| Summary: | Inference systems are a widespread framework used to define possibly
recursive predicates by means of inference rules. They allow both inductive and
coinductive interpretations that are fairly well-studied. In this paper, we
consider a middle way interpretation, called regular, which combines advantages
of both approaches: it allows non-well-founded reasoning while being finite. We
show that the natural proof-theoretic definition of the regular interpretation,
based on regular trees, coincides with a rational fixed point. Then, we provide
an equivalent inductive characterization, which leads to an algorithm which
looks for a regular derivation of a judgment. Relying on these results, we
define proof techniques for regular reasoning: the regular coinduction
principle, to prove completeness, and an inductive technique to prove
soundness, based on the inductive characterization of the regular
interpretation. Finally, we show the regular approach can be smoothly extended
to inference systems with corules, a recently introduced, generalised
framework, which allows one to refine the coinductive interpretation, proving
that also this flexible regular interpretation admits an equivalent inductive
characterisation.
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