From 00424a37c3253acf704c0273027feb5a8addef34 Mon Sep 17 00:00:00 2001 From: Eric Giovannini <ecg19@seas.upenn.edu> Date: Tue, 25 Apr 2023 10:44:04 -0400 Subject: [PATCH] work on paper (changes to intro, work on description of syntax) --- paper-new/defs.tex | 8 +- paper-new/graduality.tex | 18 + paper-new/intro.tex | 208 +++++++ paper-new/paper.pdf | Bin 548218 -> 564919 bytes paper-new/paper.tex | 964 +---------------------------- paper-new/references.bib | 19 + paper-new/semantics.tex | 196 ++++++ paper-new/syntax.tex | 494 +++++++++++++++ paper-new/technical-background.tex | 286 +++++++++ 9 files changed, 1246 insertions(+), 947 deletions(-) create mode 100644 paper-new/graduality.tex create mode 100644 paper-new/intro.tex create mode 100644 paper-new/semantics.tex create mode 100644 paper-new/syntax.tex create mode 100644 paper-new/technical-background.tex diff --git a/paper-new/defs.tex b/paper-new/defs.tex index 7f88a11..f0b8114 100644 --- a/paper-new/defs.tex +++ b/paper-new/defs.tex @@ -24,7 +24,7 @@ \newcommand{\nat}{\text{Nat}} \newcommand{\bool}{\text{Bool}} \newcommand{\ra}{\rightharpoonup} -\newcommand{\Ret}{\mathsf{Ret}} +\newcommand{\Ret}[1]{\mathsf{Ret\,}{#1}} \newcommand{\hole}[1]{\bullet \colon {#1}} @@ -44,11 +44,15 @@ \newcommand{\Lift}{\text{Lift}} -\newcommand{\rel}{\circ\hspace{-4px}-\hspace{-4px}\bullet} +%\newcommand{\rel}{\circ\hspace{-4px}-\hspace{-4px}\bullet} +\newcommand{\rel}{\mathrel{\circ\mkern-6mu-\mkern-6mu\bullet}} \newcommand{\ltdyn}{\sqsubseteq} \newcommand{\gtdyn}{\sqsupseteq} \newcommand{\equidyn}{\mathrel{\gtdyn\ltdyn}} \newcommand{\gamlt}{\Gamma^\ltdyn} +\newcommand{\deltalt}{\Delta^\ltdyn} +\newcommand{\relcomp}{\odot} + \newcommand{\hasty}[3]{{#1} \vdash {#2} \colon {#3}} \newcommand{\etmprec}[4]{{#1} \vdash {#2} \ltdyn_e {#3} \colon {#4}} \newcommand{\itmprec}[4]{{#1} \vdash {#2} \ltdyn_i {#3} \colon {#4}} diff --git a/paper-new/graduality.tex b/paper-new/graduality.tex new file mode 100644 index 0000000..b5e4f66 --- /dev/null +++ b/paper-new/graduality.tex @@ -0,0 +1,18 @@ +\section{Unary Canonicity} +Before discussing graduality, we seek to prove its ``unary'' analogue. +Namely, instead of considering inequality between terms, we start by considering equality. + + +\section{Graduality}\label{sec:graduality} +The main theorem we would like to prove is the following: + +\begin{theorem}[Graduality] + If $\cdot \vdash M \ltdyn N : \nat$, then + \begin{enumerate} + \item If $N = \mho$, then $M = \mho$ + \item If $N = `n$, then $M = \mho$ or $M = `n$ + \item If $M = V$, then $N = V$ + \end{enumerate} +\end{theorem} + + diff --git a/paper-new/intro.tex b/paper-new/intro.tex new file mode 100644 index 0000000..c0ca56c --- /dev/null +++ b/paper-new/intro.tex @@ -0,0 +1,208 @@ +\section{Introduction} + +% gradual typing, graduality +\subsection{Gradual Typing and Graduality} +One of the principal categories on which type systems of programming languages are classified +is that of static versus dynamic type discipline. +In static typing, the code is type-checked at compile time, while in a dynamic typing, +the type checking is deferred to run-time. Both approaches have benefits: with static typing, +the programmer is assured that if the program passes the type-checker, their program +is free of type errors. Meanwhile, dynamic typing allows the programmer to rapidly prototype +their application code without needing to commit to fixed type signatures for their functions. + +\emph{Gradually-typed languages} \cite{siek-taha06} allow for both disciplines +to be used in the same codebase, and support interoperability between statically-typed +and dynamically-typed code. +This flexibility allows programmers to begin their projects in a dynamic style and +enjoy the benefits of dynamic typing related to rapid prototyping and easy modification +while the codebase ``solidifies''. Over time, as parts of the code become more mature +and the programmer is more certain of what the types should be, the code can be +\emph{gradually} migrated to a statically typed style without needing to start the project +over in a completely differnt language. + +Gradually-typed languages should satisfy two intuitive properties. +First, the interaction between the static and dynamic components of the codebase +should be safe -- i.e., should preserve the guarantees made by the static types. +In particular, while statically-typed code can error at runtime in a gradually-typed language, +such an error can always be traced back to a dynamically-typed term that +violated the typing contract imposed by statically typed code. +% In other words, in the statically-typed portions of the codebase, type soundness must be preserved. +Second, gradually-typed langugaes should support the smooth migration from dynamic typing +to static typing, in that the programmer can initially leave off the +typing annotations and provide them later without altering the meaning of the +program. + +% Graduality property +Formally speaking, gradually typed languages should satisfy the +\emph{dynamic gradual guarantee}, originally defined by Siek, Vitousek, Cimini, +and Boyland \cite{siek_et_al:LIPIcs:2015:5031}. +This property is also referred to as \emph{graduality}, by analogy with parametricity. +Intuitively, graduality says that going from a dynamic to static style should not +introduce changes in the meaning of the program. +More specifically, making the types more precise by adding typing annotations +% (replacing $\dyn$ with a more specific type +%such as integers) +will either result in the same behavior as the original, less precise program, +or will result in a type error. + + +\subsection{Current Approaches to Proving Graduality} +Current approaches to proving graduality include the methods of Abstracting Gradual Typing +\cite{garcia-clark-tanter2016} and the formal tools of the Gradualier \cite{cimini-siek2016}. +These allow the language developer to start with a statically typed langauge and derive a +gradually typed language that satisfies the gradual guarantee. The downside to these approaches +is that the semantics of the resulting languages are too lazy: the frameworks consider only +the $\beta$ rules and not the $\eta$ equalities. Furthermore, while these frameworks do prove +graduality, 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 without thinking about +whether the refactoring has broken the semantics of the program. +It is the validity of the laws in the equational theory that guarantees that such refactorings are sound. +Similarly, correctness of compiler optimizations rests on the validity of the corresponding equations +from the equational theory. It is therefore important that the langages that claim to be gradually typed +have provably correct equational theories. + +%The downside is that +%not all gradually typed languages can be derived from these frameworks, and moreover, in both +%approaches the semantics is derived from the static type system as opposed to the alternative +%in which the semantics determines the type checking. Without a clear semantic interpretation of type +%dynamism, it becomes difficult to extend these techniques to new language features such as polymorphism. + +% Proving graduality via LR +New and Ahmed \cite{new-ahmed2018} +have developed a semantic approach to specifying type dynamism in terms of +\emph{embedding-projection pairs}, which allows for a particularly elegant formulation of the +gradual guarantee. +Moreover, their axiomatic account of program equivalence allows for type-based reasoning +about program equivalences. +% +In this approach, a logical relation is constructed and shown to be sound with respect to +the notion of observational approximation that specifies when one program is more precise than another. +The downside of this approach is that each new language requires a different logical relation +to prove graduality. Furthermore, the logical relations tend to be quite complicated due +to a technical requirement known as \emph{step-indexing}. +As a result, developments using this approach tend to require vast effort, with the +corresponding technical reports having 50+ pages of proofs. + +An alternative approach, which we investigate in this paper, is provided by +\emph{synthetic guarded domain theory}. +The tecnhiques of synthetic guarded domain theory allow us to internalize the +step-index reasoning normally required in logical relations proofs of graduality, +ultimately allowing us to specify the logical relation in a manner that looks nearly +identical to a typical, non-step-indexed logical relation. + +In this paper, we report on work we have done to mechanize proofs of graduality and +correctness of equational theories using SGDT techniques in Agda. +Our goal in this work is to mechanize these proofs in a reusable way, +thereby providing a framework to use to more easily and +conveniently prove that existing languages satsify graduality and have +sound equational theories. Moreover, the aim is for designers of new languages +to utlize the framework to facilitate the design of new provably-correct +gradually-typed languages with nontrivial features. + + +\subsection{Proving Graduality in SGDT} +\textbf{TODO:} This section should probably be moved to after the relevant background has been introduced. + +% TODO introduce the idea of cast calculus and explicit casts? + +In this paper, we will utilize SGDT techniques to prove graduality for a particularly +simple gradually-typed cast calculus, the gradually-typed lambda calculus. +This is the usual simply-typed lambda calculus with a dynamic type $\dyn$ such that +$A \ltdyn\, \dyn$ for all types $A$, as well as upcasts and downcasts between any types +$A$ and $B$ such that $A \ltdyn B$. The complete definition will be provided in +Section \ref{sec:GTLC}. +The graduality theorem is shown below. + + +% \begin{theorem}[Graduality] +% If $M \ltdyn N : \nat$, then either: +% \begin{enumerate} +% \item $M = \mho$ +% \item $M = N = \zro$ +% \item $M = N = \suc n$ for some $n$ +% \item $M$ and $N$ diverge +% \end{enumerate} +% \end{theorem} + + +\begin{theorem}[Graduality] + If $\cdot \vdash M \ltdyn N : \nat$, then + \begin{enumerate} + \item $M \Downarrow$ iff $N \Downarrow$ + \item If $M \Downarrow v_?$ and $N \Downarrow v'_?$ then either $v_? = \mho$, or $v_? = v'_?