Skip to content
Snippets Groups Projects
Commit ff16d612 authored by Max New's avatar Max New
Browse files

sketchy stuff in the appendix on syntax and update to the extensional model

parent a5984baa
No related branches found
No related tags found
No related merge requests found
Hide whitespace changes
Inline Side-by-side
paper-new/appendix.tex
+ 92
− 0
View file @ ff16d612
\section{Gradual Typing Syntax}
\max{decide if we want to include $\times$ or not}
\begin{align*} % TODO is hole a term?
&\text{Types } A := \dyn \alt \nat \alt A \times A \alt (A \ra A') \\
&\text{Value Contexts } \Gamma := \cdot \alt (\Gamma, x : A) \\
&\text{Terms } M, N := \upc c M \alt \dnc c M
&\quad\quad \alt \ret {V} \alt \bind{x}{M}{N} \alt V_f\, V_x \alt \dn{A}{B} M
\end{align*}
The type precision derivations $c : A \ltdyn A'$ is inductively defined by
reflexivity, transitivity, congruence for $\ra$ and $\times$, and generators
$\textsf{Inj}_\ra : (D \ra D) \ltdyn D$ and $\textsf{Inj}_{\text{nat}}
: \nat \ltdyn D$.
%
We define equivalence of type precision derivations to be inductively generated by congruence for all constructors, category laws for reflexivity and transitivity as well as functoriality laws for $\ra$ and $\times$ congruence
\[ (c_i \ra c_o)(c_i' \ra c_o) = (c_ic_i') \ra (c_oc_o') \]
\[ (c_1 \times c_2)(c_1' \times c_2') = (c_1c_1') \times (c_2c_2') \]
\begin{theorem}
\begin{enumerate}
\item For every $A$, there is a derivation $c : A \ltdyn D$
\item Any two derivations $c,c' : A \ltdyn A'$ of the same precision
are equivalent.
\end{enumerate}
\end{theorem}
\begin{proof}
\begin{enumerate}
\item See prior work \cite{prior-work}
\item We show this by showing that derivations have a canonical
form.
The following presentation of precision derivations has unique derivations
\begin{mathpar}
\inferrule{}{\textrm{refl}(D) : D \ltdyn D}
\inferrule{}{\textsf{Inj}_{\text{nat}} : \nat \ltdyn D}
\inferrule{}{\textrm{refl}(\nat) : \nat \ltdyn \nat}
\inferrule{c : A_i \ra A_o \ltdyn D\ra D}{c(\textsf{Inj}_{\text{arr}}) : A_i \ra A_o \ltdyn \nat}
\inferrule{c : A_i \ltdyn A_i' \and d : A_o \ltdyn A_o'}{c \ra d : A_i \ra A_o \ltdyn A_i'\ra A_o'}
%% \inferrule{A_1 \times A_2 \ltdyn D\times D}{A_1 \times A_2 \ltdyn \nat}
%% \inferrule{A_1 \ltdyn A_1' \and A_2 \ltdyn A_2'}{A_1 \times A_2 \ltdyn A_1'\times A_2'}
\end{mathpar}
Since it satisfies reflexivity, cut-elimination and congruence, it
is a model of the original theory. Since it is a sub-theory of the
original theory, it is equivalent.
\end{enumerate}
\end{proof}
The gradually typed lambda calculus we consider is call-by-value
gradual lambda calculus with $\ra$ and $\nat$ as the only base types.
Casts are generated by the rule
\begin{mathpar}
\inferrule
{\Gamma \vdash M : A \and c : A \ltdyn A'}
{\Gamma \vdash \upc c M : A'}
\inferrule
{\Gamma \vdash M : A' \and c : A \ltdyn A'}
{\Gamma \vdash \dnc c M : A}
\end{mathpar}
Type precision is a binary relation on typed terms. The original
gradual guarantee rules are as follows:
\begin{mathpar}
\inferrule
{\Gamma^\ltdyn \vdash M \ltdyn M' : c \and
c : A \ltdyn A'\and
c' : A \ltdyn A_2'
}
{\Gamma^\ltdyn \vdash M \ltdyn (M :: A_2') : c'}
\inferrule
{\Gamma^\ltdyn \vdash M \ltdyn M' : c \and
c : A \ltdyn A'\and
c' : A_2 \ltdyn A'
}
{\Gamma^\ltdyn \vdash (M :: A_2) \ltdyn M' : c'}
\end{mathpar}
These two rules are admissible from the following principles:
%% \begin{mathpar}
%% \inferrule
%% {}
%% {\dnc {\injarr{}} \upc {\injarr{}} M \equidyn M}
%% %% \inferrule
%% %% {}
%% %% {}
%% \end{mathpar}
\section{Call-by-push-value}
In CBPV models, all the type constructors are interpreted as functors:
......
paper-new/categorical-model.tex
+ 52
− 73
View file @ ff16d612
......@@ -222,133 +222,112 @@ such that $\upf\, c$ ``represents $c$" (we will define this shortly). % such tha
In reality, it is slightly more complicated, since we need to build composition of edges
into the definition, but the idea is similar.
