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Self-certifying Railroad Diagrams

Or: How to Teach Nondeterministic Finite Automata

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Book cover Mathematics of Program Construction (MPC 2019)

Part of the book series: Lecture Notes in Computer Science ((LNPSE,volume 11825))

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Abstract

Regular expressions can be visualized using railroad or syntax diagrams. The construction does not depend on fancy artistic skills. Rather, a diagram can be systematically constructed through simple, local transformations due to Manna. We argue that the result can be seen as a nondeterministic finite automaton with -transitions. Despite its simplicity, the construction has a number of pleasing characteristics: the number of states and the number of edges is linear in the size of the regular expression; due to sharing of sub-automata and auto-merging of states the resulting automaton is often surprisingly small. The proof of correctness relies on the notion of a subfactor. In fact, Antimirov’s subfactors (partial derivatives) appear as target states of non--transitions, suggesting a smooth path to nondeterministic finite automata without -transitions. Antimirov’s subfactors, in turn, provide a fine-grained analysis of Brzozowski’s factors (derivatives), suggesting a smooth path to deterministic finite automata. We believe that this makes a good story line for introducing regular expressions and automata.

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Notes

  1. 1.

    Actually, Conway’s axiomatization is incomplete as pointed out by Abramsky and Vickers [1]: only if the axiom \(\sum \{a\}=a\) is added, his \(\mathbf S \)-algebras are equivalent to regular algebras.

  2. 2.

    A regular algebra is really a blend of a complete join-semilattice and a monoid. It is a standard result of lattice theory that a complete join-semilattice is a complete lattice. However, a complete join-semilattice homomorphism is not necessarily a complete lattice homomorphism, as there is no guarantee that it also preserves meets.

  3. 3.

    Backhouse uses the terms matrix and graph interchangeably.

  4. 4.

    Backhouse’s definition is actually more general: a matrix is given by a function where is a fixed relation, the dimension of the matrix. This allows him to distinguish between non-existent edges and edges that are labelled with 0.

  5. 5.

    We use “correctness” and “completeness” only in this very narrow sense.

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Acknowledgements

A big thank you is due to Bernhard Möller for pointing me to Manna’s textbook on the “Mathematical Theory of Computation” [12]. Thanks are also due to Clare Martin and Sebastian Schweizer for proof-reading a preliminary version of the manuscript. The TikZ diagrams for Thompson’s construction, see Fig. 5, are due to Ian McLoughlin, who kindly granted permission to use them. As always, the anonymous referees of MPC provided detailed feedback, suggesting numerous improvements regarding presentation and organization of the material. In response to their comments I moved the section on “Railroad Diagrams” to the front and I clarified Antimirov’s contribution. Finally, I owe a particular debt of gratitude to Roland Backhouse, who provided numerous comments, pointing out, in particular, the relevance of Conway’s work. The paragraph on “linear mechanisms” was added in response to his suggestions.

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  1. Technische Universität Kaiserslautern, 67653, Kaiserslautern, Germany

    Ralf Hinze

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  1. Ralf Hinze
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Correspondence to Ralf Hinze .

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  1. University of Nottingham, Nottingham, UK

    Graham Hutton

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Hinze, R. (2019). Self-certifying Railroad Diagrams. In: Hutton, G. (eds) Mathematics of Program Construction. MPC 2019. Lecture Notes in Computer Science(), vol 11825. Springer, Cham. https://doi.org/10.1007/978-3-030-33636-3_5

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  • DOI: https://doi.org/10.1007/978-3-030-33636-3_5

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