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I was reading a paper about implementing the statistical programming language R on the JVM, and encountered this notation I haven't seen before (I'm not formally CS-trained). What is the name of notational style? I've seen it before in other papers--usually as it related to functional programming and the REPL. enter image description here

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    $\begingroup$ See here1 and Backus-Naur form. $\endgroup$ Dec 7 '13 at 7:11
  • $\begingroup$ I had seen BNF before (Odersky did it in the Functional Programming Coursera), but usually it was written out into words, so I figured this was something different. Please paste your comment into a full answer and I'll accept it. $\endgroup$ Dec 7 '13 at 7:56
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    $\begingroup$ I believe that the fraction notation for rules of inference was introduced by [R. Smullyan, Theory of Formal Systems, 1961]. See also en.wikipedia.org/wiki/Formal_system as a starting point. $\endgroup$ Dec 7 '13 at 11:16
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    $\begingroup$ actually, the fractional notation was used already in Gentzen's paper (1934) on natural deduction and sequent calculus, which I think (but am not 100% sure) is the original source. $\endgroup$ Dec 7 '13 at 13:52
  • $\begingroup$ Yes, I was wrong, sorry. Smullyan also said so here, in comments: youtube.com/watch?v=P8VBeGl9Law I had rather badly misinterpreted a remark in another paper. $\endgroup$ Dec 7 '13 at 21:57
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It is notation for Formal Proofs and specification of language grammar in Backus-Naur Form.

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The individual statements of the form $A\Rightarrow B$ are sequents and the things that look like fractions are inference rules.

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