Incorporating quotation and evaluation into Church's type theory

Abstract

cttqe 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 cttqe 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 cttqe as well as a proof system for cttqe. 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 cttqe.

HOL Light QE

“…The proof system for ctt qe consists of the axioms for Q 0 , the single rule of inference for Q 0 , and additional axioms [27, B1-B13] that define the logical constants of ctt qe (B1-B4, B5, B7), specify as an inductive type (B4, B6), state the properties of quotation and evaluation (B8, B10), and extend the rules for beta-reduction (B9, B11-13). We prove in [27] that this proof system is sound for all formulas and complete for eval-free formulas.…”

Towards Specifying Symbolic Computation

“…ctt qe [13] is a version of Church's type theory with a built-in global reflection infrastructure with global quotation and evaluation operators geared towards reasoning about the interplay of syntax and semantics and, in particular, for specifying, defining, applying, and reasoning about SBMAs. The syntax and semantics of ctt qe is presented in [13]. A proof system for ctt qe that is sound for all formulas and complete for eval-free formulas is also presented in [13].…”

“…The syntax and semantics of ctt qe is presented in [13]. A proof system for ctt qe that is sound for all formulas and complete for eval-free formulas is also presented in [13]. (An expression is eval-free if it does not contain the evaluation operator.)…”

“…Its syntax and semantics are presented in [12]. A proof system for ctt uqe is not given there, but can be straightforwardly derived by merging those for ctt qe [13] and Q u 0 [11]. The global reflection infrastructure of ctt uqe (and ctt qe ) consists of three components.…”

“…This is in contrast to local reflection which constructs an inductive type of syntactic values only for the expressions of the logic that are relevant to a particular problem. See [13] for discussion about the difference between local and global reflection infrastructures and the design challenges that stand in the way of developing a global reflection infrastructure within a logic.…”