$. + \end{enumerate} +\end{theorem} + +Details can be found in later sections, but we provide a brief explanation of the terminology and notation: + +\begin{itemize} + \item $M \ltdyn N : \nat$ means $M$ and $N$ are terms of type $\nat$ such that + $M$ is ``syntactically more precise'' than $N$, or equivalently, $N$ is + ``more dynamic'' than $M$. Intuitively this means that $M$ and $N$ are the + same except that in some places where $M$ has explicit typing annotations, + $N$ has $\dyn$ instead. + \item $\cdot \Downarrow$ is a relation on terms that is defined such that $M \Downarrow$ means + that $M$ terminates, either with a run-time error or a value $n$ of type $\nat$. + \item $\mho$ is a syntactic representation of a run-time type error, which + happens, for example, when a programmer tries to call a function with a value whose type + is found to be incompatible with the argument type of the function. + \item $v_?$ is shorthand for the syntactic representation of a term that is either equal to + $\mho$, or equal to the syntactic representation of a value $n$ of type $\nat$. +\end{itemize} + +% We also should be able to show that $\mho$, $\zro$, and $\suc\, N$ are not equal. + +Our first step toward proving graduality is to formulate an \emph{step-sensitive}, or \emph{intensional}, +gradual lambda calculus, which we call $\intlc$, in which the computation steps taken by a term are made explicit. +The ``normal'' gradual lambda calculus for which we want to prove graduality will be called the +\emph{step-insensitive}, or \emph{extensional}, gradual lambda calculus, denoted $\extlc$. +We will define an erasure function $\erase{\cdot} : \intlc \to \extlc$ which takes a program in the intensional lambda calculus +and ``forgets'' the syntactic information about the steps to produce a term in the extensional calculus. + +Every term $M_e$ in $\extlc$ will have a corresponding program $M_i$ in $\intlc$ such that +$\erase{M_i} = M_e$. Moreover, we will show that if $M_e \ltdyn_e M_e'$ in the extensional theory, +then there exists terms $M_i$ and $M_i'$ such that $\erase{M_i}=M_e$, $\erase{M_i'}=M_e'$ and +$M_i \ltdyn_i M_i'$ in the intensional theory. + +We formulate and prove an analogous graduality theorem for the intensional lambda calculus. +We define an interpretation of the intensional lambda calculus into a model in which we prove +various results. Using the observation above, given $M_e \ltdyn M_e' : \nat$, we can find +intensional programs $M_i$ and $M_i'$ that erase to them and are such that $M_i \ltdyn M_i'$. +We will then apply the intensional graduality theorem to $M_i$ and $M_i'$, and translate the result +back to $M_e$ and $M_e'$. + +\subsection{Contributions} +Our main contribution is a reusable framework in Guarded Cubical Agda for developing machine-checked proofs +of graduality of a cast calculus. +To demonstrate the feasability and utility of our approach, we have used the framework to prove +graduality for the simply-typed gradual lambda calculus. Along the way, we have developed an ``intensional" theory +of graduality that is of independent interest. + + +\subsection{Overview of Remainder of Paper} + +In Section \ref{sec:technical-background}, we provide technical background on gradually typed languages and +on synthetic guarded domain theory. +% +In Section \ref{sec:GTLC}, we introduce the gradually-typed cast calculus +for which we will prove graduality. Important here are the notions of syntactic +type precision and term precision. We introduce both the \emph{extensional} gradual lambda calculus +($\extlc$) and the \emph{intensional} gradual lambda calculus ($\intlc$). +% +In Section \ref{sec:domain-theory}, we define several fundamental constructions +internal to SGDT that will be needed when we give a denotational semantics to +our intensional lambda calculus. +This includes the notion of Predomains as well as the concept +of EP-Pairs. +% +In Section \ref{sec:semantics}, we define the denotational semantics for the +intensional gradually-typed lambda calculus using the domain theoretic constructions in the +previous section. +% +In Section \ref{sec:graduality}, we outline in more detail the proof of graduality for the +extensional gradual lambda calculus, which will make use of prove properties we prove about + the intensional gradual lambda calculus. +% +In Section \ref{sec:discussion}, we discuss the benefits and drawbacks to our approach in comparison +to the traditional step-indexing approach, as well as 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\begin{abstract} - We develop a denotational semantics for a gradually typed language - with effects that is adequate and proves the graduality theorem. + We develop a denotational semantics for a simple gradually typed language + that is adequate and proves the graduality theorem. % The denotational semantics is constructed using \emph{synthetic guarded domain theory} working in a type theory with a later @@ -68,957 +68,31 @@ \end{abstract} \maketitle - - \section{Introduction} - - % gradual typing, graduality - \subsection{Gradual Typing and Graduality} - Gradual typing allows a language to have both statically-typed and dynamically-typed terms; - the statically-typed terms are type checked at compile time, while type checking for the - dynamically-typed terms is deferred to runtime. - - Gradually-typed languages should satisfy two intuitive properties. - First, the interaction - between the static and dynamic components of the codebase should be safe -- i.e., should - preserve the guarantees made by the static types. - In other words, in the static portions of the codebase, type soundness must be preserved. - Second, gradual langugaes should support the smooth migration from dynamic typing - to static typing, in that the programmer can initially leave off the - typing annotations and provide them later without altering the meaning of the - program. - - % Graduality property - Formally speaking, gradually typed languages should satisfy the - \emph{dynamic gradual guarantee}, originally defined by Siek, Vitousek, Cimini, - and Boyland \cite{siek_et_al:LIPIcs:2015:5031}. - This property is also referred to as \emph{graduality}, by analogy with parametricity. - Intuitively, graduality says that in going from a dynamic to static style should not - introduce changes in the meaning of the program. - More specifically, making the types more precise by adding annotations - % (replacing $\dyn$ with a more specific type - %such as integers) - will either result in the same behavior as the less precise program, - or result in a type error. - - - \subsection{Current Approaches} - Current approaches to proving graduality include the methods of Abstracting Gradual Typing - \cite{garcia-clark-tanter2016} and the formal tools of the Gradualier \cite{cimini-siek2016}. - These allow the language developer to start with a statically typed langauge and derive a - gradually typed language that satisfies the gradual guarantee. The downside is that - not all gradually typed languages can be derived from these frameworks, and moreover, in both - approaches the semantics is derived from the static type system as opposed to the alternative - in which the semantics determines the type checking. Without a clear semantic interpretation of type - dynamism, it becomes difficult to extend these techniques to new language features such as polymorphism. - - % Proving graduality via LR - New and Ahmed \cite{new-ahmed2018} - have developed a semantic approach to specifying type dynamism in terms of - \emph{embedding-projection pairs}, which allows for a particularly elegant formulation of the - gradual guarantee. - Moreover, their axiomatic account of program equivalence allows for type-based reasoning - about program equivalences. -% - In this approach, a logical relation is constructed and shown to be sound with respect to - the notion of observational approximation that specifies when one program is more precise than another. - The downside of this approach is that each new language requires a different logical relation - to prove graduality. Furthermore, the logical relations tend to be quite complicated due - to a technical requirement known as \emph{step-indexing}. - As a result, developments using this approach tend to require vast effort, with the - corresponding technical reports having 50+ pages of proofs. - - An alternative approach, which we investigate in this paper, is provided by - \emph{synthetic guarded domain theory}. - The tecnhiques of synthetic guarded domain theory allow us to internalize the - step-index reasoning normally required in logical relations proofs of graduality, - ultimately allowing us to specify the logical relation in a manner that looks nearly - identical to a typical, non-step-indexed logical relation. - - In this paper, we report on work we have done to mechanize graduality proofs - using SGDT techniques in Agda. - Our hope in this work is that by mechanizing a graduality proof in a reusable way, - we will provide a framework for other language designers to use to more easily and - conveniently prove that their languages satsify graduality. - - - \subsection{Contributions} - Our main contribution is a framework in Guarded Cubical Agda for proving graduality of a cast calculus. - To demonstrate the feasability and utility of our approach, we have used the framework to prove - graduality for the simply-typed gradual lambda calculus.Along the way, we have developed an ``intensional" theory - of graduality that is of independent interest. - - - \subsection{Proving Graduality in SGDT} - \textbf{TODO:} This section should probably be moved to after the relevant background has been introduced. - - - In this paper, we will utilize SGDT techniques to prove graduality for a particularly - simple gradually-typed cast calculus, the gradually-typed lambda calculus. This is just - the usual simply-typed lambda calculus with a dynamic type $\dyn$ such that $A \ltdyn\, \dyn$ for all types $A$, - as well as upcasts and downcasts between any types $A$ and $B$ such that $A \ltdyn B$. - The complete definition will be provided in Section \ref{sec:GTLC}. - - Our main theorem is the following: - - - % \begin{theorem}[Graduality] - % If $M \ltdyn N : \nat$, then either: - % \begin{enumerate} - % \item $M = \mho$ - % \item $M = N = \zro$ - % \item $M = N = \suc n$ for some $n$ - % \item $M$ and $N$ diverge - % \end{enumerate} - % \end{theorem} - - - \begin{theorem}[Graduality] - If $\cdot \vdash M \ltdyn N : \nat$, then - \begin{enumerate} - \item If $N = \mho$, then $M = \mho$ - \item If $N = `n$, then $M = \mho$ or $M = `n$ - \item