Title:

Framework for binding operators

Binding appears in logic, programming and concurrency, e.g. it appears in Lambda Calculus, say the λabstraction. However, the binding in Lambda Calculus is unary. Certainly, we can generalize the idea of unary binding to an arbitrary finite numbers of binding. Algebraically, we can extend the framework of Universal Algebra and take arbitrary finite bindings as primitives. Therefore, operations in the new extended signature must be of second order instead of first order, and we name them as Binding Operators. The resulting framework is named as a Framework for Binding Operators, which coincides with Shapiro's diminished second order language. With a modification of Aczel's Frege Structure, we derive the algebras for Binding Operators, i.e. eBAs. The usual first order algebras, Plotkin's ωmodel of Lambda Calculus, and Girard's qualitative domains turn out to be special cases of eBAs. And also eBAs turn out to be (i) a generalization of Kechris and Moschovakis' suitable class of functionals in Recursion in Higher Types and (ii) a generalized Volken's λfamily. Following Birkhoff, we would like to equationally characterize Binding Operators. Kechris and Moschovakis' Enumeration Theorem suggests that an algebraic characterization of such might be possible. Unfortunately, eBAs and the usual satisfaction models_{eBA} of Binding Equations over these eBAs, in Birkhoff's approach, do not work. Therefore, we have to find either a remedy for it or a new semantic model for Binding Operators. We will present two solutions, one for each. (a) For a remedy, we discover a condition for Birkhoff's approach to work. This condition is necessary and sufficient, and we call it an admissible condition, which turns out weaker than Plotkin's Logical Relations in the sense that 'logical' implies 'admissible'. An admissible equational calculus vdash_{eBA} for Bindind Equations is obtained, whichis sound and complete wiyh respect to admissible satisfactionmodels_{eBA}. The relationship between Completeness and Admissible Completeness (or between satisfaction models_{eBA} and admissible satisfaction models_{eBA}) is discussed, although it is not completely clear. Other problems remain open as well, say the closedness of direct products and the admissible variety problem. (b) For a new semantic model, we will give a new binding algebra, i.e. iBA, which is intensional in contrast to the previous (extensional) one. Actually, an iBA is a generalization of Friedman's Prestructures. A sound and complete equational calculus vdash_{iBA} (in IBAs) is established. However, the derivability of vdash_{iBA} is weaker than the one of vdash_{eBA}. In other words, to share a same proof power with vdash_{eBA}, vdash_{iBA} has to use axiomatic schemas instead of pure axioms. Also, the relations between extensional models_{eBA} and intensional models_{iBA}, and between admissible vdash_{eBA} and intensional vdash_{iBA} are discussed.
