Exponential Linear Logic Research Articles

An application of parallel cut elimination in multiplicative linear logic to the Taylor expansion of proof nets

We examine some combinatorial properties of parallel cut elimination in multiplicative linear logic (MLL) proof nets. We show that, provided we impose a constraint on some paths, we can bound the size of all the nets satisfying this constraint and reducing to a fixed resultant net. This result gives a sufficient condition for an infinite weighted sum of nets to reduce into another sum of nets, while keeping coefficients finite. We moreover show that our constraints are stable under reduction. Our approach is motivated by the quantitative semantics of linear logic: many models have been proposed, whose structure reflect the Taylor expansion of multiplicative exponential linear logic (MELL) proof nets into infinite sums of differential nets. In order to simulate one cut elimination step in MELL, it is necessary to reduce an arbitrary number of cuts in the differential nets of its Taylor expansion. It turns out our results apply to differential nets, because their cut elimination is essentially multiplicative. We moreover show that the set of differential nets that occur in the Taylor expansion of an MELL net automatically satisfies our constraints. Interestingly, our nets are untyped: we only rely on the sequentiality of linear logic nets and the dynamics of cut elimination. The paths on which we impose bounds are the switching paths involved in the Danos--Regnier criterion for sequentiality. In order to accommodate multiplicative units and weakenings, our nets come equipped with jumps: each weakening node is connected to some other node. Our constraint can then be summed up as a bound on both the length of switching paths, and the number of weakenings that jump to a common node.

Monoidal-closed categories of tree automata

AbstractThis paper surveys a new perspective on tree automata and Monadic second-order logic (MSO) on infinite trees. We show that the operations on tree automata used in the translations of MSO-formulae to automata underlying Rabin’s Tree Theorem (the decidability of MSO) correspond to the connectives of Intuitionistic Multiplicative Exponential Linear Logic (IMELL). Namely, we equip a variant of usual alternating tree automata (that we call uniform tree automata) with a fibered monoidal-closed structure which in particular handles a linear complementation of alternating automata. Moreover, this monoidal structure is actually Cartesian on non-deterministic automata, and an adaptation of a usual construction for the simulation of alternating automata by non-deterministic ones satisfies the deduction rules of the !(–) exponential modality of IMELL. (But this operation is unfortunately not a functor because it does not preserve composition.) Our model of IMLL consists in categories of games which are based on usual categories of two-player linear sequential games called simple games, and which generalize usual acceptance games of tree automata. This model provides a realizability semantics, along the lines of Curry–Howard proofs-as-programs correspondence, of a linear constructive deduction system for tree automata. This realizability semantics, which can be summarized with the slogan “automata as objects, strategies as morphisms,” satisfies an expected property of witness extraction from proofs of existential statements. Moreover, it makes it possible to combine realizers produced as interpretations of proofs with strategies witnessing (non-)emptiness of tree automata.

From Linear Logic to Cyclic Sharing

We present a translation from Multiplicative Exponential Linear Logic to a simply-typed lambda calculus with cyclic sharing. This translation is derived from a simple observation on the Int-construction on traced monoidal categories. It turns out that the translation is a mixture of the call-by-name CPS translation and the Geometry of Interaction-based interpretation.

A semantic account of strong normalization in linear logic

We prove that given two cut-free nets of linear logic, by means of their relational interpretations one can determine: 1) whether or not the net obtained by cutting the two nets is strongly normalizable, 2) (in case it is strongly normalizable) the maximum length of the reduction sequences starting from that net. As a by-product of our semantic approach, we obtain a new proof of the conservation theorem for Multiplicative Exponential Linear Logic (MELL) which does not rely on confluence; this yields an alternative proof of strong normalization for MELL.

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”.

Towards a typed Geometry of Interaction

Girard's Geometry of Interaction (GoI) develops a mathematical framework for modelling the dynamics of cut elimination. We introduce a typed version of GoI, called Multiobject GoI for both multiplicative linear logic (MLL) and multiplicative exponential linear logic (MELL) with units. We present a categorical setting that includes our previous (untyped) GoI models, as well as more general models based on monoidal *-categories. Our development of multiobject GoI depends on a new theory of partial traces and trace classes, which we believe is of independent interest, as well as an abstract notion of orthogonality (which is 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 (under some restrictions). We prove soundness theorems for the MGoI interpretation (for Multiplicative and Multiplicative Exponential Linear Logic) in partially traced *-categories with an orthogonality. Finally, we briefly discuss the relationship between our GoI interpretation and other categorical interpretations of GoI.

On linear logic planning and concurrency

We present an approach to linear logic planning where an explicit correspondence between partial order plans and multiplicative exponential linear logic proofs is established. This is performed by extracting partial order plans from sound and complete encodings of planning problems in multiplicative exponential linear logic. These partial order plans exhibit a non-interleaving behavioural concurrency semantics, i.e., labelled event structures. Relying on this fact, we argue that this work is a crucial step for establishing a common language for concurrency and planning that will allow to carry techniques and methods between these two fields.

A Fully Labelled Lambda Calculus: Towards Closed Reduction in the Geometry of Interaction Machine

We investigate the possibility of performing new reduction strategies with the Geometry of Interaction Machine (GOIm). To this purpose, we appeal to Lévy's labelled lambda calculus whose labels describe: a) the path that the GOIm will follow in the graph of a term and b) the operations that the GOIm requires to compute the multiplicative part from the Multiplicative and Exponential Linear Logic encoding that the machine uses. Our goal is to unveil the missing exponential information in the structure of the labels. This will provide us with a tool to talk about strategies for computing paths with the GOIm.

Karp-Miller Trees for a Branching Extension of VASS

We study BVASS (Branching VASS) which extend VASS (Vector Addition Systems with States) by allowing addition transitions that merge two configurations. Runs in BVASS are tree-like structures instead of linear ones as for VASS. We show that the construction of Karp-Miller trees for VASS can be extended to BVASS. This entails that the coverability set for BVASS is computable. This allows us to obtain decidability results for certain classes of equational tree automata with an associative-commutative symbol. Recent independent work by de Groote et al. implies that decidability of reachability in BVASS is equivalent to decidability of provability in MELL (multiplicative exponential linear logic), which is still an open problem. Hence our results are also a step towards answering this question in the affirmative.

A Matrix Characterization for Multiplicative Exponential Linear Logic

We develop a matrix characterization of logical validity in MELL, the multiplicative fragment of propositional linear logic with exponentials and constants. To prove the correctness and completeness of our characterization, we use a purely proof-theoretical justification rather than semantical arguments. Our characterization is based on concepts similar to matrix characterizations proposed by Wallen for other nonclassical logics. It provides a foundation for developing proof search procedures for MELL by adopting techniques that are based on these concepts and also makes it possible to adopt algorithms that translate the machine-found proofs back into the usual sequent calculus for MELL.

System NEL is Undecidable

System NEL is a conservative extension of multiplicative exponential linear logic (extended by the rules mix and nullary mix) by a self-dual noncommutative connective called seq which has an intermediate position between the connectives par and times. In this paper, I will show that system NEL is undecidable by encoding two counter machines into NEL. Although the encoding is simple, the proof of the faithfulness is a little intricate because there is no sequent calculus and no phase semantics available for NEL.

Proof nets, garbage, and computations

We study the problem of local and asynchronous computation in the context of multiplicative exponential linear logic (MELL) proof nets. The main novelty is in a complete set of rewriting rules for cut-elimination in presence of weakening (which requires garbage collection). The proposed reduction system is strongly normalizing and confluent. The proofs are based on pure syntactical reasonings.