If $M = V$, then $N = V$ - \end{enumerate} - \end{theorem} - - We also should be able to show that $\mho$, $\zro$, and $\suc\, N$ are not equal. - - Our first step toward proving graduality is to formulate an \emph{intensional} gradual lambda calculus, - which we call $\intlc$, in which the computation steps taken by a term are made explicit. - The ``normal'' graudal lambda calculus for which we want to prove graduality will be called the - \emph{extensional} gradual lambda calculus, denoted $\extlc$. We will define an erasure function - $\erase{\cdot} : \intlc \to \extlc$ which takes a program in the intensional lambda calculus - and ``forgets'' the syntactic information about the steps to produce a term in the extensional calculus. - - Every term $M_e$ in $\extlc$ will have a corresponding program $M_i$ in $\intlc$ such that - $\erase{M_i} = M_e$. Moreover, we will show that if $M_e \ltdyn_e M_e'$ in the extensional theory, - then there exists terms $M_i$ and $M_i'$ such that $\erase{M_i}=M_e$, $\erase{M_i'}=M_e'$ and - $M_i \ltdyn_i M_i'$ in the intensional theory. - - We formulate and prove an analogous graduality theorem for the intensional lambda calculus. - We define an interpretation of the intensional lambda calculus into a model in which we prove - various results. Using the observation above, given $M_e \ltdyn M_e' : \nat$, we can find - intensional programs $M_i$ and $M_i'$ that erase to them and are such that $M_i \ltdyn M_i'$. - We will then apply the intensional graduality theorem to $M_i$ and $M_i'$, and translate the result - back to $M_e$ and $M_e'$. - - - \subsection{Overview of Remainder of Paper} - - In Section \ref{sec:technical-background}, we provide technical background on gradually typed languages and - on synthetic guarded domain theory. - % - In Section \ref{sec:GTLC}, we introduce the gradually-typed cast calculus - for which we will prove graduality. Important here are the notions of syntactic - type precision and term precision. We introduce both the \emph{extensional} gradual lambda calculus - ($\extlc$) and the \emph{intensional} gradual lambda calculus ($\intlc$). -% - In Section \ref{sec:domain-theory}, we define several fundamental constructions - internal to SGDT that will be needed when we give a denotational semantics to - our intensional lambda calculus. - This includes the notion of Predomains as well as the concept - of EP-Pairs. -% - In Section \ref{sec:semantics}, we define the denotational semantics for the - intensional gradually-typed lambda calculus using the domain theoretic constructions in the - previous section. -% - In Section \ref{sec:graduality}, we outline in more detail the proof of graduality for the - extensional gradual lambda calculus, which will make use of prove properties we prove about - the intensional gradual lambda calculus. -% - In Section \ref{sec:discussion}, we discuss the benefits and drawbacks to our approach in comparison - to the traditional step-indexing approach, as well as possibilities for future work. - - -\section{Technical Background}\label{sec:technical-background} - -\subsection{Gradual Typing} - - - -% Cast calculi -In a gradually-typed language, the mixing of static and dynamic code is seamless, in that -the dynamically typed parts are checked at runtime. This type checking occurs at the elimination -forms of the language (e.g., pattern matching, field reference, etc.). -Gradual languages are generally elaborated to a \emph{cast calculus}, in which the dynamic -type checking is made explicit through the insertion of \emph{type casts}. - -% Up and down casts -In a cast calculus, there is a relation $\ltdyn$ on types such that $A \ltdyn B$ means that -$A$ is a \emph{more precise} type than $B$. -There a dynamic type $\dyn$ with the property that $A \ltdyn \dyn$ for all $A$. -% -If $A \ltdyn B$, a term $M$ of type $A$ may be \emph{up}casted to $B$, written $\up A B M$, -and a term $N$ of type $B$ may be \emph{down}casted to $A$, written $\dn A B N$. -Upcasts always succeed, while downcasts may fail at runtime. -% -% Syntactic term precision -We also have a notion of \emph{syntatcic term precision}. -If $A \ltdyn B$, and $M$ and $N$ are terms of type $A$ and $B$ respectively, we write -$M \ltdyn N : A \ltdyn B$ to mean that $M$ is more precise than $N$, i.e., $M$ and $N$ -behave the same except that $M$ may error more. - -% Modeling the dynamic type as a recursive sum type? -% Observational equivalence and approximation? - -% synthetic guarded domain theory, denotational semantics therein - -\subsection{Difficulties in Prior Semantics} - - % TODO move to another section? - % Difficulties in prior semantics - % - two separate logical relations for expressions - % - transitivity - - In this work, we compare our approach to proving graduality to the approach - introduced by New and Ahmed \cite{new-ahmed2018} which constructs a step-indexed - logical relations model and shows that this model is sound with respect to their - notion of contextual error approximation. - - Because the dynamic type is modeled as a non-well-founded - recursive type, their logical relation needs to be paramterized by natural numbers - to restore well-foundedness. This technique is known as a \emph{step-indexed logical relation}. - Reasoning about step-indexed logical relations - can be tedious and error-prone, and there are some very subtle aspects that must - be taken into account in the proofs. Figure \ref{TODO} shows an example of a step-indexed logical - relation for the gradually-typed lambda calculus. - - In particular, the prior approach of New and Ahmed requires two separate logical - relations for terms, one in which the steps of the left-hand term are counted, - and another in which the steps of the right-hand term are counted. - Then two terms $M$ and $N$ are related in the ``combined'' logical relation if they are - related in both of the one-sided logical relations. Having two separate logical relations - complicates the statement of the lemmas used to prove graduality, becasue any statement that - involves a term stepping needs to take into account whether we are counting steps on the left - or the right. Some of the differences can be abstracted over, but difficulties arise for properties %/results - as fundamental and seemingly straightforward as transitivty. - - Specifically, for transitivity, we would like to say that if $M$ is related to $N$ at - index $i$ and $N$ is related to $P$ at index $i$, then $M$ is related to $P$ at $i$. - But this does not actually hold: we requrie that one of the two pairs of terms - be related ``at infinity'', i.e., that they are related at $i$ for all $i \in \mathbb{N}$. - Which pair is required to satisfy this depends on which logical relation we are considering, - (i.e., is it counting steps on the left or on the right), - and so any argument that uses transitivity needs to consider two cases, one - where $M$ and $N$ must be shown to be related for all $i$, and another where $N$ and $P$ must - be related for all $i$. This may not even be possible to show in some scenarios! - - % These complications introduced by step-indexing lead one to wonder whether there is a - % way of proving graduality without relying on tedious arguments involving natural numbers. - % An alternative approach, which we investigate in this paper, is provided by - % \emph{synthetic guarded domain theory}, as discussed below. - % Synthetic guarded domain theory allows the resulting logical relation to look almost - % identical to a typical, non-step-indexed logical relation. - -\subsection{Synthetic Guarded Domain Theory} -One way to avoid the tedious reasoning associated with step-indexing is to work -axiomatically inside of a logical system that can reason about non-well-founded recursive -constructions while abstracting away the specific details of step-indexing required -if we were working analytically. -The system that proves useful for this purpose is called \emph{synthetic guarded -domain theory}, or SGDT for short. We provide a brief overview here, but more -details can be found in \cite{birkedal-mogelberg-schwinghammer-stovring2011}. - -SGDT offers a synthetic approach to domain theory that allows for guarded recursion -to be expressed syntactically via a type constructor $\later : \type \to \type$ -(pronounced ``later''). The use of a modality to express guarded recursion -was introduced by Nakano \cite{Nakano2000}. -% -Given a type $A$, the type $\later A$ represents an element of type $A$ -that is available one time step later. There is an operator $\nxt : A \to\, \later A$ -that ``delays'' an element available now to make it available later. -We will use a tilde to denote a term of type $\later A$, e.g., $\tilde{M}$. - -% TODO later is an applicative functor, but not a monad - -There is a \emph{guarded fixpoint} operator - -\[ - \fix : \forall T, (\later T \to T) \to T. -\] - -That is, to construct a term of type $T$, it suffices to assume that we have access to -such a term ``later'' and use that to help us build a term ``now''. -This operator satisfies the axiom that $\fix f = f (\nxt (\fix f))$. -In particular, this axiom applies to propositions $P : \texttt{Prop}$; proving -a statement in this manner is known as $\lob$-induction. - -The operators $\later$, $\next$, and $\fix$ described above can be indexed by objects -called \emph{clocks}. A clock serves as a reference relative to which steps are counted. -For instance, given a clock $k$ and type $T$, the type $\later^k T$ represents a value of type -$T$ one unit of time in the future according to clock $k$. -If we only ever had one clock, then we would not need to bother defining this notion. -However, the notion of \emph{clock quantification} is crucial for encoding coinductive types using guarded -recursion, an idea first introduced by Atkey and McBride \cite{atkey-mcbride2013}. - - -% Clocked Cubical Type Theory -\subsubsection{Ticked Cubical Type Theory} -% TODO motivation for Clocked Cubical Type Theory, e.g., delayed substitutions? - -In Ticked Cubical Type Theory \cite{TODO}, there is an additional sort -called \emph{ticks}. Given a clock $k$, a tick $t : \tick k$ serves -as evidence that one unit of time has passed according to the clock $k$. -The type $\later A$ is represented as a function from ticks of a clock $k$ to $A$. -The type $A$ is allowed to depend on $t$, in which case we write $\later^k_t A$ -to emphasize the dependence. - -% TODO next as a function that ignores