G. Bellin. Proof nets for multiplicative and additive linear logic. Technical Report LFCS-91-161, LFCS, Division of Informatics, University of Edinburgh, 1991.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....logic (MLL) in which the two MLL connectives and O are allowed to occur with either positive or negative annotations and can be applied only if their arguments are appropriately polarized. This view originates from the Danos R egnier system of polarities [91] and Bellin and van de Wiele s work [17, 18]. 4.1.1 Polarization For a start, we show how to transform an IMLL formula A to a positive polarized form pAq and a negative polarized form xAy. The polarized forms are constructed from polarized atoms (a ; a ; b ; b ; and polarized connectives ( O ) by mutual recursion: paq = ....

G. Bellin. Proof nets for multiplicative and additive linear logic. Technical Report LFCS-91-161, LFCS, Division of Informatics, University of Edinburgh, 1991.

....permutabilities in G6 can be seen in Table 2.2. Permutation of inference rules in a sequent system seems to be a syntactic notion its relationship to semantics is not a straightforward issue. 2.1. 2 Linear Logic Permutation of inferences in linear logic has also been studied, notably by Bellin ([Bel93]) and by Galmiche Perrier ( GP94] These studies consider full classical linear logic with a one sided sequent presentation. In Tables 2.3 and 2.4 we present the results of Bellin and Galmiche Perrier respectively (restricting to the propositional fragment) We are interested in the ....

G. Bellin. Proof Nets for Multiplicative and Additive Linear Logic. Technical Report LFCS-91-161, University of Edinburgh, 1993.

....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 ....

G. BELLIN. "Proof Nets for Multiplicative and Additive Linear Logic". Technical Report, Department of Computer Science, Edinburgh University, 1991. ECS-LFCS 91-161.

.... of proof search (such as path, formula tree or connection) and logical concepts related to the proof net notion (as axiom link, decomposition tree, proof structure) Such correspondences were often considered implicitly in works devoted to proof construction in direct logic [33] or linear logic [7, 8, 24] and the presentation of the relationships between connections and proof nets provides an opportunity to make them more explicit, from both the automated proof search and the logical points of view. 2 Linear logic and proofs The linear logic (LL) introduced by J.Y. Girard [27] is a logic of ....

....one axiom link. De nition 3.1 A pair (S ; P) is a proof structure for MLL if each atom of S is the conclusion of exactly one axiom link. A proof net is a proof structure that satis es a graph theoretic condition that represents certain consistency conditions on the structure as a whole [7]. For instance, in [27] such condition is de ned in terms of trips over (S ; P) that visit the formulae of a proof structure in two directions, the movements being determined by the nature of the links and by arbitrary choices (switches) In [33] where the decision procedure for direct logic ....

G. Bellin. Proof nets for multiplicative and additive linear logic. Technical Report ECSLFCS 91-161, Department of Computer Science, Edinburgh University, May 1991.

....naturally associate with a proof net pn( which has exactly the same multiset of terminal formulae as the multiset of those occurring in Gamma. Given any proof net fi, there is a proof of an mll sequent such that fi = pn( Danos Regnier Graph More recently, Danos and Regnier [DR89] see [Bel93] for a very readable account) found an alternative soundness condition for proof nets. They avoid the need to reason with the bi directional flows of information particles through a formula an intuitive but somewhat complicated conceptual device which was used to great effect by Girard. ....

G. Bellin. Proof nets for multiplicative and additive linear logic. 1993. unpublished manuscript.

....additives connectives but our aim is to focus on proof nets in MLL. The rules defined above allow to associate naturally a proof net Pi to a sequential proof Pi in MLL. This notion of proof net inductively defined corresponds to the classical ones based on some information circulation criteria [4, 9]. 3.4 Example We can construct at the same time the sequential proof Pi 1 (see 3.1) and the corresponding proof net Pi 1 for the sequent A Omega B; B Omega C ; A C. Whatever order is used to apply the construction rules, the final proof net is the same. The set of input output ....

....some performance problems. But some works, like [12] emphasize the necessity to extend the work, for logic programming application, to additive and multiplicative linear logic (AMLL) and even to LL. To do this, we have two possibilities: the first one is to extend the proof net notion to AMLL [4] and to consider its automated construction by a similar approach. A second one is to consider the problem of proof construction directly in AMLL with a specific, and completely different, decision procedure [8] and to study its relationship with extension of proof nets notion. Even so, this ....