In particular, let $c : A \rel A'$ and $d : A' \rel A''$,
and let $f \in \mathcal V_f(A, A')$ and $g \in \mathcal V_f(A', A'')$.
We say that the pair $(c,d)$ is \emph{left-representable by} $(f, g)$
if the following two squares commute:
In particular, let $c : A \rel A'$ and $f \in \mathcal V_f(A, A')$. We
say that $c$ is \emph{left-representable by} $f$ if for any $c_l : A_l
\rel A$ and $c_r : A' \rel A_r$ the following squares commute\max{is ``commute'' good terminology?}:
\begin{center}
\begin{tabular}{ m{14em} m{14em} }
% UpL
\begin{tikzcd}[ampersand replacement=\&]
A \& {A'} \& {A''} \\
{A'} \&\& {A''}
A \& {A'} \& {A_r} \\
{A'} \&\& {A_r}
\arrow["f"', from=1-1, to=2-1]
%
\arrow[from=1-3, to=2-3, Rightarrow, no head]
%
\arrow["c", "\shortmid"{marking}, no head, from=1-1, to=1-2]
\arrow["d", "\shortmid"{marking}, no head, from=1-2, to=1-3]
\arrow["c_r", "\shortmid"{marking}, no head, from=1-2, to=1-3]
%
\arrow["d"', "\shortmid"{marking}, no head, from=2-1, to=2-3]
\arrow["c_r"', "\shortmid"{marking}, no head, from=2-1, to=2-3]
\end{tikzcd}
&
% UpR
\begin{tikzcd}[ampersand replacement=\&]
A \&\& {A'} \\
{A} \& {A'} \& {A''}
A_l \&\& {A} \\
{A_l} \& {A} \& {A'}
\arrow[from=1-1, to=2-1, Rightarrow, no head]
%
\arrow["g", from=1-3, to=2-3]
\arrow["f", from=1-3, to=2-3]
%
\arrow["c", "\shortmid"{marking}, no head, from=1-1, to=1-3]
\arrow["c_l", "\shortmid"{marking}, no head, from=1-1, to=1-3]
%
\arrow["c"', "\shortmid"{marking}, no head, from=2-1, to=2-2]
\arrow["d"', "\shortmid"{marking}, no head, from=2-2, to=2-3]
\arrow["c_l"', "\shortmid"{marking}, no head, from=2-1, to=2-2]
\arrow["c"', "\shortmid"{marking}, no head, from=2-2, to=2-3]
\end{tikzcd}
\end{tabular}
\end{center}
Dually, let $c \in \mathcal E_e(B, B')$ and $d \in \mathcal E_e(B', B'')$,
and let $f \in \mathcal E_f(B'', B')$ and $g \in \mathcal E_f(B', B)$.
We say the pair $(c,d)$ is \emph{right-representable by} $(f,g)$ if the
following two squares commute:
Dually, let $d \in B \rel B'$ and $\phi \in \mathcal E_f(B', B)$. We
say that $d$ is \emph{right-representable by} $\phi$ if for any $d_l :
B_l \rel B$ and $d_r : B' \rel B_r$ the following two squares commute:
\begin{center}
\begin{tabular}{ m{14em} m{14em} }
% DnR
\begin{tikzcd}[ampersand replacement=\&]
{B} \& {B'} \& {B''} \\
{B} \&\& {B'}
{B_l} \& {B} \& {B'} \\
{B_l} \&\& {B}
\arrow[from=1-1, to=2-1, Rightarrow, no head]
%
\arrow["f", from=1-3, to=2-3]
\arrow["\phi", from=1-3, to=2-3]
%
\arrow["c", "\shortmid"{marking}, no head, from=1-1, to=1-2]
\arrow["d_l", "\shortmid"{marking}, no head, from=1-1, to=1-2]
\arrow["d", "\shortmid"{marking}, no head, from=1-2, to=1-3]
%
\arrow["c"', from=2-1, to=2-3, no head]
\arrow["d_l"', from=2-1, to=2-3, no head]
\end{tikzcd}
&
% DnL
\begin{tikzcd}[ampersand replacement=\&]
{B'} \&\& {B''} \\
{B} \& {B'} \& {B''}
\arrow["g"', from=1-1, to=2-1]
{B'} \&\& {B_r} \\
{B} \& {B'} \& {B_r}
\arrow["\phi"', from=1-1, to=2-1]
%
\arrow[from=1-3, to=2-3, Rightarrow, no head]
%
\arrow["d", from=1-1, to=1-3, no head]
\arrow["d_r", from=1-1, to=1-3, no head]
%
\arrow["c"', "\shortmid"{marking}, no head, from=2-1, to=2-2]
\arrow["d"', "\shortmid"{marking}, no head, from=2-2, to=2-3]
\arrow["d"', "\shortmid"{marking}, no head, from=2-1, to=2-2]
\arrow["d_r"', "\shortmid"{marking}, no head, from=2-2, to=2-3]
\end{tikzcd}
\end{tabular}
\end{center}
Then we formulate the relationship between value relation morphisms and
Then we formulate the relationship between relation morphisms and
function morphisms as follows:
\begin{enumerate}
\item There is a functor $\upf : \mathcal V_e \to \mathcal V_f$ that is the
identity on objects, such that $\upf_*(\mathcal V_e)$, the essential image of
$\mathcal V_e$ under $\upf$, is thin.
