mostly writing updates

paper-new/appendix.tex

@@ -77,18 +77,37 @@ gradual guarantee rules are as follows:
   }
   {\Gamma^\ltdyn \vdash (M :: A_2) \ltdyn M' : c'}
 \end{mathpar}
-These two rules are admissible from the following principles:
+Where the cast $M :: A_2$ is defined to be
+\[ \dnc {dyn(A_2)}{\upc {dyn(A)} M} \]
 
+These two rules are admissible from the following principles:
 %% \begin{mathpar}
-%%   \inferrule
-%%   {}
-%%   {\dnc {\injarr{}} \upc {\injarr{}} M \equidyn M}
-
-%%   %% \inferrule
-%%   %% {}
-%%   %% {}
 %% \end{mathpar}
 
+For the first rule, we first prove that $\dnc {dyn(A_2)}\upc {dyn(A)} M = \dnc {c'}\upc{c} M$
+\begin{align*}
+  \dnc {dyn(A_2)}\upc {dyn(A)} M
+  &= \dnc {c \,\textrm{dyn}(A')}\upc{c' \,\textrm{dyn}(A')} M \tag{All derivations are equal}\\
+  &= \dnc {c}\dnc {\textrm{dyn}(A')}\upc{\textrm{dyn}(A')}\upc{c'} M\tag{functoriality}\\
+  &= \dnc {c}\upc{c'} M\tag{retraction}\\
+\end{align*}
+
+Then the rest follows by the up/dn rules above and the fact that precision derivations are all equal.
+\begin{mathpar}
+  \inferrule*
+  {c' = \textrm{dyn}(A_2)\max{todo}}
+  {\dnc {c'}\upc {c} M \ltdyn M' : c'}
+\end{mathpar}
+
+Thus the following properties are sufficient to provide an extensional
+model of gradual typing without requiring transitivity of term
+precision:
+\begin{enumerate}
+\item Every precision is representable in the above sense,
+\item The association of casts to precision is functorial
+\item Type constructors are covariant functorial with respect to
+  relational identity and composition
+\end{enumerate}
 
 \section{Call-by-push-value}
 

paper-new/categorical-model.tex

@@ -312,11 +312,11 @@ function morphisms as follows:
 
 In summary, an extensional model consists of:
 \begin{enumerate}
-  \item A CBPV model internal to reflexive graphs.
-  \item Composition and identity on value and computation relations that form a category.
+  \item A reflexive graph internal to CBPV models.
+  \item Composition of value and computation relations that form a category with the reflexive relations as identity.
   \item An identity-on-objects functor $\upf : \mathcal V_e \to \mathcal V_f$ taking each value relation to a morphism that left-represents it.
   \item An identity-on-objects functor $\dnf : \mathcal E_e^{op} \to \mathcal E_f$ taking each computation relation to a morphism that right-represents it.
-  \item The CBPV connectives $U,F,\times,\to$ are all \emph{covariant} functorial on relations
+  \item The CBPV connectives $U,F,\times,\to$ are all \emph{covariant} functorial on relations\footnote{the reflexive graph structure already requires that these functors preserve identity relations}
     \begin{itemize}
     \item $U(dd') = U(d)U(d')$
     \item $F(cc') = F(c)F(c')$
@@ -327,6 +327,9 @@ In summary, an extensional model consists of:
     $\injarr{}: U(D \to F D) \rel D$ satisfying $\dnc {\injarr{}}F(\upc{\injarr{}}) = \id$
 \end{enumerate}
 
+\subsection{Stuff about Intensional Models}
+
+
 
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

paper-new/defs.tex

@@ -36,8 +36,8 @@
 \newcommand{\up}[2]{\langle{#2}\uarrowl{#1}\rangle}
 \newcommand{\dn}[2]{\langle{#1}\darrowl{#2}\rangle}
 
