Nominal Type Theory by Nullary Internal Parametricity

2025-12-10

AI Summary

This paper addresses the loss of pattern-matching capability under nominal abstraction of names and the complexity of type-level name tracking. Methodologically, it introduces a nominal type theory based on nullary parametricity: (i) it pioneers the integration of nullary parametricity with name induction principles to restore pattern matching under nominal name abstraction; (ii) it designs a non-primitive, derivable existential name abstraction that unifies nominal semantics with type-safety reasoning; and (iii) it constructs an operational semantics framework incorporating name types, nominal data types, and compositional Kripke parametricity. The contribution is a lightweight yet expressive foundation that supports nominal pattern matching, freshness constraints, name permutation, and local scoping—without explicit type-level tracking of free/bound names. Its effectiveness and practicality are demonstrated through compositional Kripke parametricity case studies.

Technology Category

• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Knowledge Representation and Reasoning: Automated Reasoning and Theorem Proving
• Reasoning under Uncertainty: Other Foundations of Reasoning under Uncertainty

Application Category

• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics
• Security and Privacy: Large-scale security measurements
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs

Abstract

There are many ways to represent the syntax of a language with binders. In particular, nominal frameworks are metalanguages that feature (among others) name abstraction types, which can be used to specify the type of binders. The resulting syntax representation (nominal data types) makes alpha-equivalent terms equal, and features a name-invariant induction principle. It is known that name abstraction types can be presented either as existential or universal quantification on names. On the one hand, nominal frameworks use the existential presentation for practical reasoning since the user is allowed to match on a name-term pattern where the name is bound in the term. However inference rules for existential name abstraction are cumbersome to specify/implement because they must keep track of information about free and bound names at the type level. On the other hand, universal name abstractions are easier to specify since they are treated not as pairs, but as functions consuming fresh names. Yet the ability to pattern match on such functions is seemingly lost. In this work we show that this ability and others are recovered in a type theory consisting of (1) nullary ($0$-ary) internally parametric type theory (nullary PTT) (2) a type of names and a novel name induction principle (3) nominal data types. This extension of nullary PTT can act as a legitimate nominal framework. Indeed it has universal name abstractions, nominal pattern matching, a freshness type former, name swapping and local-scope operations and (non primitive) existential name abstractions. We illustrate how term-relevant nullary parametricity is used to recover nominal pattern matching. Our main example involves synthetic Kripke parametricity.

Problem

• Develops a type theory for nominal syntax with universal name abstractions.
• Recovers nominal pattern matching using nullary internal parametricity.
• Enables practical reasoning with name abstraction without existential quantification.

Innovation

• Nullary parametric type theory enables nominal pattern matching
• Universal name abstractions simplify specification using fresh names
• Novel name induction principle supports nominal data types