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In the preface to his very influential books Automata, Languages and Machines (Volumes A, B), Samuel Eilenberg tantalizingly promised Volumes C and D dealing with "a hierarchy (called the rational hierarchy) of the nonrational phenomena... using rational relations as a tool for comparison. Rational sets are at the bottom of this hierarchy. Moving upward one encounters 'algebraic phenomena,'" which lead to "to the context-free grammars and context-free languages of Chomsky, and to several related topics."

But Eilenberg never published volume C. He did leave preliminary handwritten notes for the first few chapters (http://www-igm.univ-mlv.fr/~berstel/EilenbergVolumeC.html) complete with scratchouts, question marks, side notes and gaps. But they do not reveal much beyond the beginnings of the well-known power series approach to grammars.

So, my actual question -- does anyone know of work along the same lines to possibly reconstruct what Eilenberg had in mind? If not, what material is likely closest to his ideas?

The site http://x-machines.net/ is about x-machines, one of Eilenberg's key innovations, but it deals mainly with applications of x-machines rather than further developing the theory as Eilenberg seemed to promise.

Also, anyone know why Eilenberg stopped before making much progress on Volume C? This was the late 70's, and he lived until 1998, though he did not appear to have published any math after Volume B. Yet he seemed to have the math for Volumes C and D largely done, at least in his mind.

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