-\newcommand{\upc}[2]{\text{up}\,{#1}\,{#2}}
-\newcommand{\dnc}[2]{\text{dn}\,{#1}\,{#2}}
+\newcommand{\upc}[1]{\text{up}\,{#1}\,}
+\newcommand{\dnc}[1]{\text{dn}\,{#1}\,}
 
 
 % \newcommand{\ret}{\mathsf{ret}}
@@ -45,6 +45,11 @@
 \newcommand{\zro}{\textsf{zro}}
 \newcommand{\suc}{\textsf{suc}}
 \newcommand{\lda}[2]{\lambda {#1} . {#2}}
+
+\newcommand{\Injarr}{\textsf{Inj}_\ra}
+\newcommand{\Injnat}{\textsf{Inj}_\nat}
+\newcommand{\Injtimes}{\textsf{Inj}_\times}
+
 \newcommand{\injarr}[1]{\textsf{Inj}_\ra ({#1})}
 \newcommand{\injnat}[1]{\textsf{Inj}_\text{nat} ({#1})}
 \newcommand{\casenat}[4]{\text{Case}_\text{nat} ({#1}) \{ \text{no} \to {#2} \alt \text{nat}({#3}) \to {#4} \}}
@@ -249,4 +254,3 @@
 \newcommand{\tnat}{\text{nat}}
 \newcommand{\ttimes}{\text{times}}
 \newcommand{\tfun}{\text{fun}}
-

paper-new/discussion.tex

@@ -3,6 +3,8 @@
 \subsection{Related Work}
 % Discuss Joey Eremondi's thesis on gradual dependent types
 
+% Discuss Jeremy Siek's work on graduality in Agda
+
 \subsection{Mechanization}
 % Discuss Guarded Cubical Agda and mechanization efforts
 
@@ -10,17 +12,25 @@
 
 \subsection{Synthetic Ordering}
 
-While the use of synthetic guarded domain theory allows us to very
-conveniently work with non-well-founded recursive constructions while
-abstracting away the precise details of step-indexing, we do work with
-the error ordering in a mostly analytic fashion in that gradual types
-are interpreted as sets equipped with an ordering relation, and all
-terms must be proven to be monotone.
+A key to managing the complexity of our concrete construction is in
+using a \emph{synthetic} approach to step-indexing rather than working
+analytically with presheaves. This has helped immensely in our ongoing
+mechanization in cubical Agda as it sidesteps the need to formalize
+these constructions internally. 
+%
+However, there are other aspects of the model, the bisimilarity and
+the monotonicity, which are treated analytically and are .
 %
-It is possible that a combination of synthetic guarded domain theory
-with \emph{directed} type theory would allow for an a synthetic
-treatment of the error ordering as well.
+It may be possible to utilize further synthetic techniques to reduce
+this burden as well, and have all type intrinsically carry a notion of
+bisimilarity and ordering relation, and all constructions to
+automatically preserve them.
+%
+A synthetic approach to ordering is common in (non-guarded) synthetic
+domain theory and has also been used for synthetic reasoning for cost
+models \cite{synthetic-domain-theory,decalf}.
 
 \subsection{Future Work}
 
-% Cite GrEff paper
+
\ No newline at end of file
+% Cite GrEff paper

paper-new/intro.tex

@@ -78,30 +78,37 @@ Our goal in this work is to provide a reusable semantic framework for
 gradually typed languages that can be used to prove the aforementioned
 properties: graduality and type-based reasoning. \max{TODO: say more here}
 
+\subsection{Limitations of Prior Work}
 
-\subsection{Current Approaches to Proving Graduality}
+We give an overview of current approaches to proving graduality of
+languages and why they do not meet our criteria of a reusable semantic
+framework.
 
