1,436 research outputs found
A Non-Commutative Extension of MELL
We extend multiplicative exponential linear logic (MELL) by a non-commutative, self-dual logical operator. The extended system, called NEL, is defined in the formalism of the calculus of structures, which is a generalisation of the sequent calculus and provides a more refined analysis of proofs. We should then be able to extend the range of applications of MELL, by modelling a broad notion of sequentiality and providing new properties of proofs. We show some proof theoretical results: decomposition and cut elimination. The new operator represents a significant challenge: to get our results we use here for the first time some novel techniques, which constitute a uniform and modular approach to cut elimination, contrary to what is possible in the sequent calculus
Glueability of Resource Proof-Structures: Inverting the Taylor Expansion
A Multiplicative-Exponential Linear Logic (MELL) proof-structure can be expanded into a set of resource proof-structures: its Taylor expansion. We introduce a new criterion characterizing those sets of resource proof-structures that are part of the Taylor expansion of some MELL proof-structure, through a rewriting system acting both on resource and MELL proof-structures
Unifying type systems for mobile processes
We present a unifying framework for type systems for process calculi. The
core of the system provides an accurate correspondence between essentially
functional processes and linear logic proofs; fragments of this system
correspond to previously known connections between proofs and processes. We
show how the addition of extra logical axioms can widen the class of typeable
processes in exchange for the loss of some computational properties like
lock-freeness or termination, allowing us to see various well studied systems
(like i/o types, linearity, control) as instances of a general pattern. This
suggests unified methods for extending existing type systems with new features
while staying in a well structured environment and constitutes a step towards
the study of denotational semantics of processes using proof-theoretical
methods
Relational type-checking for MELL proof-structures. Part 1: Multiplicatives
Relational semantics for linear logic is a form of non-idempotent
intersection type system, from which several informations on the execution of a
proof-structure can be recovered. An element of the relational interpretation
of a proof-structure R with conclusion acts thus as a type (refining
) having R as an inhabitant. We are interested in the following
type-checking question: given a proof-structure R, a list of formulae ,
and a point x in the relational interpretation of , is x in the
interpretation of R? This question is decidable. We present here an algorithm
that decides it in time linear in the size of R, if R is a proof-structure in
the multiplicative fragment of linear logic. This algorithm can be extended to
larger fragments of multiplicative-exponential linear logic containing
-calculus
The relational model is injective for Multiplicative Exponential Linear Logic
We prove a completeness result for Multiplicative Exponential Linear Logic
(MELL): we show that the relational model is injective for MELL proof-nets,
i.e. the equality between MELL proof-nets in the relational model is exactly
axiomatized by cut-elimination.Comment: 33 page
Context Semantics, Linear Logic and Computational Complexity
We show that context semantics can be fruitfully applied to the quantitative
analysis of proof normalization in linear logic. In particular, context
semantics lets us define the weight of a proof-net as a measure of its inherent
complexity: it is both an upper bound to normalization time (modulo a
polynomial overhead, independently on the reduction strategy) and a lower bound
to the number of steps to normal form (for certain reduction strategies).
Weights are then exploited in proving strong soundness theorems for various
subsystems of linear logic, namely elementary linear logic, soft linear logic
and light linear logic.Comment: 22 page
Taylor expansion in linear logic is invertible
Each Multiplicative Exponential Linear Logic (MELL) proof-net can be expanded
into a differential net, which is its Taylor expansion. We prove that two
different MELL proof-nets have two different Taylor expansions. As a corollary,
we prove a completeness result for MELL: We show that the relational model is
injective for MELL proof-nets, i.e. the equality between MELL proof-nets in the
relational model is exactly axiomatized by cut-elimination
The relational model is injective for Multiplicative Exponential Linear Logic (without weakenings)
We show that for Multiplicative Exponential Linear Logic (without weakenings)
the syntactical equivalence relation on proofs induced by cut-elimination
coincides with the semantic equivalence relation on proofs induced by the
multiset based relational model: one says that the interpretation in the model
(or the semantics) is injective. We actually prove a stronger result: two
cut-free proofs of the full multiplicative and exponential fragment of linear
logic whose interpretations coincide in the multiset based relational model are
the same "up to the connections between the doors of exponential boxes".Comment: 36 page
On Linear Information Systems
Scott's information systems provide a categorically equivalent, intensional
description of Scott domains and continuous functions. Following a well
established pattern in denotational semantics, we define a linear version of
information systems, providing a model of intuitionistic linear logic (a
new-Seely category), with a "set-theoretic" interpretation of exponentials that
recovers Scott continuous functions via the co-Kleisli construction. From a
domain theoretic point of view, linear information systems are equivalent to
prime algebraic Scott domains, which in turn generalize prime algebraic
lattices, already known to provide a model of classical linear logic
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