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I just read Dunn's 1979 paper Relevant Robinson's Arithmetic, and the end especially caught my interest. Following the surprising role of constant functions in collapsing "relevant Q with zero" to classical Q but leaving "relevant Q without zero" nontrivial, Dunn writes:

[T]he idea of functions that really depend on their arguments has been articulated and investigated recently by many. [...] A typical motive in such investigations has been to do for relevant logic what Lauchli did for intuitionistic logic, namely to provide a "realizability" interpretation [...] the relevant trick being to look at only those functions that depend on their arguments. [...] What is needed is a "relevant Kleene," developing a theory of "relevant partial recursive functions" that really depend on their arguments, and using these to investigate "relevant realizability interpretations" of the relevant arithmetics. ([...] It is not the case that "representability in ${\bf Q_R(1)}$" coincides with "relevant recursiveness" (perhaps what is needed is some notion of "relevant representability").)

Basically, Dunn suggests three intertwined topics - relevant realizability, relevant recursiveness, and relevant representability - which sound quite interesting to me but unfortunately go well beyond my background. My question is whether there has been any work since Dunn's paper on these topics.

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    $\begingroup$ There is work on linear realizability, of which relevant realizability would be a special case. I am not familiar with this corner of realizability, but if you look for work that cites dl.acm.org/doi/10.5555/2392389.2392430 and doi.org/10.1017/S0960129502003730, you should get an idea of what it's about. $\endgroup$ Commented Jun 3, 2024 at 20:29

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I thought this was an interesting question and so I asked some relevant logicians on Mastodon. Here's a quick summary of the answers, although the short version seems to be "No", with Shawn Standefer saying

I’m fairly confident there hasn’t been anything done on relevant recursion theory or realisability. There has been some stuff done on relevant arithmetic, but not those topics.

  • A recent special issue of the Australasian Journal of Logic gives an overview of recent work in relevant arithmetic. Of particular interest may be the paper by Logan and Leach-Krouse [1], which contains some material on representing functions. Greg Restall's thesis [2] also includes a bit about doing this in substructural arithmetics, including R#.
  • Zach Weber has a project on paraconsistent computability theory which runs until 2026. I imagine that this is a follow-up to his 2016 paper with Toby Meadows [3].
  1. S. Logan and G. Leach-Krouse. On not saying what we shouldn't have to say, Australasian Journal of Logic 18(5):524–568, 2021. https://doi.org/10.26686/ajl.v18i5.6923
  2. G. Restall. On Logics Without Contraction, PhD thesis, The University of Queensland, 1994. https://consequently.org/writing/onlogics/
  3. T. Meadows and Z. Weber. Computation in non-classical foundations? Philosophers' Imprint 16(13):1–17, 2016. https://quod.lib.umich.edu/p/phimp/3521354.0016.013/1

Thanks to Shawn Standefer and Greg Restall for their answers.

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