S4 enriched multimodal categorial grammars are context-free

Abstract

Bar-Hillel et al. [Y. Bar-Hillel, C. Gaifman, E. Shamir, On categorial and phrase structure grammars, Bulletin of the Research Council of Israel F (9) (1960) 1–16] prove that applicative categorial grammars weakly recognize the context-free languages. Buszkowski [W. Buszkowski, Generative capacity of non-associative Lambek calculus, Bulletin of the Polish Academy of Sciences: Mathematics 34 (1986) 507–518] proves that grammars based on the product-free fragment of the non-associative Lambek calculus NL recognize exactly the context-free languages. Kandulski [M. Kandulski, The equivalence of non-associative Lambek categorial grammars and context-free grammars, Zeitschrift für Mathematische Logik und Grundlagen der Mathematik 34 (1988) 41–52] furthers this result by proving that grammars based on NL also recognize exactly the context-free languages. Jäger [G. Jäger, Residuation, structural rules and context freeness, Journal of Logic, Language and Information 13 (2004) 47–59] proves that categorial grammars based on NL ♢ , the non-associative Lambek calculus enriched with residuated modalities, weakly recognize exactly the context-free languages. We extend this result, proving that categorial grammars based on NL S 4 , the enrichment of NL ♢ by the axioms 4 and T , weakly recognize exactly the context-free languages.