Zur Berechenbarkeit Primitiv-Rekursiver Funktionale Endlicher Typen

Diller, Justus

doi:10.1016/s0049-237x(08)70521-4

Cited by 6 publications

(5 citation statements)

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“…Here we prove termination of the leftmost (or standard) reduction strategy for primitive recursive terms, by means of transfinite induction of length en-Our proof is by an adaption of a method of Howard [6] to the present situation; ultimately this technique is based on ideas of Sanchis [10] and Diller [2].…”

Section: Our Conversion Rules Certainly Contain ß-Conversion (Xxr)s mentioning

confidence: 99%

“…It is shown that, for any primitive recursive term of arbitrary type, the leftmost (or standard) reduction sequence terminates. The proof is done by transfinite induction up to SQ\ it uses a method of Howard [6] which in turn is based on earlier work of Sanchis [10] and Diller [2]. Standard machinery from Recursion Theory can then be applied to obtain bounds for the length of the leftmost reduction sequence.…”

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Primitive Recursion on the Partial Continuous Functionals

Schwichtenberg¹

1991

Informatik Und Mathematik

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Section: Our Conversion Rules Certainly Contain ß-Conversion (Xxr)s mentioning

confidence: 99%

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confidence: 99%

Primitive Recursion on the Partial Continuous Functionals

Schwichtenberg¹

1991

Informatik Und Mathematik

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“…Note. As for the computability of primitive recursive functionals of finite type, see for example Diller (1968), Hinata (1967) and Hindley and others (1972).…”

Section: Construction Principlementioning

confidence: 99%

Construction principle and transfinite induction up to ε₀

Yasugi

1982

J Aust Math Soc A

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What we call here the "construction principle" is a principle on the ground of which some functional can be defined; the domain and the range of such a functional consist of some "computable" functionals of various finite types. The principle above is considered here as the basis of the functional interpretation of transfinite induction up to en. It is concretely repesented as the "term-forms", where every term-form is shown to be "computable" in some sense.

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“…Corresponding to [Di,p. 110], we can introduce primitive recursion functors g of order j (say) satisfying the following clauses.…”

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confidence: 99%

“…Let us consider the Fourth Version and include the primitive recursion functor g considered in Theorem 2.5. Where Diller [Di,p. 110] introduces a functor/by an abstraction equation/…”

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confidence: 99%

Ordinal analysis of terms of finite type

Howard

1980

J. symb. log.

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Sanchis [Sa] and Diller [Di] have introduced an interesting relation between terms for primitive recursive functionals of finite type in order to obtain a computability proof. The method is as follows. First the relation is shown to be well-founded. Then computability of each term is obtained by transfinite induction over the relation.Their relation is given by various clauses which define the (immediate) successors of a term. Hence, starting with a term H and repeatedly applying the successor relation one generates the tree of H (say). The problem is to prove that the trees are well-founded. In their proofs Sanchis and Diller use the axiom of bar induction. Diller also gives a proof using transfinite induction: each tree is shown to have a length b < φω(0) (the first ω-critical number) and the induction is over ordinals less than or equal to b.For proof-theoretic purposes it would be desirable to employ a method of proof which is more elementary than the axiom of bar induction and also to obtain a smaller ordinal bound. The purpose of the following is to provide such a method. We show that for a primitive recursive term H of finite type the tree generated by the relation of Sanchis and Diller has length less than ε0. Our method is similar to that used by Tait [Ta] in his theory of infinite terms. The immediately natural metamathematics is elementary intuitionistic analysis with the axiom of choice and transfinite induction over the relevant ordinal notations. We give several versions of the basic list of successor clauses, and for one version we show how to carry out the treatment in Skolem arithmetic (with primitive recursion of lowest type).

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Zur Berechenbarkeit Primitiv-Rekursiver Funktionale Endlicher Typen

Cited by 6 publications

References 4 publications

Primitive Recursion on the Partial Continuous Functionals

Primitive Recursion on the Partial Continuous Functionals

Construction principle and transfinite induction up to ε₀

Ordinal analysis of terms of finite type

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Zur Berechenbarkeit Primitiv-Rekursiver Funktionale Endlicher Typen

Cited by 6 publications

References 4 publications

Primitive Recursion on the Partial Continuous Functionals

Primitive Recursion on the Partial Continuous Functionals

Construction principle and transfinite induction up to ε0

Ordinal analysis of terms of finite type

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Product

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Construction principle and transfinite induction up to ε₀