Incorporating Quotation and Evaluation Into Church's Type Theory

TL;DR

This paper introduces ${\rm CTT}_{\rm qe}$, a version of Church's type theory with quotation and evaluation, enabling reasoning about syntax and semantics interplay, and formalizing syntax-based algorithms.

Contribution

It develops the syntax, semantics, and proof system for ${\rm CTT}_{\rm qe}$, integrating quotation and evaluation into Church's type theory for enhanced reasoning capabilities.

Findings

Proof system is sound for all formulas.

Proof system is complete for formulas without evaluations.

Examples demonstrate practical usefulness of quotation and evaluation.

Abstract

${\rm CTT}_{\rm qe}$ is a version of Church's type theory that includes quotation and evaluation operators that are similar to quote and eval in the Lisp programming language. With quotation and evaluation it is possible to reason in ${\rm CTT}_{\rm qe}$ about the interplay of the syntax and semantics of expressions and, as a result, to formalize syntax-based mathematical algorithms. We present the syntax and semantics of ${\rm CTT}_{\rm qe}$ as well as a proof system for ${\rm CTT}_{\rm qe}$. The proof system is shown to be sound for all formulas and complete for formulas that do not contain evaluations. We give several examples that illustrate the usefulness of having quotation and evaluation in ${\rm CTT}_{\rm qe}$.