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\begin{document}

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\section*{computational type theory}

\hypertarget{context}{}\subsubsection*{{Context}}\label{context}

\hypertarget{constructivism_realizability_computability}{}\paragraph*{{Constructivism, Realizability, Computability}}\label{constructivism_realizability_computability}

[[!include constructivism - contents]]

\hypertarget{type_theory}{}\paragraph*{{Type theory}}\label{type_theory}

[[!include type theory - contents]]

\hypertarget{deduction_and_induction}{}\paragraph*{{Deduction and Induction}}\label{deduction_and_induction}

[[!include deduction and induction - contents]]

\hypertarget{foundations}{}\paragraph*{{Foundations}}\label{foundations}

[[!include foundations - contents]]

\hypertarget{contents}{}\section*{{Contents}}\label{contents}

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


\hypertarget{idea}{}\subsection*{{Idea}}\label{idea}

The term \emph{computational type theory} has been used

\begin{enumerate}%
\item generally for [[intuitionistic type theory]] in view of its [[computation|computational]] content via the [[propositions-as-types]] and [[proofs-as-programs]] interpretation (e.g. \hyperlink{Constable02}{Constable 02}, \hyperlink{Scholarpedia}{Scholarpedia}).


\item more specifically for [[intuitionistic type theory]] with [[inductive types]] and here specifically for the dialect of the language which is implemented in the [[NuPRL]] software (\hyperlink{ConstableEtAl86}{Constable et al. 86}, \hyperlink{NuPRL05}{NuPRL 05});

\begin{quote}%
\hyperlink{Constable}{Constable, p. 6}: $[$computational type theory$]$ considerably extended Per Martin-L\"o{}f's Intuitionistic Type Theory (ITT) adding set types, quotient types, recursive types, partial object types (bar types)


\end{quote}

\item for [[modal type theory]], specifically for [[type theory]] equipped with a [[monad (in computer science)]] that preserves [[finite products]], which exhibits a kind of computation (\href{BentonBiermanDePaiva93}{Benton-Bierman-de Paiva 93}, \hyperlink{FairtloughMendler02}{Fairtlough-Mendler 02}).

The [[internal logic]] of computational type theory in this sense is also called \emph{propositional lax logic} (\hyperlink{FairtloughMendler97}{Fairtlough-Mendler 97}, \hyperlink{Crolard}{Crolard}) or \emph{computational logic}.



\end{enumerate}
\hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts}

\begin{itemize}%
\item [[computational trinitarianism]]

\end{itemize}
\hypertarget{references}{}\subsection*{{References}}\label{references}

Discussion of intuitionistic/constructive type theory (with inductive types) as such referred to as computational type theory is in

\begin{itemize}%
\item Scholarpedia, \emph{\href{http://www.scholarpedia.org/article/Computational_type_theory}{Computational type theory}}

\end{itemize}
\begin{itemize}%
\item [[Robert Constable]], \emph{Na\"i{}ve Computational Type Theory}, Proof and System-Reliability NATO Science Series Volume 62, 2002, pp 213-259

\end{itemize}
Discussion specifically in the context of [[NuPRL]] is in

\begin{itemize}%
\item [[Robert Constable]], Stuart F. Allen, H. M. Bromley, W. R. Cleaveland, J. F. Cremer, R. W. Harper, Douglas J. Howe, T. B. Knoblock, N. P. Mendler, P. Panangaden, James T. Sasaki, and Scott F. Smith. \emph{Implementing Mathematics with the Nuprl Proof Development System}. Prentice-Hall, NJ, 1986.

\end{itemize}
\begin{itemize}%
\item \emph{Innovations in Computational Type Theory using Nuprl} (\href{http://www.cs.uni-potsdam.de/ti/kreitz/PDF/05jal-nuprl.pdf}{pdf})

\end{itemize}
\begin{itemize}%
\item [[Robert Constable]], \emph{The Triumph of Types: Creating a Logic of Computational Reality} ([[ConstableTriumphOfTypes.pdf:file]])

\end{itemize}
Discussion in the sense of [[modal type theory]] where computation is exhibited by a [[monad (in computer science)]] is in

\begin{itemize}%
\item Matt Fairtlough, [[Michael Mendler]], \emph{Propositional Lax Logic}, Volume 137, Issue 1, 25 August 1997, Pages 1--33 (\href{http://www.gdi.uni-bamberg.de/personnel/mendler/research/Papers/pll.pdf}{pdf})

\end{itemize}
\begin{itemize}%
\item P. Benton, G. Bierman, [[Valeria de Paiva]], \emph{Computational types from a logical perspective}, J. Functional Programming 1 (1) January 1993 (\href{http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.36.5778}{web}, \href{http://research.microsoft.com/en-us/um/people/gmb/papers/ctflp.pdf}{pdf})

\end{itemize}
\begin{itemize}%
\item Matt Fairtlough, [[Michael Mendler]], \emph{On the Logical Content of Computational Type Theory: A Solution to Curry's Problem}, Types for Proofs and Programs, Lecture Notes in Computer Science Volume 2277, 2002, pp 63-78 ([[MendlerComputationalTypeTheory.pdf:file]])

\end{itemize}
\begin{itemize}%
\item Tristan Crolard, \emph{Monadic reflection in lax logic} \href{http://cedric.cnam.fr/cpr/crolard/publications/reflection.pdf}{pdf}

\end{itemize}
See also

\begin{itemize}%
\item Fairouz Kamareddine, Twan Laan and [[Robert Constable]] (2012) \emph{Russell's Orders in Kripke's Theory of Truth and Computational Type Theory}. In Dov. M Gabbay, Akihiro Kanamori and John Woods, (editors) \emph{Sets and Extensions in the Twentieth Century}, 6, HHL, : San Diego: North Holland, 2012, pp. 801-845.

\end{itemize}
[[!redirects computational type theories]]

[[!redirects propositional lax logic]]

[[!redirects computational logic]]



\end{document}