On Zucker's isomorphism for LJ and its extension to Pure Type Systems

It is shown how the sequent calculus LJ can be embedded into a simple extension of the -calculus by generalized applications, called J. The reduction rules of cut elimination and normalization can be precisely correlated, if explicit substitutions are added to J. The resulting system J2 is proved strongly normalizing, thus showing strong normalization for Gentzen's cut elimination steps. This re nes previous results by Zucker, Pottinger and Herbelin on the isomorphism between natural deduction and sequent calculus

Standardization for the Coinductive Lambda-Calculus

In the calculus of possibly non-wellfounded -terms, standardization is proved for a parallel notion of reduction. For this system confluence has recently been established by means of a bounding argument for the number of reductions provoked by the joining function which witnesses the confluence statement. Similarly

On Zucker’s isomorphism for LJ and its extension to pure type systems (submitted

Abstract. It is shown how the sequent calculus LJ can be embedded into a simple extension of the λ-calculus by generalized applications, called ΛJ. The reduction rules of cut elimination and normalization can be precisely correlated, if explicit substitutions are added to ΛJ. The resulting system ΛJ ✷ is proved strongly normalizing, thus showing strong normalization for Gentzen’s cut elimination steps. This refines previous results by Zucker, Pottinger and Herbelin on the isomorphism between natural deduction and sequent calculus. The concept of generalized applications extends to Pure Type Systems, so that in particular sequent calculus analogues for all systems of the λ-cube arise. Cut elimination and strong β-normalization are shown to be equivalent for all Pure Type Systems

Standardization for the Coinductive

In the calculus Λ co of possibly non-wellfounded λ-terms, standardization is proved for a parallel notion of reduction. For this system confluence has recently been established by means of a bounding argument for the number of reductions provoked by the joining function which witnesses the confluence statement. Similarly, bounds have to be introduced in order to turn the proof of standardization for the wellfounded λ-calculus into a sound coinductive argument, thus limiting the number of reduction steps arising in the process of standardization. This leads to elementary complexity bounds for the length of the resulting standard reduction sequence in terms of the length of the input sequence. A fortiori, these bounds also apply to the usual wellfounded λ-calculus, strengthening previous results by Xi

Acknowledgments

Leaning back and surveying this work, I am glad to admit that it is quite different from what I have anticipated when I started on it years ago, because this divergence reflects the influence of the people I have worked with. I owe much more than just my interest in λ-calculi and natural deduction to my Doktorvater Prof. Dr. Helmut Schwichtenberg. It is his scientific impetus that inspires his working group and so many fruitful discussions; the open-minded atmosphere that he creates is so valuable to all of us. In particular, I thank him for his confidence in my mathematical fidelity, and his stamina to endure the never-ending evolution of my confluence and normalization proofs. I hope that some of his constructive understanding of logic is mirrored in my work. Ralph Matthes is probably the person to blame most for this thesis: Without him as kind of mathematical alter ego, all my ideas would have collapsed in a woeful heap of counterexamples and errors. The basic techniques that served as a starting point for this thesis have been conceived jointly in an intellectual symbiosis, that I count among the most exciting experiences of my life

Syntactic Analysis of eta-Expansions In Pure Type Systems

By a detailed analysis of the interaction between -reduction ! and -expansion ! in the simply typed -calculus, a modular and purely syntactic proof method is devised in order to derive strong normalization of the combined reduction ! from that of ! and ! . It is shown how this technique extends to -normalizing functional Pure Type Systems with Barthe's formulation of -expansion

Confluence of the Coinductive

The coinductive λ-calculus Λ co arises by a coinductive interpretation of the grammar of the standard λ-calculus Λ and contains non-wellfounded λ-terms. An appropriate notion of reduction is analyzed and proven to be confluent by means of a detailed analysis of the usual Tait/Martin-Löf style development argument. This yields bounds for the lengths of those joining reduction sequences that are guaranteed to exist by confluence. These bounds also apply to the wellfounded λ-calculus, thus adding quantitative information to the classic result