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The Finite Model Property For Various Fragments Of Linear Logic
, 1997
"... B stand for formulas. The connectives of propositional linear logic are: ffl the multiplicatives A & B, A\Omega B, ?, 1; ffl the additives A&B, A \Phi B, ?, 0; ffl the exponentials ?A, !A. Linear negation A ? is only given for positive atoms. It is extended to all formulas by A ?? = ..."
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Cited by 28 (0 self)
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B stand for formulas. The connectives of propositional linear logic are: ffl the multiplicatives A & B, A\Omega B, ?, 1; ffl the additives A&B, A \Phi B, ?, 0; ffl the exponentials ?A, !A. Linear negation A ? is only given for positive atoms. It is extended to all formulas by A ?? = A and by (A & B) ? = A ?\Omega B ? ; ? ? = 1; (A &B) ? = A ? \Phi B ?
The Undecidability Of Second Order Linear Logic Without Exponentials
 Journal of Symbolic Logic
, 1995
"... . Recently, Lincoln, Scedrov and Shankar showed that the multiplicative fragment of second order intuitionistic linear logic is undecidable, using an encoding of second order intuitionistic logic. Their argument applies to the multiplicativeadditive fragment, but it does not work in the classical c ..."
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Cited by 12 (3 self)
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. Recently, Lincoln, Scedrov and Shankar showed that the multiplicative fragment of second order intuitionistic linear logic is undecidable, using an encoding of second order intuitionistic logic. Their argument applies to the multiplicativeadditive fragment, but it does not work in the classical case, because second order classical logic is decidable. Here we show that the multiplicativeadditive fragment of second order classical linear logic is also undecidable, using an encoding of twocounter machines originally due to Kanovich. The faithfulness of this encoding is proved by means of the phase semantics. In this paper, we write LL for the full propositional fragment of linear logic, MLL for the multiplicative fragment, MALL for the multiplicativeadditive fragment, and MELL for the multiplicativeexponential fragment. Similarly, we write ILL, IMLL, etc. for the fragments of intuitionistic linear logic, LL2, MLL2, etc. for the second order fragments of linear logic, and ILL2, IML...
Phase Semantics and Verification of Concurrent Constraint Programs
, 1998
"... The class CC of concurrent constraint programming languages and its nonmonotonic extension LCC based on linear constraint systems can be given a logical semantics in Girard's intuitionistic linear logic for a variety of observables. In this paper we settle basic completeness results and we sho ..."
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Cited by 11 (2 self)
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The class CC of concurrent constraint programming languages and its nonmonotonic extension LCC based on linear constraint systems can be given a logical semantics in Girard's intuitionistic linear logic for a variety of observables. In this paper we settle basic completeness results and we show how the phase semantics of linear logic can be used to provide simple and very concise "semantical" proofs of safety properties for CC or LCC programs.
The undecidability of boolean BI through phase semantics
 In LICS’10
, 2010
"... We solve the open problem of the decidability of Boolean BI logic (BBI), which can be considered as the core of separation and spatial logics. For this, we define a complete phase semantics for BBI and characterize it as trivial phase semantics. We deduce an embedding between trivial phase semantic ..."
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Cited by 9 (3 self)
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We solve the open problem of the decidability of Boolean BI logic (BBI), which can be considered as the core of separation and spatial logics. For this, we define a complete phase semantics for BBI and characterize it as trivial phase semantics. We deduce an embedding between trivial phase semantics for intuitionistic linear logic (ILL) and Kripke semantics for BBI. We single out a fragment of ILL which is both undecidable and complete for trivial phase semantics. Therefore, we obtain the undecidability of BBI. 1.
Nondeterministic Phase Semantics and the Undecidability of Boolean BI
, 2011
"... We solve the open problem of the decidability of Boolean BI logic (BBI), which can be considered as the “core” of Separation and Spatial Logics. For this, we define a complete phase semantics suitable for BBI and characterize it as trivial phase semantics. We deduce an embedding between trivial phas ..."
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We solve the open problem of the decidability of Boolean BI logic (BBI), which can be considered as the “core” of Separation and Spatial Logics. For this, we define a complete phase semantics suitable for BBI and characterize it as trivial phase semantics. We deduce an embedding between trivial phase semantics for intuitionistic linear logic (ILL) and Kripke semantics for BBI. We single out the elementary fragment of ILL which is both undecidable and complete for trivial phase semantics. Thus, we obtain the undecidability of BBI.
Some Observations on the Proof Theory of Second Order Propositional Multiplicative Linear Logic (Extended Abstract)
, 2007
"... We present two new aspects of the proof theory of MLL2. First, we will give a novel proof system in the framework of the calculus of structures. The main feature of the new system is the consequent use of deep inference. Due to the new freedom of permuting inference rules, we are able to observe a d ..."
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We present two new aspects of the proof theory of MLL2. First, we will give a novel proof system in the framework of the calculus of structures. The main feature of the new system is the consequent use of deep inference. Due to the new freedom of permuting inference rules, we are able to observe a decomposition theorem, which is not visible in the sequent calculus. Second, we show a new notion of (boxfree) proof nets which is inspired by the deep inference proof system. Nonetheless, the proof nets are independent from the deductive system. We have “sequentialisation” into the calculus of structures as well as into the sequent calculus. We present a notion of cut elimination which is terminating and confluent, and thus gives us a category of proof nets.
The Undecidability of Second Order Linear Affine Logic
, 1995
"... The quantiferfree propositional linear ane logic (i.e. linear logic with the weakening) is decidable. Recently, Lafont and Scedrov proved that multiplicative fragment of secondorder linear logic is undecidable. In this paper we show that the second order linear ane logic is undecidable too. At the ..."
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The quantiferfree propositional linear ane logic (i.e. linear logic with the weakening) is decidable. Recently, Lafont and Scedrov proved that multiplicative fragment of secondorder linear logic is undecidable. In this paper we show that the second order linear ane logic is undecidable too. At the same time it turns out that even its multiplicative fragment is undecidable. Moreover, we obtain the whole class of undecidability second order logics which lie between Lambek calculus (LC) and linear ane logic. The proof is based on an encoding twocounter Minsky machines in second order linear ane logic. The faithfulness of the encoding is proved by means of the phase semantic.
Simulating Computations in Second Order NonCommutative Linear Logic (Preliminary Report)
, 1995
"... this paper we prove that the combination: "left and right implications ..."
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this paper we prove that the combination: "left and right implications
Decision problems for secondorder Linear Logic
 Proceedings of the Tenth Annual IEEE Symposium on Logic in Computer Science
, 1995
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