Abstract. 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 canonicity directly in the cut-free sequent calculus by generalizing focused sequent proofs to admit multiple foci, and then considering the restricted class of maximally multifocused proofs. We validate this definition by proving a bijection to the well-known proof-nets for the unit-free multiplicative linear logic, and discuss the possibility of a similar correspondence for larger fragments.

Abstract. Probably the most significant result concerning cut-free sequent calculus proofs in linear logic is the completeness of focused proofs. This completeness theorem has a number of proof theoretic applications -e.g. in games semantics, Ludics, and proof search -and more computer science applications -e.g. logic programming, call-by-name/value evaluation. Andreoli proved this theorem for first-order linear logic 15 years ago. In the present paper, we give a new proof of the completeness of focused proofs in terms of proof transformation. The proof of this theorem is simple and modular: it is first proved for MALL and then is extended to full linear logic. Given its modular structure, we show how the proof can be extended to larger systems, such as logics with induction. Our analysis of focused proofs will employ a proof transformation method that leads us to study how focusing and cut elimination interact. A key component of our proof is the construction of a focalization graph which provides an abstraction over how focusing can be organized within a given cut-free proof. Using this graph abstraction allows us to provide a detailed study of atomic bias assignment in a way more refined that is given in Andreoli's original proof. Permitting more flexible assignment of bias will allow this completeness theorem to help establish the completeness of a number of other automated deduction procedures. Focalization graphs can be used to justify the introduction of an inference rule for multifocus derivation: a rule that should help us better understand the relations between sequentiality and concurrency in linear logic.

The λµ-calculus is an extension of the λ-calculus introduced in 1992 by Parigot [17] in order to generalize the Curry-Howard isomorphism to classical logic. Two versions of the calculus are usually considered in the literature: Parigot's original syntax and an alternative syntax introduced by de Groote. In 2001, David and Py [5] proved that the Separation Property (also referred to as Böhm theorem) fails for Parigot's λµ-calculus. By analyzing David & Py's result, we exhibit an extension of Parigot's λµ-calculus, the Λµ-calculus, for which the Separation Property holds and which is built as an intermediate language between Parigot's and de Groote's λµ-calculi. We prove the theorem and describe how Λµ-calculus can be considered as a calculus of terms and streams. We then illustrate Separation in showing how in Λµ-calculus it is possible to separate the counter-example used by David & Py.