Inference systems are a widespread framework used to define possibly recursive predicates by means of inference rules. They allow both inductive and coinductive interpre-tations 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.

Foundations of regular coinduction

Dagnino F.
2021-01-01

Abstract

Inference systems are a widespread framework used to define possibly recursive predicates by means of inference rules. They allow both inductive and coinductive interpre-tations 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.
File in questo prodotto:
File Dimensione Formato  
lmcs-2006.02887v5.pdf

accesso aperto

Descrizione: Articolo su rivista
Tipologia: Documento in Pre-print
Dimensione 374.81 kB
Formato Adobe PDF
374.81 kB Adobe PDF Visualizza/Apri

I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.

Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11567/1062700
Citazioni
  • ???jsp.display-item.citation.pmc??? ND
  • Scopus 4
  • ???jsp.display-item.citation.isi??? 1
social impact