G. Bellin and J. van de Wiele. Proof Nets and Typed Lambda Calculus, manuscript.  Home/Search   Document Not in Database   Summary   Related Articles   Check

.... of the Direct Logic tradition: chains (we shall call them causal chains, here, in order to stress the causal dependencies they create in the proof structure) This notion already appears in [KW84] A direct proof of the sequentialisation theorem from this result can be found in [Be90] see also [BW93]) Definition 4.9 Let P be a proof structure, and let A; B be two instances of formulae in P . A causal chain (of type ; #) between A and B is a path in P of the following kind: A . Omega . Omega . B All the top links are atomic; all the ....

....Ret93] We conjecture that the correspondence between proofs and deadlock free systems could be extended Retor e s system. In this respect, the main problem is to find a nice notion of Proof Structure for the additives. In [ we proposed an approach based on boolean weights over links (see [Da90, Be93, BW93] for different approaches to the same problem) In any case, the results in this paper seem to suggest the relevance of duality (w.r.t. synchronisation) in developing a formal, foundational theory for distributed computations. In particular, the fact of having a syntactical characterisation of ....

G. Bellin, J. van de Wiele. Proof Nets and Typed lambda Calculus. I. Empires and Kingdoms. Draft. 1993.

....all formula occurrence are subformulas of the conclusion (and the assumptions, if any) It is obviously useful for proof search, because it limits the set of formulas to be tried. A construction which largely avoids the bureaucracy of taxonomy are Proof Nets (for some of the developments see [62, 30, 28, 49, 77, 18, 35], for a generalization see [100] Proof nets were invented by Girard for LL, but they are not that peculiar to LL and can hopefully be adapted for other logics as well. A proof net is a graph with nodes marked by formulas (not sequents) and edges linking the premises of a rule occurrence to the ....

....together by a Cut rule. When they meet, an interaction takes place, or the two jacks are plugged together to form an information path. The study of this approach is the essence of the Geometry of Interaction [65, 66, 67, 71, 3] see [102] for a gentle introduction to the subject) Proof nets [62, 30] and interaction nets [100, 101, 102] are an important achievement of this research. 67, 71] introduces a formalism in which correctly typed programs are guaranteed to be deadlock free (which corresponds in importance to the termination condition for the sequential case) Girard has done early ....

G. Bellin and J. van de Wiele. Proof nets and typed lambda calculus. I. Empires and kingdoms. Draft, available from theory.doc.ic.ac.uk: theory/papers/Bellin/king.dvi, May 1993.

....for various logical proof reduction (normalization) strategies. A second theme of this paper is that information flow in cut elimination is related to sending receiving protocols in calculus. In Section 5, this intuition is made precise in a detailed analysis of information flow in pure nets [11, 22, 8] which is a theory of graphical networks for representing untyped lambda terms in the style of proof nets for linear logic. We shall show how to dynamically orient a proof net (assigning the symbols I (for input) and O (for output) in a manner coherent with respect to introduction and elimination ....

....input and output should be fully interchangeable. In what follows we make this precise. Following Bellin and 3 The problem of substitution for propositional variables relates to whether the proofsas processes view extends to second order linear logic. We leave that issue open. 27 Van de Wiele [8], we show that (at least for MLL) we can always assign input output directions to proof nets so that one arbitrarily chosen conclusion is the output and all the others the inputs; when this is done, we have an interpretation of the proof net as a natural deduction derivation. Conversely, all ....

[Article contains additional citation context not shown here]

G. Bellin and J. van de Wiele. Proof Nets and Typed Lambda Calculus, manuscript.

....for various logical proof reduction (normalization) strategies. A second theme of this paper is that information flow in cut elimination is related to sending receiving protocols in calculus. In Section 5, this intuition is made precise in a detailed analysis of information flow in pure nets [11, 22, 8] which is a theory of graphical networks for representing untyped lambda terms in the style of proof nets for linear logic. We shall show how to dynamically orient a proof net (assigning the symbols I (for input) and O (for output) in a manner coherent with respect to introduction and elimination ....

....input and output should be fully interchangeable. In what follows we make this precise. Following Bellin and 3 The problem of substitution for propositional variables relates to whether the proofsas processes view extends to second order linear logic. We leave that issue open. 27 Van de Wiele [8], we show that (at least for MLL) we can always assign input output directions to proof nets so that one arbitrarily chosen conclusion is the output and all the others the inputs; when this is done, we have an interpretation of the proof net as a natural deduction derivation. Conversely, all ....

[Article contains additional citation context not shown here]

G. Bellin and J. van de Wiele. Proof Nets and Typed Lambda Calculus, manuscript.

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