G. Bellin. Proof nets for multiplicative and additive linear logic. Technical Report ECS-LFCS 91-161, Department of Computer Science, Edinburgh University, May 1991.

....to additive and multiplicative linear logic (AMLL) and even to complete linear logic (LL) Then if we do not want to restrict our point to classical proof nets in MLL but to consider proofs in AMLL or LL, we have two possibilities. The first one is to extend the proof net notion to AMLL [4] and to abord its automated construction by a similar approach. Let us note that a direct extension of the presented algorithm appears problematic because it will not consist only in connecting open edges and considering the additive connectors terminal branches are not given a priori with the ....

G. Bellin. Proof nets for multiplicative and additive linear logic. Technical Report ECSLFCS 91-161, Department of Computer Science, Edinburgh University, May 1991.

.... sequent calculus, we observe the necessity to study permutability properties of inferences because they can justify non determinism reduction (inference application order) and support efficient proof search strategies proposals [13, 15] They are partially used in works on proof search analysis [3] but often without a precise definition. Intuitively, it means the possibility to invert two inferences in a proof without disturbing the rest of the proof (the parts below and above the inferences) We have proposed a corresponding formal definition and we have studied the permutability ....

G. Bellin. Proof nets for multiplicative and additive linear logic. Technical Report ECSLFCS 91-161, Department of Computer Science, Edinburgh University, May 1991.

.... sequent calculus, we observe the necessity to study permutability properties of inferences because they can justify non determinism reduction (inference application order) and support eOEcient proof search strategies proposals [13, 15] They are partially used in works on proof search analysis [3] but often without a precise de nition. Intuitively, it means the possibility to invert two inferences in a proof without disturbing the rest of the proof (the parts below and above the inferences) We have proposed a corresponding formal de nition and we have studied the permutability properties ....

G. Bellin. Proof nets for multiplicative and additive linear logic. Technical Report LFCS91 -169, University of Edimburg, May 1991.

....fragment of linear logic [12] from the fact that the sequential presentation of proofs with trees is inadequate and does not emphasize their meaning. In [11] we have investigated automatic proof net construction but the point is to be able to extend this notion to more important fragments of CLL [5] through a new appropriate representation and definition of proof nets. Having it, it would be interesting to apply the previous results on normalization directly on this concept with a view to reducing not useful redundancies. As in [16] where the uniform proof notion is essential for proof ....

G. Bellin. Proof nets for multiplicative and additive linear logic. Technical Report ECSLFCS 91-161, Department of Computer Science, Edinburgh University, May 1991.

.... the proof theory (though not necessarily lock step simulation) by different methods: ffl Modify the calculus reduction strategies to more closely mimic those inherent in logic (e.g. i) by introducing guarding of terms [25] or (ii) 3 by a version of Girard s theory of slicing of proof nets [13, 6]) ffl Modify linear logic evaluation strategies to take into account the extant theory of calculus (e.g. restriction to Geometry 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 ....

....the 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 ....

[Article contains additional citation context not shown here]

G. Bellin. Proof Nets for Multiplicative and Additive Linear Logic, Report LFCS-91-161, May 1991, Dept. of Computer Science, Univ. of Edinburgh.

.... the proof theory (though not necessarily lock step simulation) by different methods: ffl Modify the calculus reduction strategies to more closely mimic those inherent in logic (e.g. i) by introducing guarding of terms [25] or (ii) 3 by a version of Girard s theory of slicing of proof nets [13, 6]) ffl Modify linear logic evaluation strategies to take into account the extant theory of calculus (e.g. restriction to Geometry 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 ....

....the 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 : 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) ....

[Article contains additional citation context not shown here]

G. Bellin. Proof Nets for Multiplicative and Additive Linear Logic, Report LFCS-91-161, May 1991, Dept. of Computer Science, Univ. of Edinburgh.

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G. Bellin. Proof Nets for Multiplicative and Additive Linear Logic, Report LFCS-91-161, May 1991, Dept. of Computer Science, Univ. of Edinburgh.

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