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Linear Logic and Noncommutativity in the Calculus of Structures
, 2003
"... macro \clap,whichisused on almost every page, came out of such a discussion. This thesis would not exist without the support of my wife Jana. During all the time she has been a continuous source of love and inspiration. This PhD thesis has been written with the financial support of the DFGGraduiert ..."
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Cited by 38 (12 self)
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macro \clap,whichisused on almost every page, came out of such a discussion. This thesis would not exist without the support of my wife Jana. During all the time she has been a continuous source of love and inspiration. This PhD thesis has been written with the financial support of the DFGGraduiertenkolleg 334 "Spezifikation diskreter Prozesse und Prozesysteme durch operationelle Modelle und Logiken". iii iv Tab l e o f Contents Acknowledgements iii Tab l e of Contents v List of Figures vii 1Introduction 1 1.1Proof Theory andDeclarativeProgramming .................. 1 1.2LinearLogic .................................... 5 1.3Noncommutativity ................................ 8 1.4The Calculus of Structures . .......................... 9 1.5 Summary of Results............................... 12 1.6OverviewofContents.............................. 15 2LinearLogic and the Sequent Calculus 17 2.1Formulaeand Sequents . ............................. 17 2.2Rules andDerivations . .............
On the computational content of the axiom of choice
 The Journal of Symbolic Logic
, 1998
"... We present a possible computational content of the negative translation of classical analysis with the Axiom of Choice. Our interpretation seems computationally more direct than the one based on Godel's Dialectica interpretation [10, 18]. Interestingly, thisinterpretation uses a re nement of the rea ..."
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Cited by 34 (1 self)
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We present a possible computational content of the negative translation of classical analysis with the Axiom of Choice. Our interpretation seems computationally more direct than the one based on Godel's Dialectica interpretation [10, 18]. Interestingly, thisinterpretation uses a re nement of the realizibility semantics of the absurdity proposition, which is not interpreted as the empty type here. We alsoshowhow to compute witnesses from proofs in classical analysis, and how to interpret the axiom of Dependent Choice and Spector's Double Negation Shift.
On the NoCounterexample Interpretation
 J. SYMBOLIC LOGIC
, 1997
"... In [15],[16] Kreisel introduced the nocounterexample interpretation (n.c.i.) of Peano arithmetic. In particular he proved, using a complicated "substitution method (due to W. Ackermann), that for every theorem A (A prenex) of firstorder Peano arithmetic PA one can find ordinal recursive functi ..."
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Cited by 18 (10 self)
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In [15],[16] Kreisel introduced the nocounterexample interpretation (n.c.i.) of Peano arithmetic. In particular he proved, using a complicated "substitution method (due to W. Ackermann), that for every theorem A (A prenex) of firstorder Peano arithmetic PA one can find ordinal recursive functionals \Phi A of order type ! " 0 which realize the Herbrand normal form A of A. Subsequently more
Proof nets for the Lambekcalculus — an overview
 Proceedings of the Third Roma Workshop ”Proofs and Linguistic Categories
, 1996
"... 1 Introduction: the interest of proof nets for categorial grammar There are both linguistic and mathematical reasons for studying proof nets the perspective of categorial grammar. It is now well known that the Lambek calculus corresponds to intuitionnistic noncommutative multiplicative linear logic ..."
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Cited by 16 (2 self)
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1 Introduction: the interest of proof nets for categorial grammar There are both linguistic and mathematical reasons for studying proof nets the perspective of categorial grammar. It is now well known that the Lambek calculus corresponds to intuitionnistic noncommutative multiplicative linear logic — with no empty antecedent, to be absolutely precise. As natural deduction underlines the constructive contents of intuitionistic
A New Method for Establishing Conservativity of Classical Systems Over Their Intuitionistic Version
"... this paper we present such a method. Applied to I \Sigma ..."
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Cited by 16 (1 self)
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this paper we present such a method. Applied to I \Sigma
Hypersequent calculi for Gödel logics: a survey
 Journal of Logic and Computation
, 2003
"... Hypersequent calculi arise by generalizing standard sequent calculi to refer to whole contexts of sequents instead of single sequents. We present a number of results using hypersequents to obtain a Gentzenstyle characterization for the family of Gödel logics. We first describe analytic calculi for ..."
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Cited by 13 (4 self)
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Hypersequent calculi arise by generalizing standard sequent calculi to refer to whole contexts of sequents instead of single sequents. We present a number of results using hypersequents to obtain a Gentzenstyle characterization for the family of Gödel logics. We first describe analytic calculi for propositional finite and infinitevalued Gödel logics. We then show that the framework of hypersequents allows one to move straightforwardly from the propositional level to firstorder as well as propositional quantification. A certain type of modalities, enhancing the expressive power of Gödel logic, is also considered. 1
A SchütteTait style cutelimination proof for firstorder Gödel logic
 In Automated Reasoning with Tableaux and Related Methods (Tableaux’02), volume 2381 of LNAI
, 2002
"... Abstract. We present a SchütteTait style cutelimination proof for the hypersequent calculus HIF for firstorder Gödel logic. This proof allows to bound the depth of the resulting cutfree derivation by 4 d ρ(d) , where d is the depth of the original derivation and ρ(d) the maximal complexity o ..."
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Cited by 9 (6 self)
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Abstract. We present a SchütteTait style cutelimination proof for the hypersequent calculus HIF for firstorder Gödel logic. This proof allows to bound the depth of the resulting cutfree derivation by 4 d ρ(d) , where d is the depth of the original derivation and ρ(d) the maximal complexity of cutformulas in it. We compare this SchütteTait style cutelimination proof to a Gentzen style proof. 1
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 9 (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.