The Lambek Calculus with Iteration: Two Variants
Formulae of the Lambek calculus are constructed using three binary connectives, multiplication and two divisions. We extend it using a unary connective, positive Kleene iteration. For this new operation, following its natural interpretation, we present two lines of calculi. The first one is a fragment of infinitary action logic and includes an omega-rule for introducing iteration to the antecedent. We also consider a version with infinite (but finitely branching) derivations and prove equivalence of these two versions. In Kleene algebras, this line of calculi corresponds to the *-continuous case. For the second line, we restrict our infinite derivations to cyclic (regular) ones. We show that this system is equivalent to a variant of action logic that corresponds to general residuated Kleene algebras, not necessarily *-continuous. Finally, we show that, in contrast with the case without division operations (considered by Kozen), the first system is strictly stronger than the second one. To prove this, we use a complexity argument. Namely, we show, using methods of Buszkowski and Palka, that the first system is \(\varPi _1^0\)-hard, and therefore is not recursively enumerable and cannot be described by a calculus with finite derivations.
KeywordsLambek calculus Positive iteration Infinitary action logic Cyclic proofs
- 10.Kozen, D.: On action algebras. In: van Eijck, J., Visser, A. (eds.) Logic and Information Flow, pp. 78–88. MIT Press, Cambridge (1994)Google Scholar
- 11.Kozen, D., Silva, A.: Practical coinduction. Math. Struct. Comp. Sci. 1–21 (2016). FirstView. https://doi.org/10.1017/S0960129515000493
- 15.Pentus, M.: Lambek grammars are context-free. In: Proceedings of LICS 1993, pp. 429–433 (1993)Google Scholar
- 18.Pous, D., Sangiorgi, D.: Enchancements of coinductive proof methods. In: Advanced Topics in Bisimulation and Coinduction. Cambridge University Press (2011)Google Scholar
- 21.Ryzhkova, N.S.: Properties of the categorial dependencies calculus. Diploma Paper, Moscow State University (2013, unpublished). (in Russian)Google Scholar