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Dominator Trees and Fast Verification of Proof Nets
"... We consider the following decision problems: PROOFNET: Given a multiplicative linear logic (MLL) proof structure, is it a proof net? ESSNET: Given an essential net (of an intuitionistic MLL sequent), is it correct? In this paper we show that lineartime algorithms for ESSNET can be obtained by cons ..."
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We consider the following decision problems: PROOFNET: Given a multiplicative linear logic (MLL) proof structure, is it a proof net? ESSNET: Given an essential net (of an intuitionistic MLL sequent), is it correct? In this paper we show that lineartime algorithms for ESSNET can be obtained by constructing the dominator tree of the input essential net. As a corollary, by showing that PROOFNET is lineartime reducible to ESSNET (by the trip translation), we obtain a lineartime algorithm for PROOFNET. We show further that these lineartime algorithms can be optimized to simple onepass algorithms – each node of the input structure is visited at most once. As another application of dominator trees, we obtain lineartime algorithms for sequentializing proof nets (i.e. given a proof net, find a derivation for the underlying MLL sequent) and essential nets.
Towards a typed geometry of interaction
, 2005
"... We introduce a typed version of Girard’s Geometry of Interaction, called Multiobject GoI (MGoI) semantics. We give an MGoI interpretation for multiplicative linear logic (MLL) without units which applies to new kinds of models, including finite dimensional vector spaces. For MGoI (i) we develop a v ..."
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We introduce a typed version of Girard’s Geometry of Interaction, called Multiobject GoI (MGoI) semantics. We give an MGoI interpretation for multiplicative linear logic (MLL) without units which applies to new kinds of models, including finite dimensional vector spaces. For MGoI (i) we develop a version of partial traces and trace ideals (related to previous work of Abramsky, Blute, and Panangaden); (ii) we do not require the existence of a reflexive object for our interpretation (the original GoI 1 and 2 were untyped and hence involved a bureaucracy of domain equation isomorphisms); (iii) we introduce an abstract notion of orthogonality (related to work of Hyland and Schalk) and use this to develop a version of Girard’s theory of types, datum and algorithms in our setting, (iv) we prove appropriate Soundness and Completeness Theorems for our interpretations in partially traced categories with orthogonality; (v) we end with an application to completeness of (the original) untyped GoI in a unique decomposition category.
Retractile Proof Nets of the Purely Multiplicative and Additive Fragment of Linear Logic
"... Abstract. Proof nets are a parallel syntax for sequential proofs of linear logic, firstly introduced by Girard in 1987. Here we present and intrinsic (geometrical) characterization of proof nets, that is a correctness criterion (an algorithm) for checking those proof structures which correspond to p ..."
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Abstract. Proof nets are a parallel syntax for sequential proofs of linear logic, firstly introduced by Girard in 1987. Here we present and intrinsic (geometrical) characterization of proof nets, that is a correctness criterion (an algorithm) for checking those proof structures which correspond to proofs of the purely multiplicative and additive fragment of linear logic. This criterion is formulated in terms of simple graph rewriting rules and it extends an initial idea of a retraction correctness criterion for proof nets of the purely multiplicative fragment of linear logic presented by Danos in his Thesis in 1990. 1
*autonomous categories, Unique decomposition categories.
"... We analyze the categorical foundations of Girard’s Geometry of Interaction Program for Linear Logic. The motivation for the work comes from the importance of viewing GoI as a new kind of semantics and thus trying to relate it to extant semantics. In an earlier paper we showed that a special case of ..."
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We analyze the categorical foundations of Girard’s Geometry of Interaction Program for Linear Logic. The motivation for the work comes from the importance of viewing GoI as a new kind of semantics and thus trying to relate it to extant semantics. In an earlier paper we showed that a special case of Abramsky’s GoI situations–ones based on Unique Decomposition Categories (UDC’s)–exactly captures Girard’s functional analytic models in his first GoI paper, including Girard’s original Execution formula in Hilbert spaces, his notions of orthogonality, types, datum, algorithm, etc. Here we associate to a UDCbased GoI Situation a denotational model (a ∗autonomous category (without units) with additional exponential structure). We then relate this model to some of the standard GoI models via a fullyfaithful embedding into a doublegluing category, thus connecting up GoI with earlier Full Completeness
Towards a Typed Geometry of Interaction Abstract
"... Girard’s Geometry of Interaction (GoI) develops a mathematical framework for modelling the dynamics of cutelimination. We introduce a typed version of GoI, called Multiobject GoI (MGoI) for multiplicative linear logic without units in categories which include previous (untyped) GoI models, as well ..."
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Girard’s Geometry of Interaction (GoI) develops a mathematical framework for modelling the dynamics of cutelimination. We introduce a typed version of GoI, called Multiobject GoI (MGoI) for multiplicative linear logic without units in categories which include previous (untyped) GoI models, as well as models not possible in the original untyped version. The development of MGoI depends on a new theory of partial traces and trace classes, as well as an abstract notion of orthogonality (related to work of Hyland and Schalk.) We develop Girard’s original theory of types, data and algorithms in our setting, and show his execution formula to be an invariant of Cut Elimination. We prove Soundness and Completeness Theorems for the MGoI interpretation in partially traced categories with an orthogonality. Moreover, as an application of our MGoI interpretation, we prove a completeness theorem for the original untyped GoI interpretation of MLL in a traced unique decomposition category.
A Categorical Model for the Geometry of Interaction Abstract
"... We consider the multiplicative and exponential fragment of linear logic (MELL) and give a Geometry of Interaction (GoI) semantics for it based on unique decomposition categories. We prove a Soundness and Finiteness Theorem for this interpretation. We show that Girard’s original approach to GoI 1 via ..."
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We consider the multiplicative and exponential fragment of linear logic (MELL) and give a Geometry of Interaction (GoI) semantics for it based on unique decomposition categories. We prove a Soundness and Finiteness Theorem for this interpretation. We show that Girard’s original approach to GoI 1 via operator algebras is exactly captured in this categorical framework.