Lutz Straßburger

MELL in the calculus of structures

The calculus of structures is a new proof theoretical formalism, like natural deduction, the sequent calculus and proof nets, for specifying logical systems syntactically. In a rule in the calculus of structures, the premise as well as the conclusion are structures, which are expressions that share properties of formulae and sequents. In this paper, I study a system for MELL, the multiplicative exponential fragment of linear logic, in the calculus of structures. It has the following features: a local promotion rule, no non-deterministic splitting of the context in the times rule and a modular proof for the cut elimination theorem. Further, derivations have a new property, called decomposition, that cannot be observed in any other known proof theoretical formalism.

On Proof Nets for Multiplicative Linear Logic with Units

In this paper we present a theory of proof nets for full multiplicative linear logic, including the two units. It naturally extends the well-known theory of unit-free multiplicative proof nets. A linking is no longer a set of axiom links but a tree in which the axiom links are subtrees. These trees will be identified according to an equivalence relation based on a simple form of graph rewriting. We show the standard results of sequentialization and strong normalization of cut elimination. Furthermore, the identifications enforced on proofs are such that the proof nets, as they are presented here, form the arrows of the free (symmetric) *-autonomous category.

Expanding the realm of systematic proof theory

This paper is part of a general project of developing a systematic and algebraic proof theory for nonclassical logics. Generalizing our previous work on intuitionistic-substructural axioms and single-conclusion (hyper)sequent calculi, we define a hierarchy on Hilbert axioms in the language of classical linear logic without exponentials. We then give a systematic procedure to transform axioms up to the level P′3 of the hierarchy into inference rules in multiple-conclusion (hyper) sequent calculi, which enjoy cut-elimination under a certain condition. This allows a systematic treatment of logics which could not be dealt with in the previous approach. Our method also works as a heuristic principle for finding appropriate rules for axioms located at levels higher than P′3. The case study of Abelian and Łukasiewicz logic is outlined.

Proof Theory for Indexed Nested Sequents

Fitting’s indexed nested sequents can be used to give deductive systems to modal logics which cannot be captured by pure nested sequents. In this paper we show how the standard cut-elimination procedure for nested sequents can be extended to indexed nested sequents, and we discuss how indexed nested sequents can be used for intuitionistic modal logics.

Modular Focused Proof Systems for Intuitionistic Modal Logics

Focusing is a general technique for syntactically compartmentalizing nthe non-deterministic choices in a proof system, which not only nimproves proof search but also has the representational benefit of ndistilling sequent proofs into synthetic normal forms. However, since nfocusing is usually specified as a restriction of the sequent ncalculus, the technique has not been transferred to logics that lack a n(shallow) sequent presentation, as is the case for some of the logics nof the modal cube. We have recently extended the focusing technique nto classical nested sequents, a generalization of ordinary sequents. nIn this work we further extend focusing to intuitionistic nested nsequents, which can capture all the logics of the intuitionistic S5 ncube in a modular fashion. We present an internal cut-elimination nprocedure for the focused system which in turn is used to show its ncompleteness.

On the Power of Substitution in the Calculus of Structures

There are two contributions in this article. First, we give a direct proof of the known fact that Frege systems with substitution can be p-simulated by the calculus of structures (CoS) extended with the substitution rule. This is done without referring to the p-equivalence of extended Frege systems and Frege systems with substitution. Second, we then show that the cut-free CoS with substitution is p-equivalent to the cut-free CoS with extension.

On the Length of Medial-Switch-Mix Derivations

Switch and medial are two inference rules that play a central role in many deep inference proof systems. In specific proof systems, the mix rule may also be present. In this paper we show that the maximal length of a derivation using only the inference rules for switch, medial, and mix, modulo associativity and commutativity of the two binary connectives involved, is quadratic in the size of the formula at the conclusion of the derivation. This shows, at the same time, the termination of the rewrite system.