Incorporating Quotation and Evaluation into Church's Type Theory: Syntax and Semantics

TL;DR

This paper introduces a version of Church's type theory, ${\rm \small CTT}_{\rm qe}$, incorporating quotation and evaluation operators to enable reasoning about syntax and semantics, facilitating formalization of syntax-based algorithms.

Contribution

It formalizes the syntax and semantics of ${\rm \small CTT}_{\rm qe}$ with quotation and evaluation, illustrating their usefulness through examples, and sketches a potential proof system.

Findings

Demonstrates the integration of quotation and evaluation in ${\rm \small CTT}_{\rm qe}$

Provides examples illustrating reasoning about syntax and semantics

Lays groundwork for a formal proof system for ${\rm \small CTT}_{\rm qe}$

Abstract

${\rm \small CTT}_{\rm qe}$ 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 ${\rm \small CTT}_{\rm qe}$ 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 ${\rm \small CTT}_{\rm qe}$ and give several examples that illustrate the usefulness of having quotation and evaluation in ${\rm \small CTT}_{\rm qe}$. We do not give a proof system for ${\rm \small CTT}_{\rm qe}$, but we do sketch what a proof system could look like.