Helmut Schwichtenberg

An inverse of the evaluation functional for typed lambda -calculus

A functional p to e (procedure to expression) that inverts the evaluation functional for typed lambda -terms in any model of typed lambda -calculus containing some basic arithmetic is defined. Combined with the evaluation functional, p to e yields an efficient normalization algorithm. The method is extended to lambda -calculi with constants and is used to normalize (the lambda -representations of) natural deduction proofs of (higher order) arithmetic. A consequence of theoretical interest is a strong completeness theorem for beta eta -reduction. If two lambda -terms have the same value in some model containing representations of the primitive recursive functions (of level 1) then they are probably equal in the beta eta -calculus.<<ETX>>

Proof Theory: Some Applications of Cut-Elimination

Publisher Summary This chapter discusses the Cut-Elimination Theorem for first-order logic. The proof of Cut-Elimination Theorem is set up in such a way that it can be easily generalized to many other cases where a cut-elimination argument is applied. The chapter also discusses the provability and unprovability of initial cases of transfinite induction for arithmetic Z. The result is well known: given a natural well-ordering ɛ 0 , with respect to ɛ 0 , but not up to ɛ 0 itself. The underivability in Z of transfinite induction up to ɛ 0 will also follow from Godels Second Incompleteness Theorem, together with the fact that transfinite induction up to ɛ 0 suffices to prove the reflection principle for Z and hence the consistency of Z. A direct proof of this underivability result is presented using a cut-elimination argument. Technically, this provides an easy and convincing example of the usefulness of infinite derivations and the strength of the cut-elimination method when applied to infinite derivations.

Normalisation by Evaluation

We extend normalization by evaluation (FIrst presented in [4]) from the pure typed ?-calculus to general higher type term rewrite systems. This work also gives a theoretical explanation of the normalization algorithm implemented in the Minlog system.

Higher type recursion, ramification and polynomial time

Abstract It is shown how to restrict recursion on notation in all finite types so as to characterize the polynomial-time computable functions. The restrictions are obtained by using a ramified type structure, and by adding linear concepts to the lambda calculus.

Program Extraction from Classical Proofs

Different methods for extracting a program from a classical proof are investigated. A direct method based on normalization and the wellknown negative translation combined with a realizability interpretation are compared and shown to yield equal results. Furthermore, the translation method is refined in order to obtain optimized programs. An analysis of the proof translation shows that in many cases only small parts of a classical proof need to be translated. Proofs extracted from such refined translations have simpler type and control structure. The effect of the refinements is demonstrated at two examples.

Program Extraction from Normalization Proofs

This paper describes formalizations of Taits normalization proof for the simply typed λ-calculus in the proof assistants Minlog, Coq and Isabelle/HOL. From the formal proofs programs are machine-extracted that implement variants of the well-known normalization-by-evaluation algorithm. The case study is used to test and compare the program extraction machineries of the three proof assistants in a non-trivial setting.

The Warshall Algorithm and Dickson's Lemma: Two Examples of Realistic Program Extraction

By means of two well-known examples it is demonstrated that the method of extracting programs from proofs is manageable in practice and may yield efficient programs. The Warshall algorithm computing the transitive closure of a relation is extracted from a constructive proof that repetitions in a path can always be avoided. Second, we extract a program from a classical (i.e., nonconstructive) proof of a special case of Dicksons lemma, by transforming the classical proof into a constructive one. These techniques (as well as the examples) are implemented in the interactive theorem prover MINLOG developed at the University of Munich.

Terminiation of permutative conversions in intuitionistic Gentzen calculi

It is shown that permutative conversions terminate for the cut-free intuitionistic Gentzen (i.e. sequent) calculus; this proves a conjecture by Dyckhoff and Pinto. The main technical tool is a term notation for derivations in Gentzen calculi. These terms may be seen as λ-terms with explicit substitution, where the latter corresponds to the left introduction rules.

Term rewriting for normalization by evaluation

We extend normalization by evaluation (first presented in [5]) from the pure typed λ-calculus to general higher type term rewriting systems and prove its correctness w.r.t. a domain-theoretic model. We distinguish between computational rules and proper rewrite rules. The former is a rather restricted class of rules, which, however, allows for a more efficient implementation.

Proof and System-Reliability

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