Nominal Abstraction

TL;DR

This paper introduces nominal abstraction, a new relation for terms, integrated into a rich logic to improve reasoning about binding constructs in formal systems, enabling elegant proofs of properties like typing and substitution.

Contribution

It proposes nominal abstraction within a comprehensive logic framework, addressing limitations in existing logics for reasoning about dynamic binding aspects.

Findings

The logic incorporating nominal abstraction is consistent.

Examples demonstrate elegant reasoning about binding contexts.

The approach simplifies proofs of properties in typing calculi and substitution.

Abstract

Recursive relational specifications are commonly used to describe the computational structure of formal systems. Recent research in proof theory has identified two features that facilitate direct, logic-based reasoning about such descriptions: the interpretation of atomic judgments through recursive definitions and an encoding of binding constructs via generic judgments. However, logics encompassing these two features do not currently allow for the definition of relations that embody dynamic aspects related to binding, a capability needed in many reasoning tasks. We propose a new relation between terms called nominal abstraction as a means for overcoming this deficiency. We incorporate nominal abstraction into a rich logic also including definitions, generic quantification, induction, and co-induction that we then prove to be consistent. We present examples to show that this logic can provide elegant treatments of binding contexts that appear in many proofs, such as those establishing properties of typing calculi and of arbitrarily cascading substitutions that play a role in reducibility arguments.