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Subsequence versus substring constraints in sequence pattern languages

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Abstract

A family of logics for expressing patterns in sequences is investigated. The logics are all fragments of first-order logic, but they are variable-free. Instead, they can use substring and subsequence constraints as basic propositions. Propositions expressing constraints on the beginning or the end of the sequence are also available. Also wildcards can be used, which is important when the alphabet is not fixed, as is typical in database applications. The maximal logic with all four features of substring, subsequence, begin–end constraints, and wildcards, turns out to be equivalent to the family of star-free regular languages of dot-depth at most one. We investigate the lattice formed by taking all possible combinations of the above four features, and show it to be strict. For instance, we formally confirm what might intuitively be expected, namely, that boolean combinations of substring constraints are not sufficient to express subsequence constraints, and vice versa. We show an expressiveness hierarchy results from allowing multiple wildcards. We also investigate what happens with regular expressions when concatenation is replaced by subsequencing. Finally, we study the expressiveness of our logic relative to first-order logic.

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Notes

  1. If we identify a condition with the set of sequences (words) that satisfy it, a condition is indeed a formal language.

  2. Note that mixing and is the same as allowing “globbing” wildcards \(*\) in substring patterns, familiar from the Unix shell where the previous pattern would be written as \(ab*bc*c*a\).

  3. Over the one-letter alphabet, all logics we consider collapse to the family of finite or cofinite languages.

  4. We note that, motivated in part by database applications, formal language theory has now been revisited to accommodate an unknown (or infinite) alphabet [18].

  5. By the connections given in the previous section, this theorem includes as a special case the long-known fact that the family of locally testable languages is strictly included in the family of star-free regular languages.

  6. The definition of \(\mathcal {R}_{k+2}^{+}\) considered in [21] contains all words of the form (1) where each \(c_i\) does not appear in \(w_i\). However, this is merely a more restricted version of the language \(\mathcal {U}(\psi _{k+1})\). The proof in [21, Lemma 4.3] still trivially holds even if we drop the requirement that each \(c_i\) does not appear in \(w_i\).

  7. Uniformly over all alphabets, or over an infinite alphabet, this is another matter [9].

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Acknowledgements

We would like to thank the anonymous referees for their careful and helpful comments in improving our paper. We also thank Frank Neven for suggesting the connection to locally testable languages, and Jean-Eric Pin for his encouragement and help in proving Theorem 6.

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  1. Hasselt University, Hasselt, Belgium

    Steven Engels & Jan Van den Bussche

  2. National Taiwan University, Taipei City, Taiwan

    Tony Tan

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  1. Steven Engels
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  2. Tony Tan
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  3. Jan Van den Bussche
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Correspondence to Tony Tan.

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The second author acknowledges the generous financial support from Taiwan Ministry of Science and Technology under Grant No. 107-2221-E-002-026-MY2.

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Engels, S., Tan, T. & Van den Bussche, J. Subsequence versus substring constraints in sequence pattern languages. Acta Informatica 58, 35–56 (2021). https://doi.org/10.1007/s00236-019-00347-5

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