Dependently Sorted Nominal Signatures

2025-10-14

AI Summary

This paper addresses the lack of a rigorous formal foundation for raw expressions with binding that simultaneously respects α-equivalence and dependent types. It introduces the first type system integrating nominal sets with dependent sorts. The core method extends nominal multi-sorted signatures by introducing *generalised concretion*—a novel abstraction-elimination operator—and designing a sort classification mechanism that preserves α-conversion. The system natively supports abstraction and instantiation at the syntactic level, ensuring all derivations respect α-equivalence. A complete set of inference rules is provided, and key meta-theoretic properties—including type soundness, subject reduction, and α-respectful substitution—are formally established. The framework successfully encodes judgment and inference rules for diverse computational calculi, including the λ-calculus, π-calculus, and the logical framework LF. This work lays a foundational basis for building verifiable dependent type theories for binding-intensive structures.

Technology Category

• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Knowledge Representation and Reasoning: Description Logics
• Multiagent Systems: Other Foundations of Multi Agent Systems

Application Category

• Systems and Infrastructure for Web, Mobile and WoT: Experiences and lessons learnt from Web-based algorithms and system deployments
• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems

Abstract

We investigate an extension of nominal many-sorted signatures in which abstraction has a form of instantiation, called generalised concretion, as elimination operator (similarly to lambda-calculi). Expressions are then classified using a system of sorts and sort families that respects alpha-conversion (similarly to dependently-typed lambda-calculi) but not allowing names to carry abstraction sorts, thus constituting a first-order dependent sort system. The system can represent forms of judgement and rules of inference of several interesting calculi. We present rules and properties of the system as well as experiments of representation, and discuss how it constitutes a basis on which to build a type theory where raw expressions with alpha-equivalence are given a completely formal treatment.

Problem

• Extending nominal signatures with dependent sort systems
• Developing first-order dependent sort system respecting alpha-conversion
• Creating formal foundation for type theory with alpha-equivalence

Innovation

• Nominal signatures with generalised concretion operator
• First-order dependent sort system respecting alpha-conversion
• Formal treatment of raw expressions with alpha-equivalence