Cyclic Datatypes modulo Bisimulation based on Second-Order Algebraic Theories

Abstract

Cyclic data structures, such as cyclic lists, in functional programming aretricky to handle because of their cyclicity. This paper presents aninvestigation of categorical, algebraic, and computational foundations ofcyclic datatypes. Our framework of cyclic datatypes is based on second-orderalgebraic theories of Fiore et al., which give a uniform setting for syntax,types, and computation rules for describing and reasoning about cyclicdatatypes. We extract the "fold" computation rules from the categoricalsemantics based on iteration categories of Bloom and Esik. Thereby, the rulesare correct by construction. We prove strong normalisation using the GeneralSchema criterion for second-order computation rules. Rather than the fixedpoint law, we particularly choose Bekic law for computation, which is a key toobtaining strong normalisation. We also prove the property of "Church-Rossermodulo bisimulation" for the computation rules. Combining these results, wehave a remarkable decidability result of the equational theory of cyclic dataand fold.Comment: 38 page