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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 ?? = A and by ..."
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Cited by 25 (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 Multiplicative Linear Logic
, 1996
"... The multiplicative fragment of second order propositional linear logic is shown to be undecidable. Introduction Decision problems for propositional (quantifierfree) linear logic were first studied by Lincoln et al. [LMSS]. In referring to linear logic fragments, let M stand for multiplicatives, A ..."
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Cited by 14 (3 self)
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The multiplicative fragment of second order propositional linear logic is shown to be undecidable. Introduction Decision problems for propositional (quantifierfree) linear logic were first studied by Lincoln et al. [LMSS]. In referring to linear logic fragments, let M stand for multiplicatives, A for additives, E for exponentials (or modalities), 1 for first order quantifiers, 2 for second order propositional quantifiers, and I for "intuitionistic" version. In [LMSS] it was shown that full propositional linear logic is undecidable and that MALL is PSPACEcomplete. The main problems left open in [LMSS] were the NPcompleteness of MLL, the decidability of MELL, and the decidability of various fragments of propositional linear logic without exponentials but extended with second order propositional quantifiers. The decision problem for MELL is still open, but almost all the other problems have been solved: ffl The NPcompleteness of MLL has been obtained by Kanovich [K1]. Moreover, Linco...
Phase Semantics for Light Linear Logic
 Theoretical Computer Science
, 1997
"... Light linear logic [1] is a refinement of the propositionsastypes paradigm to polynomialtime computation. A semantic setting for the underlying logical system is introduced here in terms of fibred phase spaces. Strong completeness is established, with a purely semantic proof of cut elimination as ..."
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Cited by 9 (0 self)
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Light linear logic [1] is a refinement of the propositionsastypes paradigm to polynomialtime computation. A semantic setting for the underlying logical system is introduced here in terms of fibred phase spaces. Strong completeness is established, with a purely semantic proof of cut elimination as a consequence. A number of mathematical examples of fibred phase spaces are presented that illustrate subtleties of light linear logic. 1 Introduction Typed lambda calculi have long been recognized as analogous to formal logical calculi of intuitionistic logic. In technical terms this correspondence is known as the CurryHoward isomorphism or the propositionsastypes paradigm. Logic provides not only basic input/output specifications (i.e., types or formulas), but also a setting for welltyped programs (i.e., terms or formal proofs), as well as a mode of execution of welltyped programs by means of term reduction or normalization [2]. The advent of linear logic [3] with its intrinsic abil...
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
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.
Decision Problems for SecondOrder Linear Logic
 Proceedings of the Tenth Annual IEEE Symposium on Logic in Computer Science
, 1995
"... Abstract. The decision problem is studied for fragments of secondorder linear logic without modalities. It is shown that the structural rules of contraction and weakening may be simulated by secondorder propositional quantifiers and the multiplicative connectives. Among the consequences are the un ..."
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Abstract. The decision problem is studied for fragments of secondorder linear logic without modalities. It is shown that the structural rules of contraction and weakening may be simulated by secondorder propositional quantifiers and the multiplicative connectives. Among the consequences are the undecidability of the intuitionistic secondorder fragment of propositional multiplicative linear logic and the undecidability of multiplicative linear logic with firstorder and secondorder quantifiers. 1 Introduction Much of the expressive power and plasticity of linear logic may be traced to the prohibition of structural rules of Contraction and to some extent Weakening [7, 9, 28, 25, 26]. These rules are reintroduced into linear logic in a controlled fashion by the logical rules for modalities (or: exponentials), which allow, for instance, intuitionistic implication or function type A ) B to be expressed as !A \GammaffiB. Without the use of modalities, however, any reintroduction of th...
The Undecidability of Second Order Multiplicative Linear Logic
, 1996
"... The multiplicative fragment of second order propositional linear logic is shown to be undecidable. Introduction Decision problems for propositional (quantifierfree) linear logic were first studied by Lincoln et al. [LMSS]. In referring to linear logic fragments, let M stand for multiplicatives, A ..."
Abstract
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The multiplicative fragment of second order propositional linear logic is shown to be undecidable. Introduction Decision problems for propositional (quantifierfree) linear logic were first studied by Lincoln et al. [LMSS]. In referring to linear logic fragments, let M stand for multiplicatives, A for additives, E for exponentials (or modalities), 1 for first order quantifiers, 2 for second order propositional quantifiers, and I for "intuitionistic" version. In [LMSS] it was shown that full propositional linear logic is undecidable and that MALL is PSPACEcomplete. The main problems left open in [LMSS] were the NPcompleteness of MLL, the decidability of MELL, and the decidability of various fragments of propositional linear logic without exponentials but extended with second order propositional quantifiers. The decision problem for MELL is still open, but almost all the other problems have been solved: ffl The NPcompleteness of MLL has been obtained by Kanovich [K1]. Moreover, Linco...
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.