We give a graph theoretical criterion on Multiplicative Additive Linear Logic (MALL) cut-free proof structures that exactly characterizes those whose interpretation is a hyperclique in Ehrhard’s hypercoherent spaces. This criterion is strictly weaker than the one given by Hughes and van Glabbeek characterizing proof nets (i.e. desequentialized sequent calculus proofs). We thus also give the fir...

We give a graph theoretical criterion on multiplicative additive linear logic (MALL) cut-free proof structures that exactly characterizes those whose interpretation is a hyperclique in Ehrhard’s hypercoherent spaces. This criterion is strictly weaker than the one given by Hughes and van Glabbeek characterizing proof nets (i.e. desequentialized sequent calculus proofs). We thus also give the fir...

Multiplicative linear logic [6, 7] is a formal calculus for reasoning about resources, which is similar to traditional logic, except that resources cannot be duplicated or neglected in the way that propositions can. A central problem in logic is determining when two proofs should be considered equivalent. In this paper, we describe a scheme for interpreting proofs in multiplicative linear logic...

Central to the theory of free probability is the notion of summing multiplicative functionals on the lattice of non-crossing partitions. In this paper, a graph-theoretic perspective of partitions is investigated in which independent sets in graphs correspond to non-crossing partitions. By associating particular graphs with elements of “zeon” algebras (commutative subalgebras of fermion algebras...

A many-valued modal logic is introduced that combines the usual Kripke frame semantics of the modal logic K with connectives interpreted locally at worlds by lattice and group operations over the real numbers. A labelled tableau system is provided and a coNEXPTIME upper bound obtained for checking validity in the logic. Focussing on the modal-multiplicative fragment, the labelled tableau system...

Multiplicative linear logic (MLL) was introduced in Gi87] as a one-sided sequent calculus: linear negation is a notion that is deened, via De Morgan identities. One obtains proof nets for MLL by identifying derivations in the one-sided calculus that are equal up to a permutation of inference rules. In this paper we consider a similar quotient for the formulation of MLL as a two-sided sequent ca...

We investigate the question of what constitutes a proof when quantifiers and multiplicative units are both present. On the technical level this paper provides two new aspects of the proof theory of MLL2 with units. First, we 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, which allows us to obs...

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 relatio...