Über ein Problem, betreffend die Definition des Begriffes der allgemein‐rekursiven Funktion

Kalmár, László

doi:10.1002/malq.19550010204

Cited by 9 publications

(4 citation statements)

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“…Kalmár concluded his paper with his belief that certain mathematical concepts, such as 'effective calculability' or 'provability,' "cannot permit any restriction imposed by an exact mathematical definition" due to the endless development of mathematics, and thus, the possible further development of these concepts in the future. 3 Then Péter discusses Kalmár's (1955). Here, answering a question of Karl Schröter's in the negative, he showed that a system of functional equations (without restrictions on the operations to compute its values) may have a unique solution without the determined function being general recursive.…”

Section: Church's Thesis In Péter's Recursive Functionsmentioning

confidence: 99%

Péter on Church’s Thesis, Constructivity and Computers

Szabó

2021

Lecture Notes in Computer Science

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The aim of this paper is to take a look at Péter's talk Rekursivität und Konstruktivität delivered at the Constructivity in Mathematics Colloquium in 1957, where she challenged Church's Thesis from a constructive point of view. The discussion of her argument and motivations is then connected to her earlier work on recursion theory as well as her later work on theoretical computer science. * I would like to thank Marianna Antonutti-Marfori and Alberto Naibo for inviting me to the HaPoC Special Session on Church's Thesis in Constructive Mathematics. I am also indebted to Alberto for his insightful discussions on this topic and his useful comments on earlier versions of this paper. I would also like to thank Kendra Chilson, Katalin Gosztonyi, Wilfried Sieg and Kristóf Szabó for their help and the anonymous reviewers for their suggestions. The writing of this paper was supported by the UK Engineering and Physical Sciences Research Council under grant EP/R03169X/1.

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Section: Church's Thesis In Péter's Recursive Functionsmentioning

confidence: 99%

Péter on Church’s Thesis, Constructivity and Computers

Szabó

2021

Lecture Notes in Computer Science

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“…453], that E-HA ω is conservative over HA. 13 The use of classical logic, rather than constructive (intuitionistic) logic, makes here no difference, and PRA * is a sub-theory of (the classical counterpart of) E-HA ω .…”

Section: 2mentioning

confidence: 99%

“…See e.g [38]. for details and related discussions 13. I am grateful to Ulrich Kohlenbach for pointing me to that reference.…”

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Global semantic typing for inductive and coinductive computing

Leivant

2014

Logical Methods in Computer Science

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Abstract. Inductive and coinductive types are commonly construed as ontological (Churchstyle) types, with canonical semantical interpretation. When studying programs in the context of global ("uninterpreted") semantics, it is preferable to think of types as semantical properties (Curry-style). A purely logical framework for reasoning about semantic types is provided by intrinsic theories, introduced by the author in 2002, which fit tightly with syntactic, semantic, and proof theoretic fundamentals of formal logic, with potential applications in implicit computational complexity as well as extraction of programs from proofs.Intrinsic theories have been considered so far for inductive data, and we presently extend that framework to data defined using both inductive and coinductive closures. Our first main result is a Canonicity Theorem, showing that the global definition of program typing, via the usual (Tarskian) semantics of first-order logic, agrees with their operational semantics in the intended ("canonical") model. The paper's other main result is a proof theoretic calibration of intrinsic theories: every intrinsic theory is interpretable in (a conservative extension of) first-order arithmetic. This means that quantification over infinite data objects does not lead, on its own, to prooftheoretic strength beyond that of Peano Arithmetic.

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Recursive Functions

Hermes¹

1969

Enumerability · Decidability Computability

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Über ein Problem, betreffend die Definition des Begriffes der allgemein‐rekursiven Funktion

Cited by 9 publications

References 1 publication

Péter on Church’s Thesis, Constructivity and Computers

Péter on Church’s Thesis, Constructivity and Computers

Global semantic typing for inductive and coinductive computing

Recursive Functions

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