# Normalization properties of symmetric logical calculi

Abstract : It was realized in the early nineties that the Curry-Howard isomorphism can be extended to the case of the classical logic as well. Several calculi have appeared to provide a suitable background for this extension. This thesis examines some of these calculi w.r.t. various proof theoretical properties.
In the first half of the thesis the simply typed version of the $\lambda \mu$-calculus, defined by Parigot, is studied. It was already proved by Parigot that the $\lambda \mu$-calculus is strongly normalizing. Moreover, David and Nour has proved by an argument formalizable in first-order Peano-arithmetic arithmetical means that the strong normalization is preserved if we expand the calculus with the $\mu'$-rule, that is, if we consider the $\lambda \mu \mu'$-calculus. However, adding to the $\lambda \mu \mu'$-calculus another simplification rule, the $\rho$-rule, the new calculus is no more strongly normalizing. We prove that the untyped $\mu \mu' \rho$-calculus is weakly normalizing, and, as an application of this result, we demonstrate the weak normalization of the typed $\lambda \mu \mu' \rho$-calculus. Afterwards, we examine the effect of the addition of some other simplification rules to the $\mu \mu'$- and to the $\lambda \mu \mu'$-calculi, respectively.
Next, a bound for the lengths of the reduction sequences in the simply typed $\lambda \mu \rho \theta$-calculus is established. We extend a result of Xi presented for the case of the simply typed $\lambda$-calculus.
Then we give an arithmetical proof for the strong normalization of the simply typed part of the symmetric $\lambda$-calculus defined by Berardi and Barbanera.
In the final chapter we define a translation between the simply typed symmetric $\lambda$-calculus of Berardi and Barbanera and the $\overline{\lambda} \mu \widetilde{\mu} \star$-calculus, which is obtained from the $\overline{\lambda} \mu \widetilde{\mu}$-calculus defined by Curien and Herbelin by extending it with a negation. In the concluding part of the chapter, we present an arithmetical proof for the strong normalization of the $\overline{\lambda} \mu \widetilde{\mu} \star$-calculus.
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Submitted on : Thursday, September 24, 2009 - 10:09:01 AM
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• HAL Id : tel-00419492, version 1

### Citation

Peter Battyanyi. Normalization properties of symmetric logical calculi. Mathematics [math]. Université de Savoie, 2007. English. ⟨tel-00419492⟩

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