Cyclic Proofs with Ordering Constraints

Abstract

CLKID$^{\omega }$ is a sequent-based cyclic inference system able to reason on first-order logic with inductive definitions. The current approach for verifying the soundness of CLKID$^{\omega }$ proofs is based on expensive model-checking techniques leading to an explosion in the number of states.

We propose proof strategies that guarantee the soundness of a class of CLKID$^{\omega }$ proofs if some ordering and derivability constraints are satisfied. They are inspired from previous works about cyclic well-founded induction reasoning, known to provide effective sets of ordering constraints. A derivability constraint can be checked in linear time. Under certain conditions, one can build proofs that implicitly satisfy the ordering constraints.