# Cyclic hypersequents for Transitive Closure Logic

## Marianna Girlando, University of Birmingham

Transitive Closure Logic (TCL) enriches the language of first order logic with a recursive operator, expressing the transitive closure of a binary relation. Cyclic sequent calculi proofs for this logic have been introduced in the literature. In this talk, I will present a new cyclic proof system for TCL, which employs hypersequents. The main motivation behind this work is that, differently from sequent TCL proofs, hypersequent proofs for TCL successfully simulate cyclic proofs in the transitive fragment of Propositional Dynamic Logic (PDL+), without using the non-analytic rule of cut.

After introducing the cyclic hypersequent calculus for TCL, I will present two main results. First, I will sketch a proof of soundness for the calculus, which requires a more involved argument than the one used in traditional cyclic proofs. Second, I will show how PDL+ cyclic proofs can be translated into TCL hypersequent proofs. This result allows to lift the completeness result for cyclic PDL+ to the hypersequent calculus. This talk is based on joint work with Anupam Das.