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...

linTAP is a tableau prover for the multiplicative and exponential fragment M?LL of Girards linear logic. It proves the validity of a given formula by constructing an analytic tableau and ensures the linear validity using prefix unification. We present the tableau calculus used by linTAP, an algorithm for prefix unification in linear logic, the linTAP implementation, and some experimental result...

Context dependent rules are an obstacle to cut elimination. A generalised sequent style formulation using deep inferences is helpful and may indeed be essential. First this is shown for a substructural, multiplicative, pure propositional calculus. Then we consider the extra problems induced by non-logical axioms and extend the results to additive connectives and quantifiers.

We introduce proof nets and sequent calculus for the multiplicative fragment of non-commutative logic, which is an extension of both linear logic and cyclic linear logic. The two main technical novelties are a third switching position for the non-commutative disjunction, and the structure of order variety.

Interaction nets are graph rewriting systems which are a generalisation of proof nets for classical linear logic. The linear chemical abstract machine (CHAM) is a term rewriting system which corresponds to classical linear logic, via the Curry-Howard isomorphism. We can obtain a textual calculus for interaction nets which is surprisingly similar to linear CHAM based on the multiplicative fragme...

The multiplicative coalescent X(t) is a l-valued Markov process representing coalescence of clusters of mass, where each pair of clusters merges at rate proportional to product of masses. From random graph asymptotics it is known (Aldous (1997)) that there exists a standard version of this process starting with infinitesimally small clusters at time −∞. In this paper, stochastic calculus techni...