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The School of Squiggol

A History of the Bird–Meertens Formalism

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Formal Methods. FM 2019 International Workshops (FM 2019)

Part of the book series: Lecture Notes in Computer Science ((LNPSE,volume 12233))

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Abstract

The Bird–Meertens Formalism, colloquially known as “Squiggol”, is a calculus for program transformation by equational reasoning in a function style, developed by Richard Bird and Lambert Meertens and other members of IFIP Working Group 2.1 for about two decades from the mid 1970s. One particular characteristic of the development of the Formalism is fluctuating emphasis on novel ‘squiggly’ notation: sometimes favouring notational exploration in the quest for conciseness and precision, and sometimes reverting to simpler and more rigid notational conventions in the interests of accessibility. This paper explores that historical ebb and flow.

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Notes

  1. 1.

    Guido van Rossum, who worked on the ABC project in Amsterdam, was mentored by Meertens and went on to design Python [55] based on some of the ideas in ABC.

  2. 2.

    From its start, my own research career has been intimately entwined with WG2.1 and the BMF, although I came somewhat late to the party. Bird supervised my DPhil dissertation (1987–1991) [35], and Meertens was my external examiner. I have worked on and off with Bird ever since, and most of my research has been inspired by the BMF; I am Bird’s co-author on his forthcoming book [24]. I served as secretary of WG2.1 for thirteen years (1996–2009), during the Chairmanships of Doug Smith and Lambert Meertens, and then succeeded Meertens as Chair myself for the following six years (2009–2015).

  3. 3.

    It is fair to say that the Reports did not meet with universal acclaim. One reason for the mixed reception was the use of van Wijngaarden’s two-level grammar notation for describing the language, whereby a possibly infinite language grammar is generated by a finite meta-grammar. The Algol 68 experience has engendered a keen interest in notational issues within the Group ever since.

  4. 4.

    Indeed, the loop body in the first exponentiation program is a ‘specification statement’ in Morgan’s sense [49], albeit one without a frame.

  5. 5.

    In fact, Bird and Meertens had both been present at Meeting #27 in Wheeling in August 1980, although the meeting of minds evidently had to wait a bit longer.

  6. 6.

    Meertens’ preface in the Abstracto Reader, Meertens’ contemporary papers, and the WG2.1 minutes all record Bird’s paper under the title “Some Notational Suggestions...” [17]; but the technical report [16] is entitled “Notational Suggestions...”.

  7. 7.

    Applying g to every element of S and then f to every element of the result is the same as applying g then f to every element of S in a single pass.

  8. 8.

    Applying f to every element and then filtering to keep the results that satisfy P is the same as filtering first, using the predicate “P after f”, then applying f to every element that will subsequently satisfy P.

  9. 9.

    Publication of the proceedings of this conference seems to have taken frustratingly long: in a 1984 working paper [44] using the same notation, Meertens cites the Algorithmics paper [45] as appearing in the year “\({[]/}~{(1984{\le })\triangleleft }~\mathbb {U}\)”, that is, the arbitrary choice of any number at least 1984. The same joke appears in a 1985 working paper [12] but with a 1985 lower bound.

  10. 10.

    Nevertheless, Wile’s ‘sectioning’ notation (giving a binary operator one of its two arguments, as in the positivity predicate “\(({>}0)\)” and the reciprocal function “(1/)”) was discussed, and it persists today in Haskell.

  11. 11.

    In fact, Meertens says that he deliberately used a very small asterisk for ‘map’, looking from a distance or on a poor photocopy like a ragged dot, so as not to have to choose between the two notations.

  12. 12.

    Essentially the same canonical scheme is commonly used today in modern functional programming languages like Haskell:

    but for the signature of asymmetric ‘cons’ lists, rather than symmetric ‘cat’ lists. This again depends on lists being a free algebra, so the equations have a unique solution, namely the function being defined.

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Acknowledgements

I am especially indebted to Richard Bird and Lambert Meertens: for much discussion about this work, and correcting some of my misunderstandings, but more importantly for being the inspiration for essentially my entire research career. I would also like to thank Doaitse Swierstra, Hendrik Boom, and Stephen Spackman for answering my many questions, and Cezar Ionescu, Dan Shiebler, the members of IFIP WG2.1, the participants at the workshop on the History of Formal Methods in Porto in October 2019, and the anonymous reviewers for their helpful comments and enthusiastic feedback.

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    Jeremy Gibbons

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  1. McMaster University, Hamilton, ON, Canada

    Emil Sekerinski

  2. University of Porto, Porto, Portugal

    Nelma Moreira

  3. University of Minho, Braga, Portugal

    José N. Oliveira

  4. Argo Ai, Munich, Germany

    Daniel Ratiu

  5. University of Pisa, Pisa, Italy

    Riccardo Guidotti

  6. University of Liverpool, Liverpool, UK

    Marie Farrell

  7. University of Liverpool, Liverpool, UK

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  8. University of Exeter, Exeter, UK

    Diego Marmsoler

  9. University of Minho, Braga, Portugal

    José Campos

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Gibbons, J. (2020). The School of Squiggol. In: Sekerinski, E., et al. Formal Methods. FM 2019 International Workshops. FM 2019. Lecture Notes in Computer Science(), vol 12233. Springer, Cham. https://doi.org/10.1007/978-3-030-54997-8_2

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