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On the Recognizing Power of the Lambek Calculus with Brackets

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Abstract

Every language recognized by the Lambek calculus with brackets is context-free. This is shown by combining an observation by Jäger with an entirely straightforward adaptation of the method Pentus used for the original Lambek calculus. The case of the variant of the calculus allowing sequents with empty antecedents is slightly more complicated, requiring a restricted use of the multiplicative unit.

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Notes

  1. Equivalently, we may take all sequents of the form \(A {\varvec{\rightarrow }}A\) as initial sequents, as Jäger (2003) did.

  2. This is one of the two notions of recognition studied by Jäger (2003); he called this notion t-recognition.

  3. An example (adapted from Fadda and Morrill 2005) is \(\Diamond \Box ^{\downarrow }p \,\Diamond \Box ^{\downarrow }q {\varvec{\rightarrow }}\Diamond \Box ^{\downarrow }(p \bullet q)\).

  4. To be precise, Kanovich et al. (2017) were speaking of \(\mathbf {L}^{\textstyle *}\Diamond \) rather than \(\mathbf {L}\Diamond \). Jäger (2003) was dealing with \(\mathbf {L}\Diamond \) rather than \(\mathbf {L}^{\textstyle *}\Diamond \), although he did not make it entirely clear.

  5. See Kanazawa (2006) for a statement of an interpolation theorem for the implicational fragment of intuitionistic logic in terms of these links.

  6. To extend Moortgat’s (1996) proof in the presence of \(\varvec{1}\), one only need to add the reduction step

  7. Pentus’s (1999) claim of context-freeness of \(\mathbf {L}^{\textstyle *}\), as opposed to \(\mathbf {L}^{\textstyle *}_{\varvec{1}}\), is immune to this criticism since an interpolation theorem similar to Theorem 3 does hold for \(\mathbf {L}^{\textstyle *}\) and there’s no need to use \(\varvec{1}\) in converting an \(\mathbf {L}^{\textstyle *}\) grammar to a context-free grammar. The same criticism does apply to his claim about grammars based on multiplicative cyclic linear logic (CLL).

References

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  1. Department of Advanced Sciences, Faculty of Science and Engineering, Hosei University, 3–7–2 Kajino-cho, Koganei-shi, Tokyo, 184–8584, Japan

    Makoto Kanazawa

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  1. Makoto Kanazawa
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Kanazawa, M. On the Recognizing Power of the Lambek Calculus with Brackets. J of Log Lang and Inf 27, 295–312 (2018). https://doi.org/10.1007/s10849-018-9269-3

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