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Taniguchi (2022), Unprovability of Continuation-Passing Style Transformation in Lambek Calculus
Unprovabiliy of Continuation-Passing Style Transformation in Lambek Calculus
Combinatory categorial grammar (CCG) includes combinators in addition to categorial grammar (CG), to accommodate various linguistics phenomena. For example, the type-raising rule realized by a combinator in CCG to exchange the argument--functor relation; such a rule is generalized as continuation-passing style (CPS) transformation. However, there is concern that CPS may exceedingly accept ungrammatical sentences. In this paper, we investigate the expanded grammar rules of CCG in terms of Lambek Calculus (LC), that is a formal system of CG. First, we show that Barker's CPS transformation is provable in LC but Plotkin's CPS transformation is not so. Second, we show a provable subset of Plotkin's CPS transformations . Due to the complexity of proving unprovability, we formalize the proof in Isabelle/HOL and verify it. We show that this subset is a grammatical class represented in LC, and call it type-restricted CPS transformation.