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Deep Sequent Systems for Modal Logic
 ARCHIVE FOR MATHEMATICAL LOGIC
"... We see a systematic set of cutfree 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 litera ..."
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Cited by 41 (4 self)
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We see a systematic set of cutfree 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.
Exploring the gap between linear and classical logic
 Theory and Applications of Categories, 18:473–535
, 2006
"... 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 proposition ..."
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Cited by 27 (3 self)
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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.
Reducing Nondeterminism in the Calculus of Structures
, 2005
"... 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 a ..."
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Cited by 19 (5 self)
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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 nontrivial system in the calculus of structures, extending multiplicative linear logic with the rules mix, nullary mix and a self dual, noncommutative 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.
A system of interaction and structure II: the need for deep inference
 Logical Methods in Computer Science
, 2006
"... Vol. 2 (2:4) 2006, pp. 1–24 ..."
Deep inference and its normal form of derivations
 Computability in Europe 2006, volume 3988 of Lecture
, 2006
"... 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 cutfree proof in this formalism. 1 ..."
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Cited by 19 (0 self)
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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 cutfree proof in this formalism. 1
On the axiomatisation of boolean categories with and without medial
 THEORY APPL. CATEG
, 2007
"... ..."
A system of interaction and structure IV: The exponentials
 IN THE SECOND ROUND OF REVISION FOR MATHEMATICAL STRUCTURES IN COMPUTER SCIENCE
, 2007
"... We study some normalisation properties of the deepinference proof system NEL, which can be seen both as 1) an extension of multiplicative exponential linear logic (MELL) by a certain noncommutative selfdual logical operator; and 2) an extension of system BV by the exponentials of linear logic. T ..."
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Cited by 17 (9 self)
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We study some normalisation properties of the deepinference proof system NEL, which can be seen both as 1) an extension of multiplicative exponential linear logic (MELL) by a certain noncommutative selfdual logical operator; and 2) an extension of system BV by the exponentials of linear logic. The interest of NEL resides in: 1) its being Turing complete, while the same for MELL is not known, and is widely conjectured not to be the case; 2) its inclusion of a selfdual, noncommutative logical operator that, despite its simplicity, cannot be axiomatised in any analytic sequent calculus system; 3) its ability to model the sequential composition of processes. We present several decomposition results for NEL and, as a consequence of those and via a splitting theorem, cut elimination. We use, for the first time, an induction measure based on flow graphs associated to the exponentials, which captures their rather complex behaviour in the normalisation process. The results are presented in the calculus of structures, which is the first, developed formalism in deep inference.
System BV without the Equalities for Unit
, 2004
"... System BV is an extension of multiplicative linear logic with a noncommutative selfdual 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. ..."
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Cited by 14 (3 self)
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System BV is an extension of multiplicative linear logic with a noncommutative selfdual 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.
System BV is NPcomplete
, 2005
"... System BV is an extension of multiplicative linear logic (MLL) with the rules mix, nullary mix, and a selfdual, noncommutative 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, syste ..."
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Cited by 14 (4 self)
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System BV is an extension of multiplicative linear logic (MLL) with the rules mix, nullary mix, and a selfdual, noncommutative 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 NPcomplete by encoding the 3Partition 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.
Cut Elimination inside a Deep Inference System for Classical Predicate Logic
, 2005
"... Deep inference is a natural generalisation of the onesided 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 ..."
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Cited by 13 (2 self)
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Deep inference is a natural generalisation of the onesided 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 cutfree sequent calculus and it also allows for a more finegrained 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.