its input tick argument? - -The rules for tick abstraction and application are similar to those of dependent -$\Pi$ types. -In particular, if we have a term $M$ of type $\later^k A$, and we -have available in the context a tick $t' : \tick k$, then we can apply the tick to $M$ to get -a term $M[t'] : A[t'/t]$. We will also write tick application as $M_t$. -Conversely, if in a context $\Gamma, t : \tick k$ we have that $M$ has type $A$, -then in context $\Gamma$ we have $\lambda t.M$ has type $\later A$. % TODO dependent version? - -The statements in this paper have been formalized in a variant of Agda called -Guarded Cubical Agda \cite{TODO}, an implementation of Clocked Cubical Type Theory. - - -% TODO axioms (clock irrelevance, tick irrelevance)? - - -\subsubsection{The Topos of Trees Model} - -The topos of trees model provides a useful intuition for reasoning in SGDT -\cite{birkedal-mogelberg-schwinghammer-stovring2011}. -This section presupposes knowledge of category theory and can be safely skipped without -disrupting the continuity. - -The topos of trees $\calS$ is the presheaf category $\Set^{\omega^o}$. -% -We assume a universe $\calU$ of types, with encodings for operations such -as sum types (written as $\widehat{+}$). There is also an operator -$\laterhat \colon \later \calU \to \calU$ such that -$\El(\laterhat(\nxt A)) =\,\, \later \El(A)$, where $\El$ is the type corresponding -to the code $A$. - -An object $X$ is a family $\{X_i\}$ of sets indexed by natural numbers, -along with restriction maps $r^X_i \colon X_{i+1} \to X_i$ (see Figure \ref{fig:topos-of-trees-object}). -These should be thought of as ``sets changing over time", where $X_i$ is the view of the set if we have $i+1$ -time steps to reason about it. -We can also think of an ongoing computation, with $X_i$ representing the potential results of the computation -after it has run for $i+1$ steps. - -\begin{figure} - % https://q.uiver.app/?q=WzAsNSxbMiwwLCJYXzIiXSxbMCwwLCJYXzAiXSxbMSwwLCJYXzEiXSxbMywwLCJYXzMiXSxbNCwwLCJcXGRvdHMiXSxbMiwxLCJyXlhfMCIsMl0sWzAsMiwicl5YXzEiLDJdLFszLDAsInJeWF8yIiwyXSxbNCwzLCJyXlhfMyIsMl1d -\[\begin{tikzcd}[ampersand replacement=\&] - {X_0} \& {X_1} \& {X_2} \& {X_3} \& \dots - \arrow["{r^X_0}"', from=1-2, to=1-1] - \arrow["{r^X_1}"', from=1-3, to=1-2] - \arrow["{r^X_2}"', from=1-4, to=1-3] - \arrow["{r^X_3}"', from=1-5, to=1-4] -\end{tikzcd}\] - \caption{An example of an object in the topos of trees.} - \label{fig:topos-of-trees-object} -\end{figure} - -A morphism from $\{X_i\}$ to $\{Y_i\}$ is a family of functions $f_i \colon X_i \to Y_i$ -that commute with the restriction maps in the obvious way, that is, -$f_i \circ r^X_i = r^Y_i \circ f_{i+1}$ (see Figure \ref{fig:topos-of-trees-morphism}). - -\begin{figure} -% https://q.uiver.app/?q=WzAsMTAsWzIsMCwiWF8yIl0sWzAsMCwiWF8wIl0sWzEsMCwiWF8xIl0sWzMsMCwiWF8zIl0sWzQsMCwiXFxkb3RzIl0sWzAsMSwiWV8wIl0sWzEsMSwiWV8xIl0sWzIsMSwiWV8yIl0sWzMsMSwiWV8zIl0sWzQsMSwiXFxkb3RzIl0sWzIsMSwicl5YXzAiLDJdLFswLDIsInJeWF8xIiwyXSxbMywwLCJyXlhfMiIsMl0sWzQsMywicl5YXzMiLDJdLFs2LDUsInJeWV8wIiwyXSxbNyw2LCJyXllfMSIsMl0sWzgsNywicl5ZXzIiLDJdLFs5LDgsInJeWV8zIiwyXSxbMSw1LCJmXzAiLDJdLFsyLDYsImZfMSIsMl0sWzAsNywiZl8yIiwyXSxbMyw4LCJmXzMiLDJdXQ== -\[\begin{tikzcd}[ampersand replacement=\&] - {X_0} \& {X_1} \& {X_2} \& {X_3} \& \dots \\ - {Y_0} \& {Y_1} \& {Y_2} \& {Y_3} \& \dots - \arrow["{r^X_0}"', from=1-2, to=1-1] - \arrow["{r^X_1}"', from=1-3, to=1-2] - \arrow["{r^X_2}"', from=1-4, to=1-3] - \arrow["{r^X_3}"', from=1-5, to=1-4] - \arrow["{r^Y_0}"', from=2-2, to=2-1] - \arrow["{r^Y_1}"', from=2-3, to=2-2] - \arrow["{r^Y_2}"', from=2-4, to=2-3] - \arrow["{r^Y_3}"', from=2-5, to=2-4] - \arrow["{f_0}"', from=1-1, to=2-1] - \arrow["{f_1}"', from=1-2, to=2-2] - \arrow["{f_2}"', from=1-3, to=2-3] - \arrow["{f_3}"', from=1-4, to=2-4] -\end{tikzcd}\] - \caption{An example of a morphism in the topos of trees.} - \label{fig:topos-of-trees-morphism} -\end{figure} - - -The type operator $\later$ is defined on an object $X$ by -$(\later X)_0 = 1$ and $(\later X)_{i+1} = X_i$. -The restriction maps are given by $r^\later_0 =\, !$, where $!$ is the -unique map into $1$, and $r^\later_{i+1} = r^X_i$. -The morphism $\nxt^X \colon X \to \later X$ is defined pointwise by -$\nxt^X_0 =\, !$, and $\nxt^X_{i+1} = r^X_i$. It is easily checked that -this satisfies the commutativity conditions required of a morphism in $\calS$. -% -Given a morphism $f \colon \later X \to X$, i.e., a family of functions -$f_i \colon (\later X)_i \to X_i$ that commute with the restrictions in the appropriate way, -we define $\fix(f) \colon 1 \to X$ pointwise -by $\fix(f)_i = f_{i} \circ \dots \circ f_0$. -This can be visualized as a diagram in the category of sets as shown in -Figure \ref{fig:topos-of-trees-fix}. -% Observe that the fixpoint is a \emph{global element} in the topos of trees. -% Global elements allow us to view the entire computation on a global level. - - -% https://q.uiver.app/?q=WzAsOCxbMSwwLCJYXzAiXSxbMiwwLCJYXzEiXSxbMywwLCJYXzIiXSxbMCwwLCIxIl0sWzAsMSwiWF8wIl0sWzEsMSwiWF8xIl0sWzIsMSwiWF8yIl0sWzMsMSwiWF8zIl0sWzMsNCwiZl8wIl0sWzAsNSwiZl8xIl0sWzEsNiwiZl8yIl0sWzIsNywiZl8zIl0sWzAsMywiISIsMl0sWzEsMCwicl5YXzAiLDJdLFsyLDEsInJeWF8xIiwyXSxbNSw0LCJyXlhfMCJdLFs2LDUsInJeWF8xIl0sWzcsNiwicl5YXzIiXV0= -% \[\begin{tikzcd}[ampersand replacement=\&] -% 1 \& {X_0} \& {X_1} \& {X_2} \\ -% {X_0} \& {X_1} \& {X_2} \& {X_3} -% \arrow["{f_0}", from=1-1, to=2-1] -% \arrow["{f_1}", from=1-2, to=2-2] -% \arrow["{f_2}", from=1-3, to=2-3] -% \arrow["{f_3}", from=1-4, to=2-4] -% \arrow["{!}"', from=1-2, to=1-1] -% \arrow["{r^X_0}"', from=1-3, to=1-2] -% \arrow["{r^X_1}"', from=1-4, to=1-3] -% \arrow["{r^X_0}", from=2-2, to=2-1] -% \arrow["{r^X_1}", from=2-3, to=2-2] -% \arrow["{r^X_2}", from=2-4, to=2-3] -% \end{tikzcd}\] - -% \begin{figure} -% % https://q.uiver.app/?q=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 -% \[\begin{tikzcd}[ampersand replacement=\&] -% {\fix(f)_0} \& {\fix(f)_1} \& {\fix(f)_2} \& {\fix(f)_3} \\ -% 1 \& 1 \& 1 \& 1 \\ -% {X_0} \& {X_0} \& {X_0} \& {X_0} \& \cdots \\ -% \& {X_1} \& {X_1} \& {X_1} \\ -% \&\& {X_2} \& {X_2} \\ -% \&\&\& {X_3} -% \arrow["{f_0}", from=2-1, to=3-1] -% \arrow["{f_1}", from=3-2, to=4-2] -% \arrow["{f_0}", from=2-2, to=3-2] -% \arrow["{f_0}", from=2-3, to=3-3] -% \arrow["{f_1}", from=3-3, to=4-3] -% \arrow["{f_2}", from=4-3, to=5-3] -% \arrow["{f_0}", from=2-4, to=3-4] -% \arrow["{f_1}", from=3-4, to=4-4] -% \arrow["{f_2}", from=4-4, to=5-4] -% \arrow["{f_3}", from=5-4, to=6-4] -% \end{tikzcd}\] -% \caption{The first few approximations to the guarded fixpoint of $f$.} -% \label{fig:topos-of-trees-fix-approx} -% \end{figure} - - -\begin{figure} - % https://q.uiver.app/?q=WzAsNixbMywwLCIxIl0sWzAsMiwiWF8wIl0sWzIsMiwiWF8xIl0sWzQsMiwiWF8yIl0sWzYsMiwiWF8zIl0sWzgsMiwiXFxkb3RzIl0sWzIsMSwicl5YXzAiXSxbNCwzLCJyXlhfMiJdLFswLDEsIlxcZml4KGYpXzAiLDFdLFswLDIsIlxcZml4KGYpXzEiLDFdLFswLDMsIlxcZml4KGYpXzIiLDFdLFswLDQsIlxcZml4KGYpXzMiLDFdLFswLDUsIlxcY2RvdHMiLDMseyJzdHlsZSI6eyJib2R5Ijp7Im5hbWUiOiJub25lIn0sImhlYWQiOnsibmFtZSI6Im5vbmUifX19XSxbMywyLCJyXlhfMSJdLFs1LDQsInJeWF8zIl1d - \[\begin{tikzcd}[ampersand replacement=\&,column sep=2.25em] - \&\&\& 1 \\ - \\ - {X_0} \&\& {X_1} \&\& {X_2} \&\& {X_3} \&\& \dots - \arrow["{r^X_0}", from=3-3, to=3-1] - \arrow["{r^X_2}", from=3-7, to=3-5] - \arrow["{\fix(f)_0}"{description}, from=1-4, to=3-1] - \arrow["{\fix(f)_1}"{description}, from=1-4, to=3-3] - \arrow["{\fix(f)_2}"{description}, from=1-4, to=3-5] - \arrow["{\fix(f)_3}"{description}, from=1-4, to=3-7] - \arrow["\cdots"{marking}, draw=none, from=1-4, to=3-9] - \arrow["{r^X_1}", from=3-5, to=3-3] - \arrow["{r^X_3}", from=3-9, to=3-7] - \end{tikzcd}\] - \caption{The guarded fixpoint of $f$.} - \label{fig:topos-of-trees-fix} -\end{figure} - -% TODO global elements? - - -\section{GTLC}\label{sec:GTLC} - -Here we describe the syntax and typing for the graudally-typed lambda calculus. -We also give the rules for syntactic type and term precision. -We define two separate calculi: the normal gradually-typed lambda calculus, which we -call the extensional lambda calculus ($\extlc$), as well as an \emph{intensional} lambda calculus -($\intlc$) whose syntax makes explicit the steps taken by a program. - -\subsection{Syntax} - - -\begin{align*} - &\text{Types } A, B := \nat, \, \dyn, \, (A \To B) \\ - &\text{Terms } M, N := \err_A, \, \zro, \, \suc\, M, \, (\lda{x}{M}), \, (M\, N), \, (\up{A}{B} M), \, (\dn{A}{B} M) \\ - &\text{Contexts } \Gamma := \cdot, \, (\Gamma, x : A) -\end{align*} - -The typing rules are as expected, with a cast between $A$ to $B$ allowed only when $A \ltdyn B$. - -\begin{mathpar} - \inferrule*{ }{\hasty \Gamma {\err_A} A} - - \inferrule*{ }{\hasty \Gamma \zro \nat} - - \inferrule {\hasty \Gamma M \nat} {\hasty \Gamma {\suc\, M} \nat} - - \inferrule* - {\hasty {\Gamma, x : A} M B} - {\hasty \Gamma {\lda x M} {A \To B}} - - \inferrule * - {\hasty \Gamma M {A \To B} \and \hasty \Gamma N A} - {\hasty \Gamma {M \, N} B} - - \inferrule* - {A \ltdyn B \and \hasty \Gamma M A} - {\hasty \Gamma {\up A B M} B } - - \inferrule* - {A \ltdyn B \and \hasty \Gamma M B} - {\hasty \Gamma {\dn A B M} A} -\end{mathpar} - -The equational theory is also as expected, with $\beta$ and $\eta$ laws for function type. - -\subsection{Type Precision} - -The rules for type precision are as follows: - -\begin{mathpar} - \inferrule*[right = \dyn] - { }{\dyn \ltdyn \dyn} - - \inferrule*[right = \nat] - { }{\nat \ltdyn \nat} - - \inferrule*[right = $Inj_\nat$] - { }{\nat \ltdyn \dyn} - - \inferrule*[right = $\To$] - {A_i \ltdyn B_i \and A_o \ltdyn B_o } - {(A_i \To A_o) \ltdyn (B_i \To B_o)} - - \inferrule*[right=$Inj_{\To}$] - {(A_i \to A_o) \ltdyn (\dyn \To \dyn)} - {(A_i \to A_o) \ltdyn\, \dyn} - - -\end{mathpar} -% Type precision derivations -Note that as a consequence of this presentation of the type precision rules, we -have that if $A \ltdyn B$, there is a unique precision derivation that witnesses this. -As in previous work, we go a step farther and make these derivations first-class objects, -known as \emph{type precision derivations}. -Specifically, for every $A \ltdyn B$, we have a derivation $d : A \ltdyn B$ that is constructed -using the rules above. For instance, there is a derivation $\dyn : \dyn \ltdyn \dyn$, and a derivation -$\nat : \nat \ltdyn \nat$, and if $d_i : A_i \ltdyn B_i$ and $d_o : A_o \ltdyn B_o$, then -there is a derivation $d_i \To d_o : (A_i \To A_o) \ltdyn (B_i \To B_o)$. Likewise for -the remaining two rules. The benefit to making