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Parsimonious Computational Completeness

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Developments in Language Theory (DLT 2021)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 12811))

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Abstract

Throughout the history of Formal Languages, one of the research directions has always been to describe computational completeness using only a small amount of possibly scarce resources. We review some of these results in the form of an essay.

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Notes

  1. 1.

    Out of a series of 151 short videos showing interviews with Minsky within the Web of Stories - Life Stories of Remarkable People.

  2. 2.

    If somebody wonders about how one of the ‘fathers of AI’ and the philosophy behind could write one of the first textbooks on Automata Theory, let Minsky himself make the link in the preface of his book: The abstract theory—as described in this book—tells us in no uncertain terms that the machines’ potential range is enormous, and that its theoretical limitations are of the subtlest and most elusive sort. There is no reason to suppose machines have any limitations not shared by man. But this role of Minsky created some peculiar situations until today; for instance, the said book is collected within the ‘methodological books’ in the part of our university library dedicated to psychology. Presumably, this was because one part of the book explains a mathematical model of neural networks.

  3. 3.

    A further restricted form thereof Kuroda himself termed linear-bounded grammar, but our definition corresponds to what is nowadays called Kuroda normal form.

  4. 4.

    With different arguments, the fact that two such erasing non-context-free rules suffice to reach computational completeness was also explained in [41], pointing to several earlier papers. We also refer to [38, 39].

  5. 5.

    Observe that according to our definition, PCPs are collections of word pairs over a binary alphabet; of course, one could also consider such word pair collections over arbitrary alphabets, but we stick to this simpler case in our treatment.

  6. 6.

    We use \(\overleftarrow{w}\) to denote the mirror (or reversal) of word w.

  7. 7.

    For a discussion of programming languages tailored towards DNA computing, we refer to the study [49] conducted by Microsoft Research.

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  1. FB 4 – Informatikwissenschaften, Universität Trier, 54286, Trier, Germany

    Henning Fernau

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    Nelma Moreira

  2. University of Porto, Porto, Portugal

    Rogério Reis

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Fernau, H. (2021). Parsimonious Computational Completeness. In: Moreira, N., Reis, R. (eds) Developments in Language Theory. DLT 2021. Lecture Notes in Computer Science(), vol 12811. Springer, Cham. https://doi.org/10.1007/978-3-030-81508-0_2

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