This paper presents a simple and intuitive syntax for proof nets of the multiplicative cyclic fragment (McyLL) of linear logic (LL). The proposed correctness criterion for McyLL proof nets can be considered as the non-commutative counterpart of the famous Danos contraction criterion for proof nets of the pure multiplicative fragment (MLL) of LL. Unlike the most part of current syntaxes for non-...

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 s...

Lévy’s labelled λ-calculus has played an important role in the understanding of the Geometry of Interaction and its applications to the implementation of λ-evaluators: labels relate to the multiplicative information of paths. In this paper, we generalise the structure of labels, and the underlying term structure, in order to keep track of exponential information too. We first define two labelle...

The sequent calculus admits many proofs of the same conclusion that differ only by trivial permutations of inference rules. In order to eliminate this “bureaucracy” from sequent proofs, deductive formalisms such as proof nets or natural deduction are usually used instead of the sequent calculus, for they identify proofs more abstractly and geometrically. In this paper we recover permutative can...

The problem of formalising continuous change within reasoning about action systems such as the situation calculus has recently been receiving increasing attention ((Rei96], MS96]). In this paper we show that a long-existing systems theoretic reasoning methodology, that of Zeigler's DEVS (discrete event system speciication), not only subsumes the standard situation calculus but is able to elegan...

The calculus of structures is a proof theoretical formalism which generalizes the sequent calculus with the feature of deep inference: in contrast to the sequent calculus, inference rules can be applied at any depth inside a formula, bringing shorter proofs than all other formalisms supporting analytical proofs. However, deep applicability of inference rules causes greater nondeterminism than i...

In this note, we show that if a Banach space X has a predual, then every bounded linear operator on X with a continuous functional calculus admits a bounded Borel functional calculus. A consequence of this is that on such a Banach space, the classes of finitely spectral and prespectral operators coincide. We also apply this result to give some sufficient conditions for an operator with an absol...

The unification and extension of continuous calculus, discrete calculus, q-calculus, and indeed arbitrary real-number calculus to time-scale calculus, where a time scale is simply any nonempty closed set of real numbers, were first accomplished by Hilger in [4]. Since then, time-scale calculus has made steady inroads in explaining the interconnections that exist among the various calculuses, an...