these derivations explicit in the syntax is that we -can perform induction over them. -Note also that for any type $A$, we use $A$ to denote the reflexivity derivation that $A \ltdyn A$, -i.e., $A : A \ltdyn A$. -Finally, observe that for type precision derivations $d : A \ltdyn B$ and $d' : B \ltdyn C$, we -can define their composition $d' \circ d : A \ltdyn C$. This will be used in the statement -of transitivity of the term precision relation. +\input{intro} -\subsection{Term Precision} +\input{technical-background} -We allow for a \emph{heterogeneous} term precision judgment on terms $M$ of type $A$ and $N$ of type $B$, -provided that $A \ltdyn B$ holds. - -% Type precision contexts -In order to deal with open terms, we will need the notion of a type precision \emph{context}, which we denote -$\gamlt$. This is similar to a normal context but instead of mapping variables to types, -it maps variables $x$ to related types $A \ltdyn B$, where $x$ has type $A$ in the left-hand term -and $B$ in the right-hand term. We may also write $x : d$ where $d : A \ltdyn B$ to indicate this. -Another way of thinking of type precision contexts is as a zipped pair of contexts $\Gamma^l, \Gamma^r$ -with the same domain such that $\Gamma^l(x) \ltdyn \Gamma^r(x)$ for each $x$ in the domain. - -The rules for term precision come in two forms. We first have the \emph{congruence} rules, -one for each term constructor. These assert that the term constructors respect term precision. -The congruence rules are as follows: - - -\begin{mathpar} - \inferrule*[right = $\err$] - { \hasty \Gamma M A } - {\etmprec {\gamlt} {\err_A} M A} - - \inferrule*[right = Var] - { d : A \ltdyn B \and \gamlt(x) = (A, B) } { \etmprec {\gamlt} x x d } - - \inferrule*[right = Zro] - { } {\etmprec \gamlt \zro \zro \nat } - - \inferrule*[right = Suc] - { \etmprec \gamlt M N \nat } {\etmprec \gamlt {\suc\, M} {\suc\, N} \nat} - - \inferrule*[right = Lambda] - { d_i : A_i \ltdyn B_i \and - d_o : A_o \ltdyn B_o \and - \etmprec {\gamlt , x : d_i} M N {d_o} } - { \etmprec \gamlt {\lda x M} {\lda x N} (d_i \To d_o) } - - \inferrule*[right = App] - { d_i : A_i \ltdyn B_i \and - d_o : A_o \ltdyn B_o \\\\ - \etmprec \gamlt {M} {M'} (d_i \To d_o) \and - \etmprec \gamlt {N} {N'} d_i - } - { \etmprec \gamlt {M\, N} {M'\, N'} {d_o}} -\end{mathpar} - -We then have additional equational axioms, including transitivity, $\beta$ and $\eta$ laws, and -rules characterizing upcasts as least upper bounds, and downcasts as greatest lower bounds. - -We write $M \equidyn N$ to mean that both $M \ltdyn N$ and $N \ltdyn M$. - -% TODO -\begin{mathpar} - \inferrule*[right = Transitivity] - {\etmprec \gamlt M N d \and - \etmprec \gamlt N P {d'} } - {\etmprec \gamlt M P {d' \circ d} } - - \inferrule*[right = $\beta$] - { \hasty {\Gamma, x : A_i} M {A_o} \and - \hasty {\Gamma} V {A_i} } - { \etmequidyn \gamlt {(\lda x M)\, V} {M[V/x]} {A_o} } - - \inferrule*[right = $\eta$] - { \hasty {\Gamma} {V} {A_i \To A_o} } - { \etmequidyn \gamlt {\lda x (V\, x)} V {A_i \To A_o} } - - \inferrule*[right = UpR] - { d : A \ltdyn B \and - \hasty \Gamma M A } - { \etmprec \gamlt M {\up A B M} d } - - \inferrule*[right = UpL] - { d : A \ltdyn B \and - \etmprec \gamlt M N d } - { \etmprec \gamlt {\up A B M} N B } - - \inferrule*[right = DnL] - { d : A \ltdyn B \and - \hasty \Gamma M B } - { \etmprec \gamlt {\dn A B M} M d } - - \inferrule*[right = DnR] - { d : A \ltdyn B \and - \etmprec \gamlt M N d } - { \etmprec \gamlt M {\dn A B N} A } -\end{mathpar} - -% TODO explain the least upper bound/greatest lower bound rules -The rules UpR, UpL, DnL, and DnR were introduced in \cite{TODO} as a means -of cleanly axiomatizing the intended behavior of casts in a way that -doesn't depend on the specific constructs of the language. -Intuitively, rule UpR says that the upcast of $M$ is an upper bound for $M$ -in that $M$ may error more, and UpL says that the upcast is the \emph{least} -such upper bound, in that it errors more than any other upper bound for $M$. -Conversely, DnL says that the downcast of $M$ is a lower bound, and DnR says -that it is the \emph{greatest} lower bound. -% These rules provide a clean axiomatization of the behavior of casts that doesn't -% depend on the specific constructs of the language. - - -\subsection{The Intensional Lambda Calculus} - -\textbf{TODO: Subject to change!} - -Now that we have described the syntax along with the type and term precision judgments for -$\extlc$, we can now do the same for $\intlc$. One key difference between the two calculi -is that we define $\intlc$ using the constructs available to us in the language of synthetic -guarded domain theory, e.g., we use the $\later$ operator. Whereas when we defined the syntax -of the extensional lambda calculus we were working in the category of sets, when we define the -syntax of the intensional lambda calculus we will be working in the topos of trees. -% in a presheaf category known as the \emph{topos of trees}. - -More specifically, in $\intlc$, we not only have normal terms, but also terms available -``later'', which we denote by $\tilde{M}$. -We have a term constructor $\theta_s$ which takes a later-term and turns it into -a term avaialble now. -The typing and precision rules for $\theta_s$ involve the $\later$ operator, as shown below. -Observe that $\theta_s$ is a syntactic analogue to the $\theta$ constructor of the lifting monad -that we will define in the section on domain theoretic constructions (Section \ref{sec:domain-theory}), -but it is important to note that $\theta_s (\tilde{M})$ is an actual term in $\intlc$, whereas the -$\theta$ constructor is a purely semantic construction. These will be connected when we discuss the -interpretation of $\intlc$ into the semantic model. - -% Most aspects of the two calculi are the same, except that in $\intlc$ we add a new term constructor -% $\theta_s$ which represents a syntactic way of delaying a term by one computation step. - -To better understand this situation, note that $\lda{x}{(\cdot)}$ can be viewed as a function -(at the level of the ambient type theory) from terms of type $B$ under context $\Gamma, x : A$ -to terms of type $A \To B$ under context $\Gamma$. Similarly, we can view $\theta_s(\cdot)$ as a -function (in the ambient type theory, which is sythetic guarded domain theory) taking terms -$\tilde{M}$ of type $A$ avaiable \emph{later} to terms of type $A$ available \emph{now}. - -Notice that the notion of a ``term available later'' does not need to be part of the syntax -of the intensional lambda calculus, because this can be expressed in the ambient theory. -Similarly, we do not need a syntactic form to delay a term, because we can simply use $\nxt$. - - -\begin{align*} - % &\text{Later Terms } \tilde{M}, \tilde{N} - &\text{Terms } M, N := \err_A, \, \dots \, \theta_s (\tilde{M}) \\ -\end{align*} - -\begin{mathpar} - \inferrule*[] - { \later_t (\hasty \Gamma {M_t} A) } - { \hasty \Gamma {\theta_s M} {A} } -\end{mathpar} - -\begin{mathpar} - \inferrule*[] - { \later_t (\itmprec \gamlt {M_t} {N_t} d) } - { \itmprec \gamlt {\theta_s M} {\theta_s N} d } -\end{mathpar} - -Recall that $\later_t$ is a dependent form of $\later$ where the arugment is allowed -to mention $t$. In particular, here we apply the tick $t$ to the later-terms $M$ and $N$ to get -``now"-terms $M_t$ and $N_t$. - -Formally speaking, the term precision relation must be defined as a guarded fixpoint, -i.e., we assume that the function is defined later, and use it to construct a definition -that is defined ``now''. This involves applying a tick to the later-function to shift it -to a now-function. Indeed, this is what we do in the formal Agda development, but in this -paper, we will elide these issues as they are not relevant to the main ideas. - -% TODO equational theory - - -\section{Domain-Theoretic Constructions}\label{sec:domain-theory} - -In this section, we discuss the fundamental objects of the model into which we will embed -the intensional lambda calculus and inequational theory. It is important to remember that -the constructions in this section are entirely independent of the syntax described in the -previous section; the notions defined here exist in their own right as purely mathematical -constructs. In the next section, we will link the syntax and semantics via an interpretation -function. - -\subsection{The Lift Monad} - -When thinking about how to model intensional gradually-typed programs, we must consider -their possible behaviors. On the one hand, we have \emph{failure}: a program may fail -at run-time because of a type error. In addition to this, a program may ``think'', -i.e., take a step of computation. If a program thinks forever, then it never returns a value, -so we can think of the idea of thinking as a way of intensionally modelling \emph{partiality}. - -With this in mind, we can describe a semantic object that models these behaviors: a monad -for embedding computations that has cases for failure and ``thinking''. -Previous work has studied such a construct in the setting of the latter only, called the lift -monad \cite{mogelberg-paviotti2016}; here, we add the additional effect of failure. - -For a type $A$, we define the \emph{lift monad with failure} $\li A$, which we will just call -the \emph{lift monad}, as the following datatype: - -\begin{align*} - \li A &:= \\ - &\eta \colon A \to \li A \\ - &\mho \colon \li A \\ - &\theta \colon \later (\li A) \to \li A -\end{align*} - -Formally, the lift monad $\li A$ is defined as the solution to the guarded recursive type equation - -\[ \li A \cong A + 1 + \later \li A. \] - -An element of $\li A$ should be viewed as a computation that can either (1) return a value (via $\eta$), -(2) raise an error and stop (via $\mho$), or (3) think for a step (via $\theta$). -% -Notice there is a computation $\fix \theta$ of type $\li A$. This represents a computation -that thinks forever and never returns a value. - -Since we claimed that $\li A$ is a monad, we need to define the monadic operations -and show that they repect the monadic laws. The return is just $\eta$, and extend -is defined via by guarded recursion by cases on the input. -% It is instructive to give at least one example of a use of guarded recursion, so -% we show below how to define extend: -% TODO -% -% -Verifying that the monadic laws hold requires \lob-induction and is straightforward. - -The lift monad has the following universal property. Let $f$ be a function from $A$ to $B$, -where $B$ is a $\later$-algebra, i.e., there is $\theta_B \colon \later B \to B$. -Further suppose that $B$ is also an ``error-algebra'', that is, an algebra of the -constant functor $1 \colon \text{Type} \to \text{Type}$ mapping all types to Unit. -This latter statement amounts to saying that there is a map $\text{Unit} \to B$, so $B$ has a -distinguished ``error element" $\mho_B \colon B$. - -Then there is a unique homomorphism of algebras $f' \colon \li A \to B$ such that -$f' \circ \eta = f$. The function $f'(l)$ is defined via guarded fixpoint by cases on $l$. -In the $\mho$ case, we simply return $\mho_B$. -In the $\theta(\tilde{l})$ case, we will return - -\[\theta_B (\lambda t . (f'_t \, \tilde{l}_t)). \] - -Recalling that $f'$ is a guaded fixpoint, it is available ``later'' and by -applying the tick we get a function we can apply ``now''; for the argument, -we apply the tick to $\tilde{l}$ to get a term of type $\li A$. - - -\subsubsection{Model-Theoretic Description} -We can describe the lift monad in the topos of trees model as follows. - -% TODO - -\subsection{Predomains} - -The next important construction is that of a \emph{predomain}. A predomain is intended to -model the notion of error ordering that we want terms to have. Thus, we define a predomain $A$ -as a partially-ordered set, which consists of a type which we denote $\ty{A}$ and a reflexive, -transitive, and antisymmetric relation $\le_P$ on $A$. - -For each type we want to represent, we define a predomain for the corresponding semantic -type. For instance, we define a predomain for natural numbers, a predomain for the -dynamic type, a predomain for functions, and a predomain for the lift monad. We -describe each of these below. - -We define monotone functions between predomain as expected. Given predomains -$A$ and $B$, we write $f \colon A \monto B$ to indicate that $f$ is a monotone -function from $A$ to $B$, i.e, for all $a_1 \le_A a_2$, we have $f(a_1) \le_B f(a_2)$. - -% TODO -\begin{itemize} - \item There is a predomain Nat for natural numbers, where the ordering is equality. - - \item There is a predomain Dyn to represent the dynamic type. The underlying type - for this predomain is defined by guarded fixpoint to be such that - $\ty{\Dyn} \cong \mathbb{N}\, +\, \later (\ty{\Dyn} \monto \ty{\Dyn})$. - This definition is valid because the occurrences of Dyn are guarded by the $\later$. - The ordering is defined via guarded recursion by cases on the argument, using the - ordering on $\mathbb{N}$ and the ordering on monotone functions described below. - - \item For a predomain $A$, there is a predomain $\li A$ that is the ``lift'' of $A$ - using the lift monad. We use the same notation for $\li A$ when $A$ is a type - and $A$ is a predomain, since the context should make clear which one we are referring to. - The underling type of $\li A$ is simply $\li \ty{A}$, i.e., the lift of the underlying - type of $A$. - The ordering on $\li A$ is the ``lock-step error-ordering'' which we describe in - \ref{subsec:lock-step}. - - \item For predomains $A_i$ and $A_o$, we form the predomain of monotone functions - from $A_i$ to $A_o$, which we denote by $A_i \To A_o$. - The ordering is such that $f$ is below $g$ if for all $a$ in $\ty{A_i}$, we have - $f(a)$ is below $g(a)$ in the ordering for $A_o$. -\end{itemize} - - - -\subsection{Lock-step Error Ordering}\label{subsec:lock-step} - -As mentioned, the ordering on the lift of a predomain $A$ -is called the \emph{lock-step error-ordering}, the idea being that two computations -$l$ and $l'$ are related if they are in lock-step with regard to their -intensional behavior, up to $l$ erroring. -Formally, we define this ordering as follows: - -\begin{itemize} - \item $\eta\, x \ltls \eta\, y$ if $x \le_A y$. - \item $\mho \ltls l$ for all $l$ - \item $\theta\, \tilde{r} \ltls \theta\, \tilde{r'}$ if - $\later_t (\tilde{r}_t \ltls \tilde{r'}_t)$ -\end{itemize} - -We also define a heterogeneous version of this ordering between the lifts of two -different predomains $A$ and $B$, parameterized by a relation $R$ between $A$ and $B$. - -\subsection{Weak Bisimilarity Relation} - -We also define another ordering on $\li A$, called ``weak bisimilarity'', -written $l \bisim l'$. -Intuitively, we say $l \bisim l'$ if they are equivalent ``up to delays''. -We introduce the notation $x \sim_A y$ to mean $x \le_A y$ and $y \le_A x$. - -The weak bisimilarity relation is defined by guarded fixpoint as follows: - -\begin{align*} - &\mho \bisim \mho \\ -% - &\eta\, x \bisim \eta\, y \text{ if } - x \sim_A y \\ -% - &\theta\, \tilde{x} \bisim \theta\, \tilde{y} \text{ if } - \later_t (\tilde{x}_t \bisim \tilde{y}_t) \\ -% - &\theta\, \tilde{x} \bisim \mho \text{ if } - \theta\, \tilde{x} = \delta^n(\mho) \text { for some $n$ } \\ -% - &\theta\, \tilde{x} \bisim \eta\, y \text{ if } - (\theta\, \tilde{x} = \delta^n(\eta\, x)) - \text { for some $n$ and $x : \ty{A}$ such that $x \sim_A y$ } \\ -% - &\mho \bisim \theta\, \tilde{y} \text { if } - \theta\, \tilde{y} = \delta^n(\mho) \text { for some $n$ } \\ -% - &\eta\, x \bisim \theta\, \tilde{y} \text { if } - (\theta\, \tilde{y} = \delta^n (\eta\, y)) - \text { for some $n$ and $y : \ty{A}$ such that $x \sim_A y$ } -\end{align*} - - -\subsection{EP-Pairs} - -Here, we adapt the notion of embedding-projection pair as used to study -gradual typing (\cite{new-ahmed2018}) to the setting of intensional denotaional semantics. - - - - - -\section{Semantics}\label{sec:semantics} - -\subsection{Relational Semantics} - -\subsubsection{Types as Predomains} - - -\subsubsection{Terms as Monotone Functions} - - -\subsubsection{Type Precision as EP-Pairs} - - -\subsubsection{Term Precision via the Lock-Step Error Ordering} -% Homogeneous vs heterogeneous term precision - -\subsection{Logical Relations Semantics} - - -\section{Graduality}\label{sec:graduality} -The main theorem we would like to prove is the following: - -\begin{theorem}[Graduality] - If $\cdot \vdash M \ltdyn N : \nat$, then - \begin{enumerate} - \item If $N = \mho$, then $M = \mho$ - \item If $N = `n$, then $M = \mho$ or $M = `n$ - \item If $M = V$, then $N = V$ - \end{enumerate} -\end{theorem} - - -\subsection{Outline} - -\subsection{Extensional to Intensional} - -\subsection{Intensional Results} - -\subsection{Adequacy} - -\subsection{Putting it Together} +\input{syntax} +\input{semantics} +\input{graduality} \section{Discussion}\label{sec:discussion} -\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. -% -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. +% \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. +% % +% 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. \bibliographystyle{ACM-Reference-Format} diff --git a/paper-new/references.bib b/paper-new/references.bib index 8ad633f..f12e978 100644 --- a/paper-new/references.bib +++ b/paper-new/references.bib @@ -340,3 +340,22 @@ year = {2002}, month = sep, pages = {48--59} } + +@article{new-licata18, + author = {Max S. New and + Daniel R. Licata}, + title = {Call-by-name Gradual Type Theory}, + journal = {CoRR}, + year = {2018}, + url = {http://arxiv.org/abs/1802.00061}, +} + +@phdthesis{levy01:phd, +author = "Levy, Paul Blain", +title = "Call-by-Push-Value", +type = "{Ph.~D.} Dissertation", +school = "Queen Mary, University of London", +department = "Department of Computer Science", +address = "London, UK", +month = mar, +year = "2001"} \ No newline at end of file diff --git a/paper-new/semantics.tex b/paper-new/semantics.tex new file mode 100644 index 0000000..3fca4e3 --- /dev/null +++ b/paper-new/semantics.tex @@ -0,0 +1,196 @@ +\section{Domain-Theoretic Constructions}\label{sec:domain-theory} + +In this section, we discuss the fundamental objects of the model into which we will embed +the intensional lambda calculus and inequational theory. It is important to remember that +the constructions in this section are entirely independent of the syntax described in the +previous section; the notions defined here exist in their own right as purely mathematical +constructs. In the next section, we will link the syntax and semantics via an interpretation +function. + +\subsection{The Lift Monad} + +When thinking about how to model intensional gradually-typed programs, we must consider +their possible behaviors. On the one hand, we have \emph{failure}: a program may fail +at run-time because of a type error. In addition to this, a program may ``think'', +i.e., take a step of computation. If a program thinks forever, then it never returns a value, +so we can think of the idea of thinking as a way of intensionally modelling \emph{partiality}. + +With this in mind, we can describe a semantic object that models these behaviors: a monad +for embedding computations that has cases for failure and ``thinking''. +Previous work has studied such a construct in the setting of the latter, called the lift +monad \cite{mogelberg-paviotti2016}; here, we augment it with the additional effect of failure. + +For a type $A$, we define the \emph{lift monad with failure} $\li A$, which we will just call +the \emph{lift monad}, as the following datatype: + +\begin{align*} + \li A &:= \\ + &\eta \colon A \to \li A \\ + &\mho \colon \li A \\ + &\theta \colon \later (\li A) \to \li A +\end{align*} + +Unless otherwise mentioned, all constructs involving $\later$ or $\fix$ +are understood to be with repsect to a fixed clock $k$. So for the above, we really have for each +clock $k$ a type $\li^k A$ with respect to that clock. + +Formally, the lift monad $\li A$ is defined as the solution to the guarded recursive type equation + +\[ \li A \cong A + 1 + \later \li A. \] + +An element of $\li A$ should be viewed as a computation that can either (1) return a value (via $\eta$), +(2) raise an error and stop (via $\mho$), or (3) think for a step (via $\theta$). +% +Notice there is a computation $\fix \theta$ of type $\li A$. This represents a computation +that thinks forever and never returns a value. + +Since we claimed that $\li A$ is a monad, we need to define the monadic operations +and show that they repect the monadic laws. The return is just $\eta$, and extend +is defined via by guarded recursion by cases on the input. +% It is instructive to give at least one example of a use of guarded recursion, so +% we show below how to define extend: +% TODO +% +% +Verifying that the monadic laws hold requires \lob-induction and is straightforward. + +The lift monad has the following universal property. Let $f$ be a function from $A$ to $B$, +where $B$ is a $\later$-algebra, i.e., there is $\theta_B \colon \later B \to B$. +Further suppose that $B$ is also an ``error-algebra'', that is, an algebra of the +constant functor $1 \colon \text{Type} \to \text{Type}$ mapping all types to Unit. +This latter statement amounts to saying that there is a map $\text{Unit} \to B$, so $B$ has a +distinguished ``error element" $\mho_B \colon B$. + +Then there is a unique homomorphism of algebras $f' \colon \li A \to B$ such that +$f' \circ \eta = f$. The function $f'(l)$ is defined via guarded fixpoint by cases on $l$. +In the $\mho$ case, we simply return $\mho_B$. +In the $\theta(\tilde{l})$ case, we will return + +\[\theta_B (\lambda t . (f'_t \, \tilde{l}_t)). \] + +Recalling that $f'$ is a guaded fixpoint, it is available ``later'' and by +applying the tick we get a function we can apply ``now''; for the argument, +we apply the tick to $\tilde{l}$ to get a term of type $\li A$. + + +%\subsubsection{Model-Theoretic Description} +%We can describe the lift monad in the topos of trees model as follows. + + +\subsection{Predomains} + +The next important construction is that of a \emph{predomain}. A predomain is intended to +model the notion of error ordering that we want terms to have. Thus, we define a predomain $A$ +as a partially-ordered set, which consists of a type which we denote $\ty{A}$ and a reflexive, +transitive, and antisymmetric relation $\le_P$ on $A$. + +For each type we want to represent, we define a predomain for the corresponding semantic +type. For instance, we define a predomain for natural numbers, a predomain for the +dynamic type, a predomain for functions, and a predomain for the lift monad. We +describe each of these below. + +We define monotone functions between predomain as expected. Given predomains +$A$ and $B$, we write $f \colon A \monto B$ to indicate that $f$ is a monotone +function from $A$ to $B$, i.e, for all $a_1 \le_A a_2$, we have $f(a_1) \le_B f(a_2)$. + +\begin{itemize} + \item There is a predomain Nat for natural numbers, where the ordering is equality. + + \item There is a predomain Dyn to represent the dynamic type. The underlying type + for this predomain is defined by guarded fixpoint to be such that + $\ty{\Dyn} \cong \mathbb{N}\, +\, \later (\ty{\Dyn} \monto \ty{\Dyn})$. + This definition is valid because the occurrences of Dyn are guarded by the $\later$. + The ordering is defined via guarded recursion by cases on the argument, using the + ordering on $\mathbb{N}$ and the ordering on monotone functions described below. + + \item For a predomain $A$, there is a predomain $\li A$ for the ``lift'' of $A$ + using the lift monad. We use the same notation for $\li A$ when $A$ is a type + and $A$ is a predomain, since the context should make clear which one we are referring to. + The underling type of $\li A$ is simply $\li \ty{A}$, i.e., the lift of the underlying + type of $A$. + The ordering on $\li A$ is the ``step-sensitive error-ordering'' which we describe in + \ref{subsec:lock-step}. + + \item For predomains $A_i$ and $A_o$, we form the predomain of monotone functions + from $A_i$ to $A_o$, which we denote by $A_i \To A_o$. + The ordering is such that $f$ is below $g$ if for all $a$ in $\ty{A_i}$, we have + $f(a)$ is below $g(a)$ in the ordering for $A_o$. +\end{itemize} + + + +\subsection{Step-Sensitive Error Ordering}\label{subsec:lock-step} + +As mentioned, the ordering on the lift of a predomain $A$ is called the +\emph{step-sensitive error-ordering} (also called ``lock-step error ordering''), +the idea being that two computations $l$ and $l'$ are related if they are in +lock-step with regard to their intensional behavior, up to $l$ erroring. +Formally, we define this ordering as follows: + +\begin{itemize} + \item $\eta\, x \ltls \eta\, y$ if $x \le_A y$. + \item $\mho \ltls l$ for all $l$ + \item $\theta\, \tilde{r} \ltls \theta\, \tilde{r'}$ if + $\later_t (\tilde{r}_t \ltls \tilde{r'}_t)$ +\end{itemize} + +We also define a heterogeneous version of this ordering between the lifts of two +different predomains $A$ and $B$, parameterized by a relation $R$ between $A$ and $B$. + +\subsection{Step-Insensitive Relation} + +We define another ordering on $\li A$, called the ``step-insensitive ordering'' or +``weak bisimilarity'', written $l \bisim l'$. +Intuitively, we say $l \bisim l'$ if they are equivalent ``up to delays''. +We introduce the notation $x \sim_A y$ to mean $x \le_A y$ and $y \le_A x$. + +The weak bisimilarity relation is defined by guarded fixpoint as follows: + +\begin{align*} + &\mho \bisim \mho \\ +% + &\eta\, x \bisim \eta\, y \text{ if } + x \sim_A y \\ +% + &\theta\, \tilde{x} \bisim \theta\, \tilde{y} \text{ if } + \later_t (\tilde{x}_t \bisim \tilde{y}_t) \\ +% + &\theta\, \tilde{x} \bisim \mho \text{ if } + \theta\, \tilde{x} = \delta^n(\mho) \text { for some $n$ } \\ +% + &\theta\, \tilde{x} \bisim \eta\, y \text{ if } + (\theta\, \tilde{x} = \delta^n(\eta\, x)) + \text { for some $n$ and $x : \ty{A}$ such that $x \sim_A y$ } \\ +% + &\mho \bisim \theta\, \tilde{y} \text { if } + \theta\, \tilde{y} = \delta^n(\mho) \text { for some $n$ } \\ +% + &\eta\, x \bisim \theta\, \tilde{y} \text { if } + (\theta\, \tilde{y} = \delta^n (\eta\, y)) + \text { for some $n$ and $y : \ty{A}$ such that $x \sim_A y$ } +\end{align*} + +\subsection{Error Domains} + + +\subsection{Globalization} + +Recall that in the above definitions, any occurrences of $\later$ were with +repsect to a fixed clock $k$. Intuitively, this corresponds to a step-indexed set. +It will be necessary to consider the ``globalization'' of these definitions, +i.e., the ``global'' behavior of the type over all potential time steps. +This is accomplished in the type theory by \emph{clock quantification} \cite{atkey-mcbride2013}, +whereby given a type $X$ parameterized by a clock $k$, we consider the type +$\forall k. X[k]$. This corresponds to leaving the step-indexed world and passing to +the usual semantics in the category of sets. + + +\section{Semantics}\label{sec:semantics} + + +\subsection{Relational Semantics} + +\subsubsection{Term Precision via the Step-Sensitive Error Ordering} +% Homogeneous vs heterogeneous term precision + +% \subsection{Logical Relations Semantics} \ No newline at end of file diff --git a/paper-new/syntax.tex b/paper-new/syntax.tex new file mode 100644 index 0000000..d4d156e --- /dev/null +++ b/paper-new/syntax.tex @@ -0,0 +1,494 @@ +\section{GTLC}\label{sec:GTLC} + +Here we describe the syntax and typing for the gradually-typed lambda calculus. +We also give the rules for syntactic type and term precision. +% We define four separate calculi: the normal gradually-typed lambda calculus, which we +% call the extensional or \emph{step-insensitive} lambda calculus ($\extlc$), +% as well as an \emph{intensional} lambda calculus +% ($\intlc$) whose syntax makes explicit the steps taken by a program. + +Before diving into the details, let us give a brief overview of what we will define. +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 show that composition of type precision derivations is admissible, as is +heterogeneous transitivity for term precision, so it will suffice to consider a new +language ($\extlcm$) in which we don't have composition of type precision derivations +or heterogeneous transitivity of term precision. + +We then 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 will be 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 a \emph{step-sensitive} (also called \emph{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{Syntax} + +The language is based on Call-By-Push-Value \cite{levy01:phd}, and as such it has two kinds of types: +\emph{value types}, representing pure values, and \emph{computation types}, represting +potentially effectful computations. +In the language, all computation types have the form $\Ret A$ for some value type $A$. +Given a value $V$ of type $A$, the term $\ret V$ views $V$ as a term of computation type $\Ret A$. +Given a term $M$ of computation type $B$, the term $\bind{x}{M}{N}$ should be thought of as +running $M$ to a value $V$ and then continuning as $N$, with $V$ in place of $x$. + + +We also have value contexts and computation contexts, where the latter can be viewed +as a pair consisting of (1) a stoup $\Sigma$, which is either empty or a hole of type $B$, +and (2) a (potentially empty) value context $\Gamma$. + +\begin{align*} + &\text{Value Types } A := \nat \alt \dyn \alt (A \ra A') \\ + &\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 \up{A}{B} V \\ + &\text{Terms } M, N := \err_B \alt \ret {V} \alt \bind{x}{M}{N} + \alt \lda{x}{M} \alt V_f\, V_x \alt + \alt \dn{A}{B} M +\end{align*} + +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 value contexts by the following rules: + +\begin{mathpar} + \inferrule* + { \gamma : \Gamma' \to \Gamma \and + \hasty{\Gamma'}{V}{A}} + { (\gamma , V/x ) \colon \Gamma' \to \Gamma , x : A } + + \inferrule* + {} + {\cdot \colon \cdot \to \cdot} +\end{mathpar} + +We define substitution for computation contexts by the following rules: + +\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 expected, with a cast between $A$ to $B$ allowed only when $A \ltdyn B$. +Notice that the upcast of a value is a value, since it always succeeds, while the downcast +of a value is a computation, since it may fail. + + \begin{mathpar} + % Err + \inferrule*{ }{\hasty {\cdot, \Gamma} {\err_B} B} + + % Zero and suc + \inferrule*{ }{\hasty \Gamma \zro \nat} + + \inferrule*{\hasty \Gamma V \nat} {\hasty \Gamma {\suc\, V} \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} {V_f \, V_x} {\Ret A'}} + + % Ret + \inferrule* + {\hasty \Gamma V A} + {\hasty {\cdot , \Gamma} {\ret V} {\Ret A}} + % TODO should this involve a Delta? + + + % 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}} + + % Upcast + \inferrule* + {A \ltdyn A' \and \hasty \Gamma V A} + {\hasty \Gamma {\up A {A'} V} {A'} } + + % Downcast + \inferrule* + {A \ltdyn A' \and \hasty {\Gamma} V {A'}} + {\hasty {\cdot, \Gamma} {\dn A {A'} V} {\Ret A}} + % TODO are the contexts correct? + + % \inferrule* + % {A \ltdyn A' \and \hasty {\Delta|_V} V {A'}} + % {\hasty {\Delta} {\dn A {A'} V} {\Ret A}} + + \end{mathpar} + + +In the equational theory, we have $\beta$ and $\eta$ laws for function type, +as well a $\beta$ and $\eta$ law for $\Ret A$. + +% TODO do we need to add a substitution rule here? +\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}} + + % 