Equivalence of bar induction and bar recursion for continuous functions with continuous moduli

Fujiwara, Makoto; Kawai, Tatsuji

doi:10.1016/j.apal.2019.04.001

Cited by 5 publications

(4 citation statements)

References 4 publications

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“…Whereas bar induction is a principle of logic (a method to obtain proofs), bar recursion is an principle of mathematical construction (a method to define functions) proposed by Spector (1962). In results obtained subsequent to those described in our own paper, Fujiwara & Kawai (2019) show that bar induction for decidable bars is equivalent to the existence of a bar recursor for every functional of type (N → N) → N that has a continuous modulus of continuity. This result is closely related to that of Schwichtenberg (1979), as well as those of Escardó and ourselves, considering Escardó's proof that every definable such functional in System T has a definable modulus of continuity-and hence a continuous modulus of continuity (Escardó, 2013).…”

Section: Bar Recursion and Continuous Modulimentioning

confidence: 84%

Higher order functions and Brouwer’s thesis

Sterling¹

2021

J. Funct. Prog.

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Extending Martín Escardó’s effectful forcing technique, we give a new proof of a well-known result: Brouwer’s monotone bar theorem holds for any bar that can be realized by a functional of type (ℕ→ℕ)→ℕ in Gödel’s System T. Effectful forcing is an elementary alternative to standard sheaf-theoretic forcing arguments, using ideas from programming languages, including computational effects, monads, the algebra interpretation of call-by-name λ-calculus, and logical relations. Our argument proceeds by interpreting System T programs as well-founded dialogue trees whose nodes branch on a query to an oracle of type ℕ→ℕ, lifted to higher type along a call-by-name translation. To connect this interpretation to the bar theorem, we then show that Brouwer’s famous “mental constructions” of barhood constitute an invariant form of these dialogue trees in which queries to the oracle are made maximally and in order.

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Section: Bar Recursion and Continuous Modulimentioning

confidence: 84%

Higher order functions and Brouwer’s thesis

Sterling¹

2021

J. Funct. Prog.

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“…Before we start a general discussion, let us comment on one very specific point. While there are known constructions of self-modulating moduli from continuous moduli of continuity [FK19], we are not aware of one that translates to our setting. This is because all such constructions we are aware of directly work with the natural numbers and make use of their ordering.…”

Section: Discussionmentioning

confidence: 99%

Continuous and monotone machines

Konečný¹,

Steinberg²,

Thies³

2020

Preprint

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We investigate a variant of the fuel-based approach to modeling diverging computation in type theories and use it to abstractly capture the essence of oracle Turing machines. The resulting objects we call continuous machines. We prove that it is possible to translate back and forth between such machines and names in the standard function encoding used in computable analysis. Put differently, among the operators on Baire space, exactly the partial continuous ones are implementable by continuous machines and the data that such a machine provides is a description of the operator as a sequentially realizable functional.Continuous machines are naturally formulated in type theories and we have formalized our findings in Coq. Continuous machines, their equivalence to the standard encoding and correctness of basic operations are now part of Incone, a Coq library for computable analysis. While the correctness proofs use a classical meta-theory with countable choice, the translations and algorithms that are proven correct are all fully executable. Along the way we formally prove some known results such as existence of a self-modulating moduli of continuity for partial continuous operators on Baire space.To illustrate their versatility we use continuous machines to specify some algorithms that operate on objects that cannot be fully described by finite means, such as real numbers and functions. We present particularly simple algorithms for finding the multiplicative inverse of a real number and for composition of partial continuous operators on Baire space. Some of the simplicity is achieved by utilizing the fact that continuous machines are compatible with multivalued semantics. We also connect continuous machines to the construction of precompletions and completions of represented spaces, topics that have recently caught the attention of the computable analysis community.

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“…It is straightforward to show that for each k , the function

G_{k}

is a continuous modulus of itself (cf. [12, Lemma 2.2]). We show that G is a modulus of f .…”

Section: The Extension Of Ucc To Real‐valued Functionsmentioning

confidence: 99%

“…In this paper, we work in Heyting arithmetic in all finite types

{HA}^{ω}

[21, 1.6.15] with the axiom scheme

QF - {sans-serifAC}^{1, 0}

of quantifier free choice from sequences to numbers (cf., e.g., [12, § 1.4]). For readability, we write our proofs informally in this system.…”

Section: Introductionmentioning

confidence: 99%

Decidable fan theorem and uniform continuity theorem with continuous moduli

Fujiwara

Kawai

2021

Mathematical Logic Qtrly

Self Cite

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The uniform continuity theorem (UCT) states that every pointwise continuous real‐valued function on the unit interval is uniformly continuous. In constructive mathematics, sans-serifUCT is strictly stronger than the decidable fan theorem (DFT), but Loeb [17] has shown that the two principles become equivalent by encoding continuous real‐valued functions as type‐one functions. However, the precise relation between such type‐one functions and continuous real‐valued functions (usually described as type‐two objects) has been unknown. In this paper, we introduce an appropriate notion of continuity for a modulus of a continuous real‐valued function on [0, 1], and show that real‐valued functions with continuous moduli are exactly those functions induced by Loeb's codes. Our characterisation relies on two assumptions: (1) real numbers are represented by regular sequences (equivalently Cauchy sequences with explicitly given moduli); (2) the continuity of a modulus is defined with respect to the product metric on the regular sequences inherited from the Baire space. Our result implies that sans-serifDFT is equivalent to the statement that every pointwise continuous real‐valued function on [0, 1] with a continuous modulus is uniformly continuous. We also show that sans-serifDFT is equivalent to a similar principle for real‐valued functions on the Cantor space false{0,1false}N. These results extend Berger's [2] characterisation of sans-serifDFT for integer‐valued functions on false{0,1false}N and unify some characterisations of sans-serifDFT in terms of functions having continuous moduli.

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Equivalence of bar induction and bar recursion for continuous functions with continuous moduli

Cited by 5 publications

References 4 publications

Higher order functions and Brouwer’s thesis

Higher order functions and Brouwer’s thesis

Continuous and monotone machines

Decidable fan theorem and uniform continuity theorem with continuous moduli

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