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26
A system of interaction and structure
 ACM TRANSACTIONS ON COMPUTATIONAL LOGIC
, 2004
"... This paper introduces a logical system, called BV, which extends multiplicative linear logic by a noncommutative selfdual logical operator. This extension is particularly challenging for the sequent calculus, and so far it is not achieved therein. It becomes very natural in a new formalism, call ..."
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Cited by 109 (19 self)
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This paper introduces a logical system, called BV, which extends multiplicative linear logic by a noncommutative selfdual logical operator. This extension is particularly challenging for the sequent calculus, and so far it is not achieved therein. It becomes very natural in a new formalism, called the calculus of structures, which is the main contribution of this work. Structures are formulae subject to certain equational laws typical of sequents. The calculus of structures is obtained by generalising the sequent calculus in such a way that a new topdown symmetry of derivations is observed, and it employs inference rules that rewrite inside structures at any depth. These properties, in addition to allowing the design of BV, yield a modular proof of cut elimination.
A system of interaction and structure IV: The exponentials
 IN THE SECOND ROUND OF REVISION FOR MATHEMATICAL STRUCTURES IN COMPUTER SCIENCE
, 2007
"... We study some normalisation properties of the deepinference proof system NEL, which can be seen both as 1) an extension of multiplicative exponential linear logic (MELL) by a certain noncommutative selfdual logical operator; and 2) an extension of system BV by the exponentials of linear logic. T ..."
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Cited by 16 (9 self)
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We study some normalisation properties of the deepinference proof system NEL, which can be seen both as 1) an extension of multiplicative exponential linear logic (MELL) by a certain noncommutative selfdual logical operator; and 2) an extension of system BV by the exponentials of linear logic. The interest of NEL resides in: 1) its being Turing complete, while the same for MELL is not known, and is widely conjectured not to be the case; 2) its inclusion of a selfdual, noncommutative logical operator that, despite its simplicity, cannot be axiomatised in any analytic sequent calculus system; 3) its ability to model the sequential composition of processes. We present several decomposition results for NEL and, as a consequence of those and via a splitting theorem, cut elimination. We use, for the first time, an induction measure based on flow graphs associated to the exponentials, which captures their rather complex behaviour in the normalisation process. The results are presented in the calculus of structures, which is the first, developed formalism in deep inference.
Pomset logic as a calculus of directed cographs
 DYNAMIC PERSPECTIVES IN LOGIC AND LINGUISTICS
, 1999
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A system of interaction and structure V: The exponentials and splitting
, 2009
"... System NEL is the mixed commutative/noncommutative linear logic BV augmented with linear logic’s exponentials, or, equivalently, it is MELL augmented with the noncommutative selfdual connective seq. System NEL is Turingcomplete, it is able to directly express process algebra sequential compositio ..."
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Cited by 10 (4 self)
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System NEL is the mixed commutative/noncommutative linear logic BV augmented with linear logic’s exponentials, or, equivalently, it is MELL augmented with the noncommutative selfdual connective seq. System NEL is Turingcomplete, it is able to directly express process algebra sequential composition and it faithfully models causal quantum evolution. In this paper, we show cut elimination for NEL, based on a property that we call splitting. NEL is presented in the calculus of structures, which is a deepinference formalism, because no Gentzen formalism can express it analytically. The splitting theorem shows how and to what extent we can recover a sequentlike structure in NEL proofs. Together with the decomposition theorem, proved in the previous paper of the series, this immediately leads to a cutelimination theorem for NEL. 1
Sequentiality by Linear Implication and Universal Quantification
 In Jorg Desel, editor, Structures in Concurrency Theory, Workshops in Computing
, 1995
"... In this paper we address the issue of understanding sequential and parallel composition of agents from a logical viewpoint. We use the methodology of abstract logic programming in linear logic, where computations are proof searches in a suitable fragment of linear logic. While parallel composition h ..."
