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0000000..3bf1848 --- /dev/null +++ b/notes/technical-outline/technical-outline.tex @@ -0,0 +1,332 @@ +\documentclass{article} +\usepackage{amssymb} +\usepackage{amsmath} +\usepackage{amsthm} +\usepackage{tikz-cd} +\usepackage{mathpartir} +\usepackage{stmaryrd} +\usepackage{parskip} +\usepackage{tikz-cd} + + +\newtheorem{theorem}{Theorem}[section] +\newtheorem{nonnum-theorem}{Theorem}[section] +\newtheorem{corollary}{Corollary}[section] +\newtheorem{definition}{Definition}[section] +\newtheorem{lemma}{Lemma}[section] + +\input{../../paper-new/defs} + +\begin{document} + +\title{Technical Outline} +\author{Eric Giovannini} + +\maketitle + +\section{The Overall Plan} + +The goal is to show graduality of the extensional gradual lambda calculus. +The main theorem is the following: + +\begin{theorem}[Graduality] +If $M \ltdyn M'$, then + +\begin{itemize} + \item $M \Downarrow$ iff $M' \Downarrow$. + \item $M \Downarrow v_?$ iff $M' \Downarrow v_?'$, where $v_? = \err$ or $v_? = v_?'$. +\end{itemize} +\end{theorem} + + +\section{Syntax} + + % TODO details on admissibility: we can translate equations of the original logic + % into the new one and show that they hold there + + We begin with a gradually-typed lambda calculus $(\extlc)$, which is similar to + the normal call-by-value gradually-typed lambda calculus, but differs in that it + is actually a fragment of call-by-push-value specialized such that there are no + non-trivial computation types. We do this for convenience, as either way + we would need a distinction between values and effectful terms; the framework of + of call-by-push-value gives us a convenient langugae to define what we need. + + We then observe that composition of type precision derivations is admissible, + as is heterogeneous transitivity for term + precision (via casts), so it will suffice to consider a new language ($\extlcm$) + in which we don't have composition of type precision derivations or transitivity + of term precision. + + We further observe that all casts, except those between $\nat$ and $\dyn$ + and between $\dyn \ra \dyn$ and $\dyn$, are admissible + (we can define the cast of a function type + functorially using the casts for its domain and codomain). + + This means it is sufficient to consider a new language ($\extlcmm$) in which + instead of having arbitrary casts, we have injections from $\nat$ and + $\dyn \ra \dyn$ into $\dyn$, and case inspections from $\dyn$ to $\nat$ and + $\dyn$ to $\dyn \ra \dyn$. + + From here, we define an intensional GSTLC, so-named because it makes the + intensional stepping behavior of programs explicit in the syntax. + This is acocmplished by adding a syntactic ``later'' type and a + syntactic $\theta$ that maps terms of type later $A$ to terms of type $A$. + + \subsection{Extensional ($\extlcmm$)} + + \begin{align*} + &\text{Value Types } A := \nat \alt \dyn \alt (A \ra B) \\ + &\text{Computation Types } B := \Ret A \\ + &\text{Value Contexts } \Gamma := \cdot \alt (\Gamma, x : A) \\ + &\text{Computation Contexts } \Delta := \cdot \alt \hole B \alt \Delta , x : A \\ + &\text{Values } V := \zro \alt \suc\, V \alt \injnat V \alt \injarr V \\ + &\text{Terms } M, N := \err_B \alt \Ret {V} \alt \bind{x}{M}{N} % \alt \suc\, M, + \alt \lda{x}{M} \alt V_f\, V_x \alt + %\\ & \quad\quad \injnat