # as a graph: a disjoint union of parse trees, with literals above. For a linking # on # obtain the graph # of # from the additive resolution # # # (a subgraph of #) by adding each axiom link a of # as a ....

V. DANOS & L. REGNIER (1989): The structure of multiplicatives. Arch. Math. Logic 28, pp. 181--203.

....logic, contains too many redundancies to be useful for e# cient proof search. Attempts to remove permutabilities from sequent proofs [1, 10] and to add proof strategies [23] have provided significant improvements. But because of the use of sequent calculi some redundancies remain. Proof nets [7], on the other hand, can handle only a fragment of the logic. Matrix characterizations of logical validity, originally developed as foundation of the connection method for classical logic [2, 3, 5] avoid many kinds of redundancies contained in sequent calculi and yield a compact representation ....

V. Danos & L. Regnier. The structure of the multiplicatives. Arch. Math. Logic 28:181--203, 1989.

....nets of linear logic [4] in particular, an MNL proof net is a graph which satis es certain speci c trip condition. We now introduce a correctness criterion for MNL proof nets which can be considered as the non commutative counterpart to the Danos Regnier criterion for proof nets of linear logic [3], based on the idea that a proof net Equipe Logique de la Programmation, Institut de Math ematiques de Luminy, CNRS. 163, avenue de Luminy, Case 907, 13288 Marseille Cedex 9, France. e mail: maieli iml.univ mrs.fr Research supported by the EU TMR Research Programme Linear Logic and Theoretical ....

....allows the mapping of any proof net onto a sequent proof via cut elimination. This new criterion aims to develop a simple theory of modules [1] for MNL which could represent the natural non commutative counterpart of some linear correctness criteria like that one based on switching partitions [3]. The rest of this paper is organized as follows: Section 2 gives the basic de nitions and properties of order varieties (Section 2.1) and the sequent calculus of MNL (Section 2.2) Proof nets and cut free sequentialization are presented, respectively, in Section 3 and Section 4. Sequentialization ....

V. Danos and L. Regnier. The structure of multiplicatives. Arch. Math. Logic, 28(3):181-203, 1989.

....A B being partitions of A B. D is equivalent to the free symmetric monoidal category generated by a relational Frobenius object (see [36] D is in fact already compact closed. For maps p : A B and q : B A in D, that is partitions p and q of A B, the Danos Regnier orthogonality [22] is p q if and only if the graph induced by p and q is connected and acyclic. iii) Let D be the traced monoidal category of nite sets and relations with as tensor product. Given relations u : A B and x : B A we can set u x if and only if u x is nilpotent if and only if x u is ....

V. Danos and L. Regnier. The structure of the multiplicatives. Arch. Math. Logic, 28:181-203, 1989.

....X, A # B, Y [#] # X, A # B, Y # X, A, B [ # X, A B # t [t] # X [f] # X, f # X, # [#] # X, A [ # X, A # X, A [ # X, A # X [K ] # X, A # X, A, A [WI ] # X, A 2.9.2 Proof Nets [ This section must be added. Relevant citations will be from among [26, 43, 57, 66, 107, 117, 119]] 2.10 Curry Howard Some logicians have found that it is possible to analyse proofs more closely by giving them names. After all, if proofs are first class entities, we will be better off if we can distinguish different proofs. I can illustrate this by looking at an example from intuitionistic ....

VINCENT DANOS AND LAURENT REGINER. "The Structures of Multiplicatives". AoML, 28:181--203, 1986.

....putting axiom links on the proof frame. Testing the long trip condition as it stands is not attractive computationally since in a proof structure with i links and j Omega links there are 2 i j switchings to be tried. The situation is improved with the correctness criterion as formulated by Danos and Regnier (1989), which considers only switchings of links. For any given switching, a certain graph results by removing from an expanded proof net the edges between each conclusion and its closed premise. The result of Danos and Regnier is that a proof structure is a proof net if and only if for every ....

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

.... of Interaction style evaluation strategies [17, 18] One of the theses of this work is that the Abramsky style translations (of linear logic) into the process world actually have less to do with logic than one might think: they are essentially only about the abstract pluggings in proof structures [13, 16, 12], and we formulate many of our results in this more general setting. This phenomenon may be already expected by experts in concurrency theory, who view process algebras as a theory of hand shaking protocols (which have nothing to do with logic) but it came as a surprise to us. What seems ....

....role of the units in all that follows. For simplicity, we also consider only atomic axiom links. Much of the proof theory used in this paper is standard; further details for the case of linear logic are contained in Girard s original paper [13] Troelstra s recent book [32] as well as in [6, 7] [12], etc. 9 3.1 The Abramsky Translation: the multiplicatives Logical Rule translation x : A; y : A Ixy = x(a)yhai . F w : Gamma; x : A . G v : Delta; y : B w : Gamma; v : Delta; z : A Omega B Omega x;y O z (F; G) w vz = xy(zhxyi(F wx k G vy) ....

