Simple Type Theory with Undefinedness, Quotation, and Evaluation

TL;DR

This paper introduces a version of simple type theory, ${\cal Q}^{\rm uqe}_{0}$, that formalizes undefinedness, quotation, and evaluation, enabling reasoning about syntax and semantics of expressions, with applications demonstrated.

Contribution

It extends Church's type theory with undefinedness, quotation, and evaluation, providing a formal framework for syntax-semantics interplay and reasoning about mathematical algorithms.

Findings

Proof system is sound for all formulas.

Proof system is complete for formulas without evaluation.

Enables formal reasoning about syntax-based algorithms.

Abstract

This paper presents a version of simple type theory called ${\cal Q}^{\rm uqe}_{0}$ that is based on ${\cal Q}_0$, the elegant formulation of Church's type theory created and extensively studied by Peter B. Andrews. ${\cal Q}^{\rm uqe}_{0}$ directly formalizes the traditional approach to undefinedness in which undefined expressions are treated as legitimate, nondenoting expressions that can be components of meaningful statements. ${\cal Q}^{\rm uqe}_{0}$ is also equipped with a facility for reasoning about the syntax of expressions based on quotation and evaluation. Quotation is used to refer to a syntactic value that represents the syntactic structure of an expression, and evaluation is used to refer to the value of the expression that a syntactic value represents. With quotation and evaluation it is possible to reason in ${\cal Q}^{\rm uqe}_{0}$ about the interplay of the syntax and semantics of expressions and, as a result, to formalize in ${\cal Q}^{\rm uqe}_{0}$ syntax-based mathematical algorithms. The paper gives the syntax and semantics of ${\cal Q}^{\rm uqe}_{0}$ as well as a proof system for ${\cal Q}^{\rm uqe}_{0}$. The proof system is shown to be sound for all formulas and complete for formulas that do not contain evaluations. The paper also illustrates some applications of ${\cal Q}^{\rm uqe}_{0}$.