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Cited by 7 (3 self)
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In this paper we address the issue of understanding sequential and parallel composition of agents from a logical viewpoint. We use the methodology of abstract logic programming in linear logic, where computations are proof searches in a suitable fragment of linear logic. While parallel composition has a straightforward treatment in this setting, sequential composition is much more difficult to be obtained. We define and study a logic programming language, SMR, in which the causality relation among agents forms a seriesparallel order; top agents are recursively rewritten by seriesparallel structures of new agents. We show a declarative and simple treatment of sequentialization, which smoothly integrates with parallelization, by translating SMR into linear logic in a complete way. This means that we obtain a full two ways correspondence between proofs in linear logic and computations in SMR; thus we have full correspondence between the two formalisms. Our case study is very general per ...
On the Relation Between Coherence Semantics and Multiplicative Proof Nets
, 1994
"... It is known that (mix) proof nets admit a coherence semantics, computed as a set of experiments. We prove here the converse: a proof structure is shown to be a proof net whenever its set of experiments is a semantical object  a clique of the corresponding coherence space. Moreover the interpretat ..."
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Cited by 6 (4 self)
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It is known that (mix) proof nets admit a coherence semantics, computed as a set of experiments. We prove here the converse: a proof structure is shown to be a proof net whenever its set of experiments is a semantical object  a clique of the corresponding coherence space. Moreover the interpretation of atomic formulae can be restricted to a given coherent space with four tokens in its web. This is done by transforming cutlinks into tensorlinks. Dealing directly with noncutfree proof structure we characterise the deadlock freeness of the proof structure. These results are especially convenient for Abramsky 's proof expressions, and are extended to the pomset calculus.
Resource logics and minimalist grammars
 Proceedings ESSLLI’99 workshop (Special issue Language and Computation
, 2002
"... This ESSLLI workshop is devoted to connecting the linguistic use of resource logics and categorial grammar to minimalist grammars and related generative grammars. Minimalist grammars are relatively recent, and although they stem from a long tradition of work in transformational grammar, they are lar ..."
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Cited by 5 (0 self)
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This ESSLLI workshop is devoted to connecting the linguistic use of resource logics and categorial grammar to minimalist grammars and related generative grammars. Minimalist grammars are relatively recent, and although they stem from a long tradition of work in transformational grammar, they are largely informal apart from a few research papers. The study of resource logics, on the other hand, is formal and stems naturally from a long logical tradition. So although there appear to be promising connections between these traditions, there is at this point a rather thin intersection between them. The papers in this workshop are consequently rather diverse, some addressing general similarities between the two traditions, and others concentrating on a thorough study of a particular point. Nevertheless they succeed in convincing us of the continuing interest of studying and developing the relationship between the minimalist program and resource logics. This introduction reviews some of the basic issues and prior literature. 1 The interest of a convergence What would be the interest of a convergence between resource logical investigations of
Yet another correctness criterion for Multiplicative Linear Logic with MIX
 In Logic at St Petersburg '94, Symposium on Logical Foundations of Computer Science, LNCS 813
, 1994
"... A new correctness criterion for discriminating Proof Nets among Proof Structures of Multiplicative Linear Logic with MIX rule is provided. This criterion is inspired by an original interpretation of Proof Structures as distributed systems, and logical formulae as processes. The computation inside a ..."
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Cited by 5 (0 self)
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A new correctness criterion for discriminating Proof Nets among Proof Structures of Multiplicative Linear Logic with MIX rule is provided. This criterion is inspired by an original interpretation of Proof Structures as distributed systems, and logical formulae as processes. The computation inside a system corresponds to the logical flow of information inside a proof, that is, roughly, a distributed version of Girard's token trip. Proof Nets are then characterised as deadlock free Proof Structures (deadlock free distributed systems). This result follows by considering the causal dependencies among logical formulae inside proofs, and it provides a new understanding of notions like acyclicity, chains, and empires in terms of concurrent computations. 1 Introduction Proof Structures and Proof Nets (see next section) have been one of the most innovative and provocative contribution of Girard's Linear Logic [Gi86]. They provide a nice, graphical representation for Logical Proofs (a sort of ...