[Article contains additional citation context not shown here]

V. Danos and L. Regnier. The Structure of Multiplicatives, Arch. Math. Logic , 1989, 28, pp. 181-203.

....logic, contains too many redundancies to be useful for ef cient proof search. Attempts to remove permutabilities from sequent proofs [1,10] and to add proof strategies [23] have provided signi cant improvements. But because of the use of sequent calculi some redundancies remain. Proof nets [7], on the other hand, can handle only a fragment of the logic. Matrix characterizations of logical validity, originally developed as foundation of the connection method for classical logic [2,3,5] avoid many kinds of redundancies contained in sequent calculi and yield a compact representation of ....

V. Danos & L. Regnier. The structure of the multiplicatives. Arch. Math. Logic 28:181-203, 1989.

....structure then the DR graph is acyclic and connected. The following fundamental result relates provability (and sequent calculus) and proof nets in MLL [27] Theorem 3.1 S is provable in MLL if and only if there exists a set of axiom links P such that (S ; P) is a proof net. Proof 3. 1 See [11]. 8 This result gives another way of characterizing the MLL provability through the existence of a proof net and more precisely the existence of a set of so called axioms links. The relationships between such a set and a path (as previously de ned) are important to understand the connections ....

V. Danos and L. Regnier. The structures of multiplicatives. Arch. Math. Logic, 28:181203, 1989.

.... sequent, is in fact a proof structure that corresponds to legal proofs [21] There are several equivalent definitions of what a proof net is, that are generally based on criteria that characterize the proof structures that are proof nets [21] like for instance the Danos Regnier characterization [12]. The construction of a proof net can be seen as the search of the connections (axiom links) that characterize the provability of a given sequent [15] 18 LABELLED DEDUCTION To simplify the different representations, we often omit the formulae labelling the nodes and only represents the ....

V. Danos and L. Regnier. The structures of multiplicatives. Arch. Math. Logic, 28:181--203, 1989.

....circle, or as labelling radial lines on a disk. Since we will only consider such structures up to a rotation, then we will not need any explicit representation of the cyclic exchange rule [38, 31] It is possible to represent nets by an inductive procedure analogous to that of the commutative case [17, 15, 16]. Rather than describe the construction in detail, we present an example of a cyclic net in figure 1, and refer the reader to [38, 31] for the details of the definition. The net presented corresponds to a deduction of the following sequent : ffi ; ff Omega ffi ) Omega ff; ff ....

V. Danos, L. Regnier, The Structure of Multiplicatives, Arch. Math. Logic 28 (1989) pp. 181203

.... of Interaction style evaluation strategies [17, 18] One of the theses of this work is that the Abramsky style translations (of linear logic) into the process world actually have less to do with logic than one might think: they are essentially only about the abstract pluggings in proof structures [13, 16, 12], and we formulate many of our results in this more general setting. This phenomenon may be already expected by experts in concurrency theory, who view process algebras as a theory of hand shaking protocols (which have nothing to do with logic) but it came as a surprise to us. What seems ....

....of the units in all that follows. For simplicity, we also consider only atomic axiom links. Much of the proof theory used in this paper is standard; further details for the case of linear logic are contained in Girard s original paper [13] Troelstra s recent book [32] as well as in [6, 7] [12], etc. 9 3.1 The Abramsky Translation: the multiplicatives Logical Rule translation x : A; y : A Ixy = x(a)yhai . F w : 0; x : A . G v : 1; y : B w : 0; v : 1; z : A Omega B Omega x;y O z (F; G) w vz = xy(zhxyi(F wx k G vy) F ....

[Article contains additional citation context not shown here]

V. Danos and L. Regnier. The Structure of Multiplicatives, Arch. Math. Logic , 1989, 28, pp. 181-203.

....essential for the proof net construction consists in finding the good links between dual leaves. In fact, a proof structure is composed by the decomposition tree with a set of axioms links covering the leaves and there are various criteria to verify if a given proof structure is also a proof net [4,6,11]. A first approach could consist in building a proof structure and in verifying a posteriori if it is a proof net. Here we want to consider the proof net construction from an automated deduction point of view and then, from a given sequent, to directly and automatically construct a corresponding ....

.... It is well known that the proof nets of MLL are characterized by a simple and elegant graph theoretic condition, saying that any Danos Regnier graph (D R graph) is a proof net of MLL if and only if it is acyclic and connected under a choice of link switching 2 (switching condition) see [6] for more details) But what about the non commutative proof nets, firstly defined in [2] We now present the inductive definition of non commutative proof nets and the graph theoretic characterization from which we can justify the properties of our algorithm. Inductive non commutative proof ....

