We see a systematic set of cut-free axiomatisations for all the basic normal modal logics formed by some combination the axioms d,t,b,4, 5. They employ a form of deep inference but otherwise stay very close to Gentzen’s sequent calculus, in particular they enjoy a subformula property in the literal sense. No semantic notions are used inside the proof systems, in particular there is no use of labels. All their rules are invertible and the rules cut, weakening and contraction are admissible. All systems admit a straightforward terminating proof search procedure as well as a syntactic cut elimination procedure.

Abstract. The Medial rule was first devised as a deduction rule in the Calculus of Structures. In this paper we explore it from the point of view of category theory, as additional structure on a ∗-autonomous category. This gives us some insights on the denotational semantics of classical propositional logic, and allows us to construct new models for it, based on suitable generalizations of the theory of coherence spaces. 1.

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 in the sequent calculus regarding proof search. In this paper, we introduce a new technique which reduces nondeterminism without breaking proof theoretical properties, and provides a more immediate access to shorter proofs. We present our technique on system BV, the smallest technically non-trivial system in the calculus of structures, extending multiplicative linear logic with the rules mix, nullary mix and a self dual, non-commutative logical operator. Since our technique exploits a scheme common to all the systems in the calculus of structures, we argue that it generalizes to these systems for classical logic, linear logic and modal logics.

System BV is an extension of multiplicative linear logic with a non-commutative self-dual operator. We first map derivations of system BV of the calculus of structures to rewritings in a term rewriting system modulo equality, and then express this rewriting system as a Maude system module. This results in an automated proof search implementation for this system, and provides a recipe for implementing existing calculus of structures systems for other logics. Our result is interesting from the view of applications, specially, where sequentiality is essential, e.g., planning and natural language processing. In particular, we argue that we can express plans as logical formulae by using the sequential operator of BV and reason on them in a purely logical way.

System BV is an extension of multiplicative linear logic with a non-commutative self-dual operator. In this paper we present systems equivalent to system BV where equalities for unit are oriented from left to right and new structural rules are introduced to preserve completeness.

In its most general meaning, a Boolean category is to categories what a Boolean algebra is to posets. In a more specific meaning a Boolean category should provide the abstract algebraic structure underlying the proofs in Boolean Logic, in the same sense as a Cartesian closed category captures the proofs in intuitionistic logic and a *-autonomous category captures the proofs in linear logic. However, recent work has shown that there is no canonical axiomatisation of a Boolean category. In this work, we will see a series (with increasing strength) of possible such axiomatisations, all based on the notion of *-autonomous category. We will particularly focus on the medial map, which has its origin in an inference rule in KS, a cut-free deductive system for Boolean logic in the calculus of structures. Finally, we will present a category proof nets as a particularly well-behaved example of a Boolean category.

www.iam.unibe.ch / ∼ kai/ Abstract. We see a notion of normal derivation for the calculus of structures, which is based on a factorisation of derivations and which is more general than the traditional notion of cut-free proof in this formalism. 1

Deep inference is a natural generalisation of the one-sided sequent calculus where rules are allowed to apply deeply inside formulas, much like rewrite rules in term rewriting. This freedom in applying inference rules allows to express logical systems that are di#cult or impossible to express in the cut-free sequent calculus and it also allows for a more fine-grained analysis of derivations than the sequent calculus. However, the same freedom also makes it harder to carry out this analysis, in particular it is harder to design cut elimination procedures. In this paper we see a cut elimination procedure for a deep inference system for classical predicate logic.

We begin by showing how to faithfully encode the Classical Modal Display Logic (CMDL) of Wansing into the Calculus of Structures (CoS) of Guglielmi. Since every CMDL calculus enjoys cut-elimination, we obtain a cut-elimination theorem for all corresponding CoS calculi. We then show how our result leads to a minimal cut-free CoS calculus for modal logic S5. No other existing CoS calculi for S5 enjoy both these properties simultaneously.

System BV is an extension of multiplicative linear logic (MLL) with the rules mix, nullary mix, and a self-dual, non-commutative logical operator, called seq. While the rules mix and nullary mix extend the deductive system, the operator seq extends the language of MLL. Due to the operator seq, system BV extends the applications of MLL to those where sequential composition is crucial, e.g., concurrency theory. System FBV is an extension of MLL with the rules mix and nullary mix. In this paper, by relying on the fact that system BV is a conservative extension of system FBV, I show that system BV is NP-complete by encoding the 3-Partition problem in FBV. I provide a simple completeness proof of this encoding by resorting to a novel proof theoretical method for reducing the nondeterminism in proof search, which is also of independent interest.