# The Lambek Calculus with Iteration: Two Variants

## Abstract

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.

### Keywords

Lambek calculus Positive iteration Infinitary action logic Cyclic proofs## Notes

### Acknowledgments

The author is grateful to Arnon Avron, Lev Beklemishev, Michael Kaminski, Max Kanovich, Glyn Morrill, Fedor Pakhomov, Mati Pentus, Nadezhda Ryzhkova, Andre Scedrov, Daniyar Shamkanov, and Stanislav Speranski for fruitful discussions. The author is also grateful to the anonymous referees for their comments.

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