# The Lambek calculus with iteration: two variants

## TL;DR

This paper extends the Lambek calculus with a unary iteration operator, presenting two variants of calculi, analyzing their equivalence, strength, and complexity, and relating them to Kleene algebras and action logic.

## Contribution

It introduces two new variants of Lambek calculus with iteration, compares their expressive power, and establishes their computational complexity and relation to Kleene algebras.

## Key findings

- The first calculus variant is strictly stronger than the second.
- The first system is $ ext{Pi}_1^0$-hard and not recursively enumerable.
- The two variants are equivalent in certain infinite derivation frameworks.

## 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 $\Pi_1^0$-hard, and therefore is not recursively enumerable and cannot be described by a calculus with finite derivations.