Abstract
\({\textsc {ctt}}_\mathrm{qe}\) is a version of Church’s type theory with global quotation and evaluation operators that is engineered to reason about the interplay of syntax and semantics and to formalize syntax-based mathematical algorithms. \({\textsc {ctt}}_\mathrm{uqe}\) is a variant of \({\textsc {ctt}}_\mathrm{qe}\) that admits undefined expressions, partial functions, and multiple base types of individuals. It is better suited than \({\textsc {ctt}}_\mathrm{qe}\) as a logic for building networks of theories connected by theory morphisms. This paper presents the syntax and semantics of \({\textsc {ctt}}_\mathrm{uqe}\), defines a notion of a theory morphism from one \({\textsc {ctt}}_\mathrm{uqe}\) theory to another, and gives two simple examples involving monoids that illustrate the use of theory morphisms in \({\textsc {ctt}}_\mathrm{uqe}\).
This research was supported by NSERC.
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Notes
- 1.
Theory morphisms are also known as immersions, realizations, theory interpretations, translations, and views.
- 2.
A logic without support for partial functions and undefinedness—such as \({\textsc {ctt}}_\mathrm{qe}\) or the logic of HOL [10]—can interpret \(\alpha \) by a type \(\beta '\) that is isomorphic to a subset of \(\beta \). However, this approach is more complicated and farther from standard mathematics practice than interpreting \(\alpha \) directly by a subset of \(\beta \).
- 3.
It would be more natural for the second argument of an evaluation to be a type, but that would lead to an infinite family of evaluation operators, one for every type, since type variables are not available in \({\textsc {ctt}}_\mathrm{uqe}\) (as well as in \({\textsc {ctt}}_\mathrm{qe}\) and \(\mathcal{Q}_0\)).
- 4.
We write \(V^\mathcal{M}_{\varphi }(\mathbf D _\delta )\) instead of \(V^\mathcal{M}(\varphi ,\mathbf D _\delta )\).
- 5.
Technically, \(e'_\iota \) is a constant chosen from \(\mathcal {C}{\setminus } \mathcal {C}_{M}\). There is no harm is assuming that such a constant already exists in \(\mathcal {C}\).
- 6.
The definition of \(*'_{\iota \rightarrow \iota \rightarrow \iota }\) can be simplified by noting that \(e_\iota *_{\iota \rightarrow \iota \rightarrow \iota } e_\iota \) equals \(e_\iota \).
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Farmer, W.M. (2017). Theory Morphisms in Church’s Type Theory with Quotation and Evaluation. In: Geuvers, H., England, M., Hasan, O., Rabe, F., Teschke, O. (eds) Intelligent Computer Mathematics. CICM 2017. Lecture Notes in Computer Science(), vol 10383. Springer, Cham. https://doi.org/10.1007/978-3-319-62075-6_11
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