The objects of $\upf_*(\mathcal V_e)$ are the objects of $\mathcal V_f$, and
the hom-set
$\upf_*(\mathcal V_e)(A, A') = \{ f \in \mathcal V_f(A,A') \mid \exists c \in \mathcal V_e(A,A'). \upf(c) = f \}$.
\item Every pair of morphisms $(c,d) \in \mathcal V_e(A, A') \times \mathcal V_e(A', A'')$ is
left-representable by $(\upf(c), \upf(d))$.
\end{enumerate}
And likewise for computations:
\begin{enumerate}
\item There is a functor $\dnf : \mathcal E_e^{op} \to \mathcal V_f$ that is
the identity on objects, such that the essential image of $\mathcal E_e$ under
$\dnf$ is thin.
\item Every pair of morphisms $(c,d) \in \mathcal E_e(B, B') \times \mathcal E_e(B',B'')$
is right-representable by $(\dnf(d), \dnf(c))$.
\item There is an identity-on-objects functor $\upf : \mathcal V_e \to
\mathcal V_f$ such that every $c$ is left-representable by $\upf(c)$.
\item There is an identity-on-objects functor $\dnf : \mathcal
E_e^{op} \to \mathcal \mathcal E_f$ such that every $d$ is
right-representable by $\dnf(d)$.
\end{enumerate}
\textbf{TODO: do we still need this?}
We also want something like
\[ F_c : \mathcal V_u^{op} \to \mathcal E_d \]
\[ U_c : \mathcal E_d^{op} \to \mathcal V_u \]
which ensures that if $R$ is a value edge equivalent to $A(u,-)$ then
\[ F(R) = F(A(u,-)) = (F A)(-,F u) \]
%% \textbf{TODO: do we still need this?}
%% We also want something like
%% \[ F_c : \mathcal V_u^{op} \to \mathcal E_d \]
%% \[ U_c : \mathcal E_d^{op} \to \mathcal V_u \]
%% which ensures that if $R$ is a value edge equivalent to $A(u,-)$ then
%% \[ F(R) = F(A(u,-)) = (F A)(-,F u) \]
In summary, an extensional model consists of:
\begin{enumerate}
\item CBPV models $\mathcal M_f$ and $\mathcal M_{sq}$
\item CBPV morphisms $r : \mathcal M_f \to \mathcal M_{sq}$ and $s, t : \mathcal M_{sq} \to \mathcal M_f$
\item Thinness: there is at most one square with a given boundary
\item A ``horizontal composition" operation on value relations and on computation relations
(from which we can define the categories $\ve$ and $\ee$ of value types/relations and computation types/relations, repsectively).
\item The categories $\ve$ and $\ee$ are thin up to an identity square
\item A functor $\upf : \mathcal V_e \to \mathcal V_f$ that is the identity on objects,
such that $\upf_*(\mathcal V_e)$, the essential image of $\mathcal V_e$ under $\upf$, is thin.
\item Every pair of morphisms $(c,d) \in \mathcal V_e(A, A') \times \mathcal V_e(A', A'')$ is
left-representable by $(\upf(c), \upf(d))$.
\item A functor $\dnf : \mathcal E_e^{op} \to \mathcal V_f$ that is
the identity on objects, such that the essential image of $\mathcal E_e$ under $\dnf$ is thin.
\item Every pair of morphisms $(c,d) \in \mathcal E_e(B, B') \times \mathcal E_e(B',B'')$
is right-representable by $(\dnf(d), \dnf(c))$.
\item A CBPV model internal to reflexive graphs.
\item Composition and identity on value and computation relations that form a category.
\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
\begin{itemize}
\item $U(dd') = U(d)U(d')$
\item $F(cc'') = F(c)F(c')$
\item $(cc') \to (dd') = (c \to d)(c' \to d')$
\item $(c_1c_1') \times (c_2c_2') = (c_1 \times c_2)(c_1'\times c_2')$
\end{itemize}
\item A distinguished value type $D$ with distinguished relation
$\injarr{}: U(D \to F D) \rel D$ satisfying $\dnc {\injarr{}}F(\upc{\injarr{}}) = \id$
\end{enumerate}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
......
0% or .
You are about to add 0 people to the discussion. Proceed with caution.
Finish editing this message first!
Please register or to comment