 \subsubsection{From Static to Gradual}
 
-Current approaches to constructing languages that satisfy the graduality property
-include the methods of Abstracting Gradual Typing
-\cite{garcia-clark-tanter2016} and the formal tools of the Gradualizer \cite{cimini-siek2016}.
-These allow the language developer to start with a statically typed language and derive a
-gradually typed language that satisfies the gradual guarantee. The main downside to
-these approaches lies in their inflexibility: since the process in entirely mechanical,
-the language designer must adhere to the predefined framework.
-Many gradually typed languages do not fit into either framework, e.g., Typed Racket
-\cite{tobin-hochstadt06, tobin-hochstadt08}.
+Current approaches to constructing languages that satisfy the
+graduality property include the methods of Abstracting Gradual Typing
+\cite{garcia-clark-tanter2016} and the formal tools of the Gradualizer
+\cite{cimini-siek2016}.  These allow the language developer to start
+with a statically typed language and derive a gradually typed language
+that satisfies the gradual guarantee. The main downside to these
+approaches lies in their inflexibility: since the process in entirely
+mechanical, the language designer must adhere to the predefined
+framework.  Many gradually typed languages do not fit into either
+framework, e.g., Typed Racket \cite{tobin-hochstadt06,
+  tobin-hochstadt08} and the semantics produced is not always the
+desired one.
 %
-Furthermore, while these frameworks do prove graduality of the resulting languages,
-they do not show the correctness of the equational theory, which is equally important
-to sound gradual typing.
-For example, programmers often refactor their code, and in so doing they rely
-implicitly on the validity of the laws in the equational theory.
-Similarly, correctness of compiler optimizations rests on the validity of the
-corresponding equations from the equational theory. It is therefore important
-that the languages that claim to be gradually typed have provably correct
-equational theories.
+Furthermore, while these frameworks do prove graduality of the
+resulting languages, they do not show the correctness of the
+equational theory, which is equally important to sound gradual typing.
+
+%% For example, programmers often refactor their code, and in so doing they rely
+%% implicitly on the validity of the laws in the equational theory.
+%% Similarly, correctness of compiler optimizations rests on the validity of the
+%% corresponding equations from the equational theory. It is therefore important
+%% that the languages that claim to be gradually typed have provably correct
+%% equational theories.
 
 % The approaches are too inflexible... the fact that the resulting semantics are too lazy
 % is a consequence of that inflexibility.
@@ -116,25 +123,51 @@ equational theories.
 
 % [Eric] I moved the next two paragraphs from the technical background section
 % to here in the intro.
-\subsubsection{Classical Domain Models}
-
-New and Licata \cite{new-licata18} developed an axiomatic account of the
-graduality relation on cast calculus terms and gave a denotational
-model of this calculus using classical domain theory based on
-$\omega$-CPOs. This semantics has scaled up to an analysis of a
-dependently typed gradual calculus in \cite{asdf}. This meets our
-criterion of being a reusable mathematical theory, as general semantic
-theorems about gradual domains can be developed independent of any
-particular syntax and then reused in many different denotational
-models. However, it is widely believed that such classical domain
-theoretic techniques cannot be extended to model higher-order store, a
-standard feature of realistic gradually typed languages such as Typed
-Racket. Thus if we want a reusable mathematical theory of gradual
-typing that can scale to realistic programming languages, we need to
-look elsewhere to so-called ``step-indexed'' techniques.
-
-\subsubsection{Operational Reduction Proofs}
-
+\subsubsection{Double Categorical Semantics}
+
+New and Licata \cite{new-licata18} developed an axiomatic account of
+the graduality relation on a call-by-name cast calculus terms and
+showed that the graduality proof could be modeled using semantics in
+certain kinds of \emph{double categories}, categories internal to the
+category of categories. A double category extends a category with a
+second notion of morphism, often a notion of ``relation'' to be paired
+with the notion of functional morphism, as well as a notion of
+functional morphisms preserving relations. In gradual typing the
+notion of relation models type precision and the squares model the
+term precision relation. This approach was influenced by Ma and
+Reynolds semantics of parametricity using reflexive graph categories:
+reflexive graph categories are essentially double categories without a
+notion of relational composition. In addition to capturing the notions
+of type and term precision, the double categorical approach allows for
+a \emph{universal property} for casts: upcasts are the
+\emph{universal} way to turn a relation arrow into a function in a
+forward direction and downcasts are the dual universal arrow.  Later,
+New, Licata and Ahmed \cite{new-licata-ahmed19} extended this
+axiomatic treatment from call-by-name to call-by-value as well by
+giving an axiomatic theory of type and term precision in
+call-by-push-value. This left implicit any connection to a ``double
+call-by-push-value'', which we make explicit in
+Section~\ref{sec:cbpv}.
+
+With this notion of abstract categorical model in hand, denotational
+semantics is then the work of constructing concrete models that
+exhibit the categorical construction. New and Licata
+\cite{new-licata18} present such a model using categories of
+$\omega$-CPOs, and this model was extended by Lennon-Bertrand,
+Maillard, Tabareau and Tanter to prove graduality of a gradual
+dependently typed calculus $\textrm{CastCIC}^{\mathcalG}$. This
+domain-theoretic approach meets our criteria of being a semantic
+framework for proving graduality, but suffers from the limitations of
+classical domain theory: the inability to model viciously
+self-referential structures such as higher-order extensible state and
+similar features such as runtime-extensible dynamic types. Since these
+features are quite common in dynamically typed languages, we seek a
+new denotational framework that can model these type system features.
+
+The standard alternative to domain theory that scales to essentially
+arbitrary self-referential definitions is \emph{step-indexing} or its
+synthetic form of \emph{guarded recursion}.
+\max{todo: adapt the next paragraph to fit in with this one}
 A series of works \cite{new-ahmed2018, new-licata-ahmed2019, new-jamner-ahmed19}
 developed step-indexed logical relations models of gradually typed
 languages based on operational semantics. Unlike classical domain
@@ -149,6 +182,26 @@ these reusable mathematical principles from these tedious models to
 make formalization of realistic gradual languages tractible.
 