[Article contains additional citation context not shown here]

V. Danos and L. Regnier. The structures of multiplicatives. Arch. Math. Logic, 28:181--203, 1989.

....produced in the end, some rather similiar programming schemes. On the one hand, the connectionist paradigm (Hillis (1985) gave rise to the connection graphs of Bawden (1986) On the other, linear logic (Girard (1987) gave rise to the notion of multiplicative proof nets (Girard (1988) Danos and Regnier (1989)) and ultimately to the interaction nets of Lafont (1990, 1991) These two schemes are closely related since one can view interaction nets as special cases of connection graphs, possessing additional structure that enables powerful safety properties to be proved. The purpose of this paper is ....

....occurrences as a:w . Thus the threading in Fig. 14 starting at the point yields the longtrip a:t , a:t , b:x , b:y , b:z , b:z , b:y , b:x . The correspondence between such longtrips and families of nonintersecting chords of a circle has been pointed out by Danos and Regnier (1989). Definition 5.4 A threaded pattern or net (B, F) is a pattern or net B equipped with a set of threadings F (by possibly different hyperforests) Likewise for a specially threaded net, bearing in mind the additional constraint on cardinalities. Definition 5.5 Threaded patterns form a category TP. ....

[Article contains additional citation context not shown here]

Danos V., Regnier L. (1989), The Structure of Multiplicatives. Archive for Mathematical Logic 28, 181-203.

....proof structure. There is a proof net with roots A ffi ; A 1 ffl ; An ffl iff A 1 ; An ) A is a valid sequent. 2 This criterion, adapted from that of Lecomte and Retore (1995) derives from Girard s (1987) long trip condition , which is a highly involved mathematical result. Danos and Regnier (1989) express it in terms of acyclicity and connectivity of certain subgraphs. Intuitively, acyclicity assures that the subformulas of ii links (binary rules) occur in different subproofs, whereas connectivity assures that the subformulas of i links (unary rules) occur in the same subproofs (attributed ....

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

....of linear derivation (and thus different from the trivial inductive characterisation) for recognising proof nets among all proof structures. Soundness criteria for (variant of) Multiplicative Linear Logic have been proposed by Girard [Gi86] the well known long trip condition) Danos and Regnier [DR89, Da90], Roorda [Ro90] and Bellin [Be90] We briefly recall Danos and Regnier s acyclicity condition, since we shall use it to provide a first proof of the correctness of our condition. we shall be a bit informal here; see [Da90] for a detailed exposition) Definition 3.1 Let P be a proof structure. ....

V. Danos, L. Regnier. The Structure of Multiplicatives. Arch. Math. Logic, 28. 1989.

....McyLL. Definition 2.10 (Long trips and bilateral trips) Let be a proof structure of MNL and s a switching for . A trip v in s( is a long trip if v is a cycle and in v every occurrence of formula A in occurs twice, once as A once as A # . A cycle v in s( is bilateral (see [5]) if v is not of the form A x ; B y ; A x ; B y ; A x where A and B are occurrences of formulas in . Definition 2.11 (Proof nets) is a proof net (of MNL) iff is a proof structure of MNL and for every switching s for : 1) there is exactly one ....

.... is a proof net, there is exactly one cycle oe, whence oe is a long trip. Conversely, assume that there is a long trip in s( for every switching s for : if s is a switching for , the long trip oe in s( is the unique cycle in s( and satisfies (2) obvious) and (3) see Danos R egnier [5]) ii) If has no r link, the result is obvious. If has r links, and l is a r link in with conclusion ArB, then for every switching s, no conclusion occurs in the unique cycle oe in s( between B and A # : indeed, otherwise, by taking the switching s 0 such that s 0 (l) r3 and ....

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V. Danos and L. R'egnier. The structure of multiplicatives. Arch. for Math. Logic, 28(3):181--203, 1989.

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V. Danos and L. Regnier. The structure of multiplicatives. Arch. Math. Logic, 28:181--203, 1989.

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V. Danos and L. Regnier. 1989. The structure of multiplicatives. Arch. Math. Logic, 28:181--203.

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DANOS, V. AND REGNIER, L. 1989. The structure of multiplicatives. Archive for Mathematical Logic 28, 181--203.

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V. Danos and L. Regnier. The structure of multiplicatives. Arch. Math. Logic, 26, 1989. Referenced on pp. 7

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V. Danos and L. Regnier, The structure of multiplicatives. Arch. for Math. Logic, 28(3):181-203, 1989.

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

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V. Danos and L. Regnier. The structure of multiplicatives. Arch. Math. Logic, 26, 1989.

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V. DANOS & L. REGNIER (1989): The structure of multiplicatives. Arch. Math. Logic 28, pp. 181--203.

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