Semantics of templates in a compositional framework for building logics

TL;DR

This paper introduces a semantic framework for templates in logic specification languages, viewing them as second order definitions, ensuring modularity and preserving complexity.

Contribution

It extends FO(.) with second order definitions and a compositional framework, providing a semantic basis for templates as macros.

Findings

Templates as macros are semantically correct under certain conditions.

Adding templates does not increase the descriptive complexity of the logic.

The framework enables modular logic construction with second order definitions.

Abstract

There is a growing need for abstractions in logic specification languages such as FO(.) and ASP. One technique to achieve these abstractions are templates (sometimes called macros). While the semantics of templates are virtually always described through a syntactical rewriting scheme, we present an alternative view on templates as second order definitions. To extend the existing definition construct of FO(.) to second order, we introduce a powerful compositional framework for defining logics by modular integration of logic constructs specified as pairs of one syntactical and one semantical inductive rule. We use the framework to build a logic of nested second order definitions suitable to express templates. We show that under suitable restrictions, the view of templates as macros is semantically correct and that adding them does not extend the descriptive complexity of the base logic, which is in line with results of existing approaches.