Bar induction: The good, the bad, and the ugly

Rahli, Vincent; Bickford, Mark; Constable, Robert L.

doi:10.1109/lics.2017.8005074

Cited by 8 publications

(13 citation statements)

References 51 publications

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“…We can now go on to show that Object and Arrangement are well defined. We do so by producing a model within set theory that fulfils most of the constraints in section 2.1., which are written out formally in Appendix D. 16 This proof requires that the reader pick some appropriate values for Symbol, Pos, Pointer, , ǫ and B. The definition of these sets in Appendix D is adequate.…”

Section: B Proof Of Key Resultsmentioning

confidence: 99%

What Does This Notation Mean Anyway? BNF-style notation as it is actually used

Feller¹,

Wells²,

Carlier³

et al. 2018

EasyChair Preprints

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Following the introduction of BNF notation by Backus for the Algol 60 report and subsequent notational variants, a metalanguage involving formal “grammars” has developed for discussing structured objects in Computer Science and Mathematical Logic. We refer to this offspring of BNF as Math-BNF or MBNF, to the original BNF and its notational variants just as BNF, and to aspects common to both as BNF-style. MBNF is sometimes called abstract syntax, but we avoid that name because MBNF is in fact a concrete form and has a more abstract form. What all BNF-style notations share is the use of production rules roughly of this form: ◯ ⩴ □₁ | ⋯ | □ₙ Normally, such a rule says “every instance of □ᵢ for i ∈ {1, ..., n} is also an instance of ◯”.MBNF is distinct from BNF in the entities and operations it allows. Instead of strings, MBNF builds arrangements of symbols that we call math-text. Sometimes “syntax” is defined by interleaving MBNF production rules and other mathematical definitions that can contain chunks of math-text.There is no clear definition of MBNF. Readers do not have a document which tells them how MBNF is to be read and must learn MBNF through a process of cultural initiation. To the extent that MBNF is defined, it is largely through examples scattered throughout the literature.This paper gives MBNF examples illustrating some of the differences between MBNF and BNF. We propose a definition of syntactic math text (SMT) which handles many (but far from all) uses of math-text and MBNF in the wild. We aim to balance the goal of being accessible and not requiring too much prerequisite knowledge with the conflicting goal of providing a rich mathematical structure that already supports many uses and has possibilities to be extended to support more challenging cases.

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Section: B Proof Of Key Resultsmentioning

confidence: 99%

What Does This Notation Mean Anyway? BNF-style notation as it is actually used

Feller¹,

Wells²,

Carlier³

et al. 2018

EasyChair Preprints

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“…For example, the Axiom of Open Data (more precisely, the continuity principle for numbers which follows from it) was used by Brouwer in order to prove that all real-valued functions on the unit interval are uniformly continuous [13,Thm.3]. The bar induction principle, which is a powerful intuitionistic induction principle (equivalent to the classical transfinite induction) is another consequence of the introduction of free choice sequences [11,10], which has also been explored in the context of the Nuprl proof assistant [28].…”

Section: The Creative Subject and Choice Sequencesmentioning

confidence: 99%

On Expanding Standard Notions of Constructivity

Cohen¹,

Kellison²

2018

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Brouwer developed the notion of mental constructions based on his view of mathematical truth as experienced truth. These constructions extend the traditional practice of constructive mathematics, and we believe they have the potential to provide a broader and deeper foundation for various constructive theories. We here illustrate mental constructions in two well-studied theories – computability theory and plane geometry – and discuss the resulting extended mathematical structures. Further, we demonstrate how these notions can be embedded in an implemented formal framework, namely the constructive type theory of the Nuprl proof assistant. Additionally, we point out several similarities in both the theory and implementation of the extended structures.

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“…In [39] the assumption of existence of choice sequences was exploited to establish Bar Induction, a key intuitionistic principle, in Nuprl. However, choice sequences were there used only as an instrumental tool in the metatheory, not embedded into the theory itself.…”

Section: Introductionmentioning

confidence: 99%

“…Choice sequences were generated using Coq functions, including such that use non-computable axioms. As noted in [39]: "choice sequences do not have to be-and are not-part of the syntax of Nuprl definitions and proofs, i.e., the syntax visible to users". This approach had some undesired consequences.…”

Section: Introductionmentioning

confidence: 99%

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Computability Beyond Church-Turing via Choice Sequences

Bickford

Cohen

Constable

et al. 2018

Proceedings of the 33rd Annual ACM/IEEE Symposium on Logic in Computer Science

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Church-Turing computability was extended by Brouwer who considered non-lawlike computability in the form of free choice sequences. Those are essentially unbounded sequences whose elements are chosen freely, i.e. not subject to any law. In this work we develop a new type theory BITT, which is an extension of the type theory of the Nuprl proof assistant, that embeds the notion of choice sequences. Supporting the evolving, non-deterministic nature of these objects required major modifications to the underlying type theory. Even though the construction of a choice sequence is non-deterministic, once certain choices were made, they must remain consistent. To ensure this, BITT uses the underlying library as state and store choices as they are created. Another salient feature of BITT is that it uses a Beth-like semantics to account for the dynamic nature of choice sequences. We formally define BITT and use it to interpret and validate essential axioms governing choice sequences. These results provide a foundation for a fully intuitionistic version of Nuprl.

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Bar induction: The good, the bad, and the ugly

Cited by 8 publications

References 51 publications

What Does This Notation Mean Anyway? BNF-style notation as it is actually used

What Does This Notation Mean Anyway? BNF-style notation as it is actually used

On Expanding Standard Notions of Constructivity

Computability Beyond Church-Turing via Choice Sequences

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