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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 7 (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
"... 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
Craig Interpolation in Displayable Logics
"... Abstract. We give a general prooftheoretic method for establishing Craig interpolation for displayable logics, based upon an analysis of the individual proof rules of their display calculi. Using this uniform method, we establish interpolation for a spectrum of display calculi differing in their st ..."
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Abstract. We give a general prooftheoretic method for establishing Craig interpolation for displayable logics, based upon an analysis of the individual proof rules of their display calculi. Using this uniform method, we establish 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 at the level of proof rules also provides new insights into the reasons why interpolation fails, or seems likely to fail, in many substructural logics. Specifically, we identify contraction as being particularly problematic for interpolation except in special circumstances. 1