M \alt \injarr M + \\ & \quad\quad \casenat{V}{M_{no}}{n}{M_{yes}} + \alt \casearr{V}{M_{no}}{f}{M_{yes}} + \end{align*} + + % TODO is err a value? What about injnat and injarr? + % TODO should suc be a term and a value? What about injnat and injarr? + + The value typing judgment is written $\hasty{\Gamma}{V}{A}$ and + the computation typing judgment is written $\hasty{\Delta}{M}{B}$. + + We define substitution for computation contexts as follows: + \begin{mathpar} + \inferrule* + { \delta : \Delta' \to \Delta \and + \hasty{\Delta'|_V}{V}{A}} + { (\delta , V/x ) \colon \Delta' \to \Delta , x : A } + + \inferrule* + {} + {\cdot \colon \cdot \to \cdot} + + \inferrule* + {\hasty{\Delta'}{M}{B}} + {M \colon \Delta' \to \hole{B}} + \end{mathpar} + + The typing rules are as follows: + \begin{mathpar} + % Err + \inferrule*{ }{\hasty {\cdot, \Gamma} {\err_B} B} + + % Zero and suc + \inferrule*{ }{\hasty \Gamma \zro \nat} + + \inferrule*{\hasty \Gamma M \nat} {\hasty \Gamma {\suc\, M} \nat} + + % Lambda + \inferrule* + {\hasty {\cdot, \Gamma, x : A} M {\Ret A'}} + {\hasty \Gamma {\lda x M} {A \ra A'}} + + % App + \inferrule* + {\hasty \Gamma {V_f} {A \ra A'} \and \hasty \Gamma {V_x} A} + {\hasty {\cdot , \Gamma} {M \, N} {\Ret A'}} + + % Ret + \inferrule* + {\hasty \Gamma V A} + {\hasty {\cdot , \Gamma} {\ret V} {\Ret A}} + + % Bind + \inferrule* + {\hasty \Delta M {\Ret A} \and \hasty{\cdot , \Delta|_V , x : A}{N}{B} } % Need x : A in context + {\hasty{\Delta}{\bind{x}{M}{N}}{B}} + + % inj-nat + \inferrule* + {\hasty \Gamma M \nat} + {\hasty \Gamma {\injnat M} \dyn} + + % inj-arr + \inferrule* + {\hasty \Gamma M (\dyn \ra \dyn)} + {\hasty \Gamma {\injarr M} \dyn} + + % Case nat + \inferrule* + {\hasty{\Gamma}{V}{\dyn} \and + \hasty{\Delta , x : \nat }{M_{yes}}{B} \and + \hasty{\Delta}{M_{no}}{B}} + {\hasty {\Delta} {\casenat{M}{M_{no}}{n}{M_{yes}}} {B}} + + % Case arr + \inferrule* + {\hasty{\Gamma}{V}{\dyn} \and + \hasty{\Delta , x : (\dyn \ra \dyn) }{M_{yes}}{B} \and + \hasty{\Delta}{M_{no}}{B}} + {\hasty {\Delta} {\casearr{M}{M_{no}}{f}{M_{yes}}} {B}} + \end{mathpar} + + + The equational theory is as follows: + \begin{mathpar} + % Function Beta and Eta + \inferrule* + {\hasty {\cdot, \Gamma, x : A} M {\Ret A'} \and \hasty \Gamma V A} + {(\lda x M)\, V = M[V/x]} + + \inferrule* + {\hasty \Gamma V {A \ra A}} + {\Gamma \vdash V = \lda x {V\, x}} + + % Case-nat Beta + \inferrule* + {\hasty \Gamma V \nat} + {\casenat {\injnat {V}} {M_{no}} {n} {M_{yes}} = M_{yes}[V/n]} + + \inferrule* + {\hasty \Gamma V {\dyn \ra \dyn} } + {\casenat {\injarr {V}} {M_{no}} {n} {M_{yes}} = M_{no}} + + % Case-nat Eta + \inferrule* + {} + {\Gamma , x :\, \dyn \vdash M = \casenat{x}{M}{n}{M[(\injnat{n}) / x]} } + + + % Case-arr Beta + + + % Case-arr Eta + + + % Ret Beta and Eta + \inferrule* + {} + {\bind{x}{\ret\, V}{N} = N[V/x]} + + \inferrule* + {\hasty {\hole{\Ret A} , \Gamma} {M} {B}} + {\hole{\Ret A}, \Gamma \vdash M = \bind{x}{\bullet}{M[\ret\, x]}} + + \end{mathpar} + + Equivalent terms in the equational theory are considered equal. + + + We now discuss type and term precision. In our language, we do not have + ``term precision'' but rather arbitrary monotone relations on types, + which we denote by $A \rel B$. We have relations on value types, as well + as on computation types. + % + In addition, because we don't have