Combinatory Completeness in Structured Multicategories
Ivan Kuzmin, Chad Nester, Ülo Reimaa, Sam Speight
TL;DR
The paper develops a general framework for combinatory completeness parameterized by faithful cartesian clubs, unifying eight distinct notions of applicative systems within structured multicategories. By introducing $ rak{S}$-polynomials and a club-dependent sequent calculus, it shows that eight corresponding algebraic structures (BI, BCI, BKI, BWI, BCKI, BCWI, BKWI, and BCKWI) precisely capture when an applicative system is $ rak{S}$-combinatory complete. In the classical setting of $ extbf{Fun}$-multicategories (i.e., ordinary sets and functions), the framework recovers the standard characterization of combinatory algebras as $ extbf{Fun}$-combinatory complete or equivalently as $ extsf{BCKWI}$-algebras. The work thus provides a systematic, categorical method to analyze and derive combinatory completeness across a spectrum of structured multicategories, with potential extensions to partial, braided, monadic, and planar variants of combinatory algebras. This lays groundwork for future abstract characterizations and broader generalizations of computability in categorical contexts.
Abstract
We give a general notion of combinatory completeness with respect to a faithful cartesian club and use it systematically to obtain characterisations of a number of different kinds of applicative system. Each faithful cartesian club determines a notion of structured multicategory, with the different notions of structured multicategory obtained in this way giving different notions of polynomial over an applicative system, which in turn give different notions of combinatory completeness. We obtain the classical characterisation of combinatory algebras as combinatory complete applicative systems as a specific instance.