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Classical Modal Display Logic . . .
, 2007
"... 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 cutelimination, we obtain a cutelimination theorem for all corresponding CoS calculi. We then show how our result le ..."
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Cited by 8 (5 self)
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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 cutelimination, we obtain a cutelimination theorem for all corresponding CoS calculi. We then show how our result leads to a minimal cutfree CoS calculus for modal logic S5. No other existing CoS calculi for S5 enjoy both these properties simultaneously.
A New Machinechecked Proof of Strong Normalisation for Display Logic
 Electronic Notes in Theoretical Computer Science
, 2002
"... We use a deep embedding of the display calculus for relation algebras #RA in the logical framework Isabelle/HOL to formalise a new, machinechecked, proof of strong normalisation and cutelimination for #RA which does not use measures on the size of derivations. Our formalisation generalises easily ..."
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Cited by 6 (2 self)
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We use a deep embedding of the display calculus for relation algebras #RA in the logical framework Isabelle/HOL to formalise a new, machinechecked, proof of strong normalisation and cutelimination for #RA which does not use measures on the size of derivations. Our formalisation generalises easily to other display calculi and can serve as a basis for formalised proofs of strong normalisation for the classical and intuitionistic versions of a vast range of substructural logics like the Lambek calculus, linear logic, relevant logic, BCKlogic, and their modal extensions. We believe this is the first full formalisation of a strong normalisation result for a sequent system using a logical framework.
Tools and Techniques for Formalising Structural Proof Theory
, 2009
"... Whilst results from Structural Proof Theory can be couched in many formalisms, it is the sequent calculus which is the most amenable of the formalisms to metamathematical treatment. Constructive syntactic proofs are filled with bureaucratic details; rarely are all cases of a proof completed in the l ..."
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Whilst results from Structural Proof Theory can be couched in many formalisms, it is the sequent calculus which is the most amenable of the formalisms to metamathematical treatment. Constructive syntactic proofs are filled with bureaucratic details; rarely are all cases of a proof completed in the literature. Two intermediate results can be used to drastically reduce the amount of effort needed in proofs of Cut admissibility: Weakening and Invertibility. Indeed, whereas there are proofs of Cut admissibility which do not use Invertibility, Weakening is almost always necessary. Use of these results simply shifts the bureaucracy, however; Weakening and Invertibility, whilst more easy to prove, are still not trivial. We give a framework under which sequent calculi can be codified and analysed, which then allows us to prove various results: for a calculus to admit Weakening and for a rule to be invertible in a calculus. For the latter, even though many calculi are investigated, the general condition is simple and easily verified. The results have been applied to G3ip, G3cp, G3c, G3s, G3LC and G4ip. Invertibility is important in another respect; that of proofsearch. Should all rules in a calculus be invertible, then terminating rootfirst proof search gives a decision procedure
Tool support for reasoning in display calculi
"... Abstract. We present a tool for reasoning in and about propositional sequent calculi. One aim is to support reasoning in calculi that contain a hundred rules or more, so that even relatively small pen and paper derivations become tedious and error prone. As an example, we implement the display calcu ..."
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Abstract. We present a tool for reasoning in and about propositional sequent calculi. One aim is to support reasoning in calculi that contain a hundred rules or more, so that even relatively small pen and paper derivations become tedious and error prone. As an example, we implement the display calculus D.EAK of dynamic epistemic logic. Second, we provide embeddings of the calculus in the theorem prover Isabelle for formalising proofs about D.EAK. As a case study we show that the solution of the muddy children puzzle is derivable for any number of muddy children. Third, there is a set of metatools, that allows us to adapt the tool for a wide variety of user defined calculi. 1
Tools for the Investigation of Substructural, Intermediate and
"... iii Acknowledgements First and foremost I want to thank my advisor Agata Ciabattoni, who introduced me to the world of logic and research. I am grateful that I had the possibility to work with her and have her as a teacher. I learned a lot from Agata — impossible to enumerate everything — like the w ..."
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iii Acknowledgements First and foremost I want to thank my advisor Agata Ciabattoni, who introduced me to the world of logic and research. I am grateful that I had the possibility to work with her and have her as a teacher. I learned a lot from Agata — impossible to enumerate everything — like the way to approach logical problems, solve them and present them to an audience in a proper way. She understood my way of working, always challenging me anew, and allowed me to grow in various ways. Mille grazie. I want to thank Anna Zamansky and Ori Lahav for their continuous support and valuable comments on the thesis. Working with them was always a pleasure and I learned a lot from them during (and after) our collaborations. Toda. I am very thankful to Gernot Salzer for his support, especially when I was paralyzed with one of my Prolog tools. I was glad to be part of the Theory And Logic Group and to participate in many interesting discussions with Alex, Bernhard, Chris, Matthias and Rudi. I also want to say thanks to Franzi and Elisabeth, who helped me with all the
A New Machinechecked Proof of Strong Normalisation for Display Logic
"... 1 Introduction Sequent calculi provide a rigorous basis for metatheoretic studies of logics. The central theorem is cutelimination which states that detours through lemmata can be avoided, and it can be used to show many important logical properties like consistency, interpolation, and Beth defina ..."
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1 Introduction Sequent calculi provide a rigorous basis for metatheoretic studies of logics. The central theorem is cutelimination which states that detours through lemmata can be avoided, and it can be used to show many important logical properties like consistency, interpolation, and Beth definability. Cutfree sequent calculi are also useful for automated deduction [14], nonclassical extensions of logic programming [22], and studying deep connections between cut elimination, lambda calculi and functional programming. Sequent calculi, and their extensions, therefore play an important role in theoretical computer science.
Abstract A New Machinechecked Proof of Strong Normalisation for Display Logic
"... We use a deep embedding of the display calculus for relation algebras δRA in the logical framework Isabelle/HOL to formalise a new, machinechecked, proof of strong normalisation and cutelimination for δRA which does not use measures on the size of derivations. Our formalisation generalises easily ..."
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We use a deep embedding of the display calculus for relation algebras δRA in the logical framework Isabelle/HOL to formalise a new, machinechecked, proof of strong normalisation and cutelimination for δRA which does not use measures on the size of derivations. Our formalisation generalises easily to other display calculi and can serve as a basis for formalised proofs of strong normalisation for the classical and intuitionistic versions of a vast range of substructural logics like the Lambek calculus, linear logic, relevant logic, BCKlogic, and their modal extensions. We believe this is the first full formalisation of a strong normalisation result for a sequent system using a logical framework. 1
Craig Interpolation in Displayable Logics
"... Abstract. We give a general prooftheoretic method for proving Craig interpolation for displayable logics, based on an analysis of the individual proof rules of their display calculi. Using this uniform method, we prove interpolation for a spectrum of display calculi differing in their structural ru ..."
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Abstract. We give a general prooftheoretic method for proving Craig interpolation for displayable logics, based on an analysis of the individual proof rules of their display calculi. Using this uniform method, we prove interpolation for a spectrum of display calculi differing in their structural rules, including those for multiplicative linear logic, multiplicative additive linear logic and ordinary classical logic. Our analysis of proof rules also provides new insights into why interpolation fails, or seems likely to fail, in many substructural logics. Specifically, contraction appears particularly problematic for interpolation except in special circumstances. 1