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John Tromp defines a version of the lambda calculus that is encoded in binary: https://tromp.github.io/cl/cl.html

a) Does there exist a concatenative version of this language (or its combinatory equivalent)?

b) Does there exist a concatenative version restricted to the primitive recursive functions (e.g. as per Combinators for the Primitive Recursive Functions)?

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    $\begingroup$ I'm not sure what you mean by concatenative. Do you mean something like here: en.wikipedia.org/wiki/Concatenative_programming_language or a language in which the concatenation of any two programs is a valid program? $\endgroup$ – cody Aug 18 '15 at 14:32
  • $\begingroup$ @cody - Actually, it looks like just the condition that the concatenation of any two programs is a valid program suffices. If anyone can say what languages with this weaker condition are called, I'll edit the question/title accordingly. $\endgroup$ – NietzscheanAI Aug 18 '15 at 14:37
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    $\begingroup$ With this weaker condition, I believe that Iota, Jot and Zot are examples of a): semarch.linguistics.fas.nyu.edu/barker/Iota/zot.html $\endgroup$ – NietzscheanAI Aug 18 '15 at 14:38
  • $\begingroup$ You should turn your comment into an answer. $\endgroup$ – cody Aug 18 '15 at 14:40
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    $\begingroup$ This is a related post. cstheory.stackexchange.com/questions/31883/… $\endgroup$ – Joshua Herman Aug 19 '15 at 0:11
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As per @cody's comment above, if we require only that "the concatenation of any two programs is a valid program" then I believe that Iota, Jot and Zot (semarch.linguistics.fas.nyu.edu/barker/Iota/zot.html) are examples of a).

However, IMO the more interesting question part of the question is b)...

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