 
+
+%% \subsubsection{Classical Domain Models}
+
+%%  and gave a denotational
+%% model of this calculus using classical domain theory based on
+%% $\omega$-CPOs. This semantics has scaled up to an analysis of a
+%% dependently typed gradual calculus in \cite{asdf}. This meets our
+%% criterion of being a reusable mathematical theory, as general semantic
+%% theorems about gradual domains can be developed independent of any
+%% particular syntax and then reused in many different denotational
+%% models. However, it is widely believed that such classical domain
+%% theoretic techniques cannot be extended to model higher-order store, a
+%% standard feature of realistic gradually typed languages such as Typed
+%% Racket. Thus if we want a reusable mathematical theory of gradual
+%% typing that can scale to realistic programming languages, we need to
+%% look elsewhere to so-called ``step-indexed'' techniques.
+
+%% \subsubsection{Operational Reduction Proofs}
+
+
 % Alternative phrasing:
 \begin{comment}
 \subsubsection{Embedding-Projection Pairs}
@@ -204,7 +257,16 @@ complex features.
 
 
 \subsection{Contributions}
+
+The main contribution of this work is a categorical and denotational
+semantics for gradually typed langauges that models not just the term
+language but the graduality property as well.
+\begin{enumerate}
+\item First, we give a rational reconstruction \max{todo: more shit.}
+\end{enumerate}
+
 Our main contribution is a compositional denotational semantics for step-aware gradual typing,
+
 analogous to Gradual Type Theory but now in the ``intensional" setting.
 In parallel, we are developing a reusable framework in
 Guarded Cubical Agda for developing machine-checked proofs of graduality of a cast calculus.

paper-new/paper.tex

@@ -112,7 +112,7 @@
 %%     domain used to model the dynamic type, and type precision is modeled
 %%     as simply a subset relation.
 %%     %
-  \end{abstract}
+\end{abstract}
 
 \maketitle
 
@@ -142,15 +142,17 @@
 
 \input{technical-background}
 
-% \input{syntax}
+\input{syntax}
+
+\input{extensional-model}
 
 % \input{semantics}
 
 \input{terms}
 
-\input{ordering}
+%% \input{ordering}
 
-\input{graduality}
+%% \input{graduality}
 
 \input{categorical-model}