casts in our language, we do not have + the usual cast rules specifying that casts are least upper bounds and greatest + lower bounds. Instead, we have rules for our injection and case terms. + + \begin{align*} + &\text{Value Relations } R := \nat \alt \dyn \alt (R \ra R) \alt \alpha_1 \leq_A \alpha_2 \\ + &\text{Computation Relations } S := \Lift R \\ + \end{align*} + + % We define + + % \begin{mathpar} + % \inferrule* + % {\alpha_1 : A_1 \mid \alpha_2 : A_2 \vdash R \text{ rel} \and + % \alpha_1' : A_1' \vdash V_1 : A_1 \and + % \alpha_2' : A_2' \vdash V_2 : A_2 } + % {\alpha_1' : A_1' \mid \alpha_2' : A_2' \vdash R (V_1/\alpha_1, V_2/\alpha_2)} + % \end{mathpar} + + % TODO term precision + + + \subsection{Intensional} + In the intensional syntax, we add a type constructor for later, as well as a + syntactic $\theta$ term and a syntactic $\nxt$ term. + We add rules for each of these, and also modify the rules for inj-arr and + case-arr, since now + the function is not $\Dyn \ra \Dyn$ but rather $\later (\Dyn \ra \Dyn)$. + + % TODO show changes + + We define an erasure function from intensional syntax to extensional syntax + by induction on the intensional types and terms. + The basic idea is that the syntactic type $\later A$ erases to $A$, + and $\nxt$ and $\theta$ erase to the identity. + + + + +\section{Semantics} +To give a semantics of the extensional, quotiented syntax ($\extlcmm$), we proceed +in a few steps: + +\begin{enumerate} + \item We first give a denotational semantics of the intensional syntax $\intlc$ + using synthetic guarded domain theory, i.e., working internal to the topos of trees. + The denotation of a value type $A$ will be a poset $\sem{A}$, and the denotation of a computation + type $\Ret A$ will be the lift $\li \sem{A}$, (i.e., the free error domain on $\sem{A}$). % Ret A is our only computation type + We refer to this semantics as the \emph{step-indexed semantics}. + + %The denotation of a closed term of type $B$ will be as an element of $\li \sem{B}$. + + \item We then show how to go from the above step-indexed semantics to a set-based + semantics which is still intensional in that the denotations of terms that differ only in + their number of ``steps'' will not be equal. The denotations here are a type of + coinductive ``Machine'' that at each step of computation can either return a result, + error, or continue running. We call this the \emph{Machine semantics}. + % + The passage from step-indexed semantics to the coinductive semantics + will make use of clock quantification; see below for details. + + \item Finally, we collapse the above intensional Machine-based semantics to an + extensional semantics, so-called because at this point the information about the + precise number of steps a Machine has taken has been lost. We care only whether the + Machine runs to a value or error, or whether it diverges. Of course, we cannot + in general decide whether a given Machine will halt, so the semantic objects here + are pairs of a proposition $P$ and a function taking a proof of $P$ to an element of + the poset. + +\end{enumerate} + +The benefit to working synthetically in step 1 above is that the construction of the logical +relation that proves canonicity for the intensional syntax can be carried out much like +any normal, non-step-indexed logical relation. + + +\section{Unary Canonicity} + +We first want to show soundness: + +\begin{lemma}[Soundness] + In the Machine semantics, none of the following are equal: $\mho, \zro, \suc M, \omega$, + where $\omega$ represents a diverging program. +\end{lemma} + +Next we use a logical relations argument to establish the following: + +\begin{lemma} + $M_i \Dwn^n v_?