Abstract
The count invariance of van Benthem (1991[16]) is that for a sequent to be a theorem of the Lambek calculus, for each atom, the number of positive occurrences equals the number of negative occurrences. (The same is true for multiplicative linear logic.) The count invariance provides for extensive pruning of the sequent proof search space. In this paper we generalize count invariance to categorial grammar (or linear logic) with additives and bracket modalities. We define by mutual recursion two counts, minimum count and maximum count, and we prove that if a multiplicative-additive sequent is a theorem, then for every atom, the minimum count is less than or equal to zero and the maximum count is greater than or equal to zero; in the case of a purely multiplicative sequent, minimum count and maximum count coincide in such a way as to together reconstitute the van Benthem count criterion. We then define in the same way a bracket count providing a count check for bracket modalities. This allows for efficient pruning of the sequent proof search space in parsing categorial grammar with additives and bracket modalities.
Research partially supported by an ICREA Acadèmia 2012 to the third author, and by BASMATI MICINN project (TIN2011-27479-C04-03) and SGR2009-1428 (LARCA). Many thanks to Josefina Sierra and to three Formal Grammar referees for comments and suggestions. Particular thanks to the referee who pointed towards the simplification of the proposal in the appendix which we have used in the main text. Any errors are our own.
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Valentín, O., Serret, D., Morrill, G. (2013). A Count Invariant for Lambek Calculus with Additives and Bracket Modalities. In: Morrill, G., Nederhof, MJ. (eds) Formal Grammar. FG FG 2013 2012. Lecture Notes in Computer Science, vol 8036. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-39998-5_17
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