$ iff $M_i = \delta^n(\qte{v_?})$ +\end{lemma} + +We will need the following key property relating erasure to the semantics: + +\begin{lemma} + If $\erase{M_i} = M$ and $\erase{M_i'} = M'$, and $M = M'$, + then $| \sem{M_i} | = | \sem{M_i'} |$. +\end{lemma} + +The idea of the proof is that since $M_i$ and $M_i'$ have equal erasures +(in the extensional equational theory), we can recover from the proof that their +erasures are equal a series of equational rules (e.g., $\beta$ equality), and +apply the intensional analogues of those rules to $M_i$ to ensure that $M_i$ and $M_i'$ +differ syntactically only in their number of $\theta$'s. + + + +\section{Graduality} + + + + + + +\end{document} \ No newline at end of file diff --git a/paper-new/defs.tex b/paper-new/defs.tex index f1a99a9..7f88a11 100644 --- a/paper-new/defs.tex +++ b/paper-new/defs.tex @@ -1,10 +1,13 @@ \newcommand{\To}{\Rightarrow} \newcommand{\inl}{\mathsf{inl}} \newcommand{\inr}{\mathsf{inr}} +\newcommand{\alt}{\mathrel{\bf \,\mid\,}} -\newcommand{\extlc}{\text{Ext-}\lambda C} -\newcommand{\intlc}{\text{Int-}\lambda C} +\newcommand{\extlc}{\text{Ext-}\lambda} +\newcommand{\extlcm}{\text{Ext-}\lambda^{-\text{trans}}} +\newcommand{\extlcmm}{\text{Ext-}\lambda^{-\text{trans}-\text{cast}}} +\newcommand{\intlc}{\text{Int-}\lambda} \newcommand{\erase}[1]{\lfloor {#1} \rfloor} @@ -20,14 +23,28 @@ \newcommand{\dyn}{?} \newcommand{\nat}{\text{Nat}} \newcommand{\bool}{\text{Bool}} +\newcommand{\ra}{\rightharpoonup} +\newcommand{\Ret}{\mathsf{Ret}} +\newcommand{\hole}[1]{\bullet \colon {#1}} + +\newcommand{\up}[2]{\langle{#2}\uarrowl{#1}\rangle} +\newcommand{\dn}[2]{\langle{#1}\darrowl{#2}\rangle} + +\newcommand{\ret}{\mathsf{ret}} \newcommand{\err}{\mho} \newcommand{\zro}{\textsf{zro}} \newcommand{\suc}{\textsf{suc}} \newcommand{\lda}[2]{\lambda {#1} . {#2}} -\newcommand{\up}[2]{\langle{#2}\uarrowl{#1}\rangle} -\newcommand{\dn}[2]{\langle{#1}\darrowl{#2}\rangle} +\newcommand{\injarr}[1]{\textsf{Inj}_\to ({#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} \}} +\newcommand{\casearr}[4]{\text{Case}_\to ({#1}) \{ \text{no} \to {#2} \alt \text{fun}({#3}) \to {#4} \}} +\newcommand{\bind}[3]{\text{var } {#1} = {#2} \text{ in } {#3}} +\newcommand{\Lift}{\text{Lift}} + +\newcommand{\rel}{\circ\hspace{-4px}-\hspace{-4px}\bullet} \newcommand{\ltdyn}{\sqsubseteq} \newcommand{\gtdyn}{\sqsupseteq} \newcommand{\equidyn}{\mathrel{\gtdyn\ltdyn}} @@ -38,7 +55,8 @@ \newcommand{\etmequidyn}[4]{{#1} \vdash {#2} \equidyn_e {#3} \colon {#4}} \newcommand{\itmequidyn}[4]{{#1} \vdash {#2} \equidyn_i {#3} \colon {#4}} - +\newcommand{\Dwn}{\Downarrow} +\newcommand{\qte}[1]{\text{quote}({#1})} % SGDT and Intensional Stuff @@ -70,8 +88,8 @@ \newcommand{\ltls}{\lesssim} \newcommand{\bisim}{\approx} -\newcommand{\injarr}{\textsf{Inj}_\to} -\newcommand{\injnat}{\textsf{Inj}_\mathbb{N}} +%\newcommand{\injarr}{\textsf{Inj}_\to} +%\newcommand{\injnat}{\textsf{Inj}_\mathbb{N}} \newcommand{\id}{\mathsf{id}} \newcommand{\ep}{\leadsto} -- GitLab