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} + +\subsection{Type Precision} + +The type precision rules specify what it means for a type $A$ to be more precise than $A'$. +We have reflexivity rules for $\dyn$ and $\nat$, as well as rules that $\nat$ is more precise than $\dyn$ +and $\dyn \ra \dyn$ is more precise than $\dyn$. +We also have a transitivity rule for composition of type precision, +and also a rule for function types stating that given $A_i \ltdyn A'_i$ and $A_o \ltdyn A'_o$, we can prove +$A_i \ra A_o \ltdyn A'_i \ra A'_o$. +Finally, we can lift a relation on value types $A \ltdyn A'$ to a relation $\Ret A \ltdyn \Ret A'$ on +computation types. + +\begin{mathpar} + \inferrule*[right = \dyn] + { }{\dyn \ltdyn\, \dyn} + + \inferrule*[right = \nat] + { }{\nat \ltdyn \nat} + + \inferrule*[right = $Inj_\nat$] + { }{\nat \ltdyn\, \dyn} + + \inferrule*[right = $\To$] + {A_i \ltdyn A'_i \and A_o \ltdyn A'_o } + {(A_i \To A_o) \ltdyn (A'_i \To A'_o)} + + \inferrule*[right=$Inj_{\To}$] + { } + {(\dyn \ra \dyn) \ltdyn\, \dyn} + + \inferrule*[right=ValComp] + {A \ltdyn A' \and A' \ltdyn A''} + {A \ltdyn A''} + + \inferrule*[right=CompComp] + {B \ltdyn B' \and B' \ltdyn B''} + {B \ltdyn B''} + + \inferrule*[right=$\Ret{}$] + {A \ltdyn A'} + {\Ret {A} \ltdyn \Ret {A'}} + + % TODO are there other rules needed for computation types? + + +\end{mathpar} + +% Type precision derivations +Note that as a consequence of this presentation of the type precision rules, we +have that if $A \ltdyn A'$, there is a unique precision derivation that witnesses this. +As in previous work, we go a step farther and make these derivations first-class objects, +known as \emph{type precision derivations}. +Specifically, for every $A \ltdyn A'$, we have a derivation $c : A \ltdyn A'$ that is constructed +using the rules above. For instance, there is a derivation $\dyn : \dyn \ltdyn \dyn$, and a derivation +$\nat : \nat \ltdyn \nat$, and if $c_i : A_i \ltdyn A_i$ and $c_o : A_o \ltdyn A'_o$, then +there is a derivation $c_i \To c_o : (A_i \To A_o) \ltdyn (A'_i \To A'_o)$. Likewise for +the remaining rules. The benefit to making these derivations explicit in the syntax is that we +can perform induction over them. +Note also that for any type $A$, we use $A$ to denote the reflexivity derivation that $A \ltdyn A$, +i.e., $A : A \ltdyn A$. +Finally, observe that for type precision derivations $c : A \ltdyn A'$ and $c' : A' \ltdyn A''$, we +can define (via the rule ValComp) their composition $c \relcomp c' : A \ltdyn A''$. +The same holds for computation type precision derivations. +This notion will be used below in the statement of transitivity of the term precision relation. + +\subsection{Term Precision} + +We allow for a \emph{heterogeneous} term precision judgment on terms values $V$ of type +$A$ and $V'$ of type $A'$ provided that $A \ltdyn A'$ holds. Likewise, for computation +types $B \ltdyn B'$, if $M$ has type $B$ and $M'$ has type $B'$, we can form the judgment +that $M \ltdyn M'$. + +% Type precision contexts +In order to deal with open terms, we will need the notion of a type precision \emph{context}, which we denote +$\gamlt$. This is similar to a normal context but instead of mapping variables to types, +it maps variables $x$ to related types $A \ltdyn B$, where $x$ has type $A$ in the left-hand term +and $B$ in the right-hand term. We may also write $x : d$ where $d : A \ltdyn B$ to indicate this. +Another way of thinking of type precision contexts is as a zipped pair of contexts $\Gamma^l, \Gamma^r$ +with the same domain such that $\Gamma_l(x) \ltdyn \Gamma_r(x)$ for each $x$ in the domain. +Similarly, we have computation type precision contexts $\Delta^\ltdyn$. Similar to ``normal'' computation +type precision contexts $\Delta$, these consist of (1) a stoup $\Sigma$ which is either empty or +has a hole $\hole{d}$ for some computation type precision derivation $d$, and (2) a value type precision context +$\Gamma^\ltdyn$. +% + +As with type precision derivations, we write $\Gamma$ to mean the context of reflexivity derivations +$\Gamma(x) \ltdyn \Gamma(x)$. Likewise for computation type precision contexts. +Furthermore, we write $\gamlt_1 \relcomp \gamlt_2$ to denote the ``composition'' of $\gamlt_1$ and $\gamlt_2$ +--- that is, the precision context whose value at $x$ is the type precision derivation +$\gamlt_1(x) \relcomp \gamlt_2(x)$. This of course assumes that each of the type precision +derivations is composable, i.e., that the RHS of $\gamlt_1(x)$ is the same as the left-hand side of $\gamlt_2(x)$. +We define the same for computation type precision contexts $\deltalt_1$ and $\deltalt_2$, +provided that both the computation type precision contexts have the same ``shape'', which is defined as +(1) either the stoup is empty in both, or the stoup has a hole in both, say $\hole{d}$ and $\hole{d'}$ +where $d$ and $d'$ are composable, and (2) their value type precision contexts are composable as described above. +% TODO reflexivity contexts and composition of contexts + +The rules for term precision come in two forms. We first have the \emph{congruence} rules, +one for each term constructor. These assert that the term constructors respect term precision. +The congruence rules are as follows: + +\begin{mathpar} + + \inferrule*[right = Var] + { c : A \ltdyn B \and \gamlt(x) = (A, B) } { \etmprec {\gamlt} x x c } + + \inferrule*[right = Zro] + { } {\etmprec \gamlt \zro \zro \nat } + + \inferrule*[right = Suc] + { \etmprec \gamlt V {V'} \nat } {\etmprec \gamlt {\suc\, V} {\suc\, V'} \nat} + + \inferrule*[right = Lambda] + { c_i : A_i \ltdyn A'_i \and + c_o : A_o \ltdyn A'_o \and + \etmprec {\cdot , \gamlt , x : c_i} {M} {M'} {\Ret c_o} } + { \etmprec \gamlt {\lda x M} {\lda x {M'}} {(c_i \ra c_o)} } + + \inferrule*[right = App] + { c_i : A_i \ltdyn A'_i \and + c_o : A_o \ltdyn A'_o \\\\ + \etmprec \gamlt {V_f} {V_f'} {(c_i \ra c_o)} \and + \etmprec \gamlt {V_x} {V_x'} {c_i} + } + { \etmprec {\cdot , \gamlt} {V_f\, V_x} {V_f'\, V_x'} {\Ret {c_o}}} + + \inferrule*[right = Ret] + {\etmprec {\gamlt} V {V'} c} + {\etmprec {\cdot , \gamlt} {\ret V} {\ret V'} {\Ret c}} + + \inferrule*[right = Bind] + {\etmprec {\deltalt} {M} {M'} {\Ret c} \and + \etmprec {\cdot , \deltalt|_V , x : c} {N} {N'} {d} } + {\etmprec {\deltalt} {\bind {x} {M} {N}} {\bind {x} {M'} {N'}} {d}} +\end{mathpar} + +We then have additional equational axioms, including transitivity, $\beta$ and $\eta$ laws, and +rules characterizing upcasts as least upper bounds, and downcasts as greatest lower bounds. + +We write $M \equidyn N$ to mean that both $M \ltdyn N$ and $N \ltdyn M$. + +% TODO adapt these for value/computation distinction +% TODO substitution rule +\begin{mathpar} + \inferrule*[right = $\err$] + { \hasty {\deltalt_l} M B } + {\etmprec {\Delta} {\err_B} M B} + + \inferrule*[right = Transitivity] + { d : B \ltdyn B' \and d' : B' \ltdyn B'' \\\\ + \etmprec {\deltalt_1} {M} {M'} {d} \and + \etmprec {\deltalt_2} {M'} {M''} {d'} } + {\etmprec {\deltalt_1 \relcomp \deltalt_2} {M} {M''} {d \relcomp d'} } + + \inferrule*[right = $\beta$-fun] + { \hasty {\Gamma, x : A_i} M {A_o} \and + \hasty {\Gamma} V {A_i} } + { \etmequidyn \gamlt {(\lda x M)\, V} {M[V/x]} {A_o} } + + \inferrule*[right = $\eta$-fun] + { \hasty {\Gamma} {V} {A_i \To A_o} } + { \etmequidyn \gamlt {\lda x (V\, x)} V {A_i \To A_o} } + + \inferrule*[right = $\beta$-ret] + {} + {\bind{x}{\ret\, V}{N} = N[V/x]} + + \inferrule*[right = $\eta$-ret] + {\hasty {\hole{\Ret A} , \Gamma} {M} {B}} + {\hole{\Ret A}, \Gamma \vdash M = \bind{x}{\bullet}{M[\ret\, x]}} + + \inferrule*[right = UpR] + { d : A \ltdyn B \and + \hasty \Gamma M A } + { \etmprec \gamlt M {\up A B M} d } + + \inferrule*[right = UpL] + { d : A \ltdyn B \and + \etmprec \gamlt M N d } + { \etmprec \gamlt {\up A B M} N B } + + \inferrule*[right = DnL] + { d : A \ltdyn B \and + \hasty \Gamma M B } + { \etmprec \gamlt {\dn A B M} M d } + + \inferrule*[right = DnR] + { d : A \ltdyn B \and + \etmprec \gamlt M N d } + { \etmprec \gamlt M {\dn A B N} A } +\end{mathpar} + +% TODO explain the least upper bound/greatest lower bound rules +The rules UpR, UpL, DnL, and DnR were introduced in \cite{new-licata18} as a means +of cleanly axiomatizing the intended behavior of casts in a way that +doesn't depend on the specific constructs of the language. +Intuitively, rule UpR says that the upcast of $M$ is an upper bound for $M$ +in that $M$ may error more, and UpL says that the upcast is the \emph{least} +such upper bound, in that it errors more than any other upper bound for $M$. +Conversely, DnL says that the downcast of $M$ is a lower bound, and DnR says +that it is the \emph{greatest} lower bound. +% These rules provide a clean axiomatization of the behavior of casts that doesn't +% depend on the specific constructs of the language. + + +\subsection{Removing Transitivity} + +The first observation we make is that transitivity of type precision, and heterogeneous +transitivity of term precision, are admissible. That is, consider a related language which +is the same as $\extlc$ excpet that we have removed the composition rule for type precision and +the heterogeneous transitivity rule for type precision. Denote this language by $\extlcm$. +We claim that in this new language, the rules we removed are derivable from the remaining rules. +More specifically, consider the following situation in $\extlc$: + + +\begin{figure} +% https://q.uiver.app/?q=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 +\[\begin{tikzcd}[ampersand replacement=\&] + \Gamma \&\& A \\ + {\Gamma'} \&\& {A'} \\ + {\Gamma''} \&\& {A''} + \arrow["{M''}", from=3-1, to=3-3] + \arrow["{M'}", from=2-1, to=2-3] + \arrow["M", from=1-1, to=1-3] + \arrow["{\ltdyn_c}"{marking}, draw=none, from=1-3, to=2-3] + \arrow["{\ltdyn_{c'}}"{marking}, draw=none, from=2-3, to=3-3] + \arrow["{\ltdyn_{\Gamma_c}}"{marking}, draw=none, from=1-1, to=2-1] + \arrow[draw=none, from=2-1, to=3-1] + \arrow["{\ltdyn_{\Gamma_{c'}}}"{marking}, draw=none, from=2-1, to=3-1] +\end{tikzcd}\] + \caption{Heterogeneous transitivity.} + \label{fig:transitivity} + \end{figure} + + + \textbf{TODO} + + + +\subsection{Removing Casts} + +% We now observe that all casts, except those between $\nat$ and $\dyn$ +% and between $\dyn \ra \dyn$ and $\dyn$, are admissible, in the sense that +% we can start from $\extlcm$, remove casts except the aforementioned ones, +% and in the resulting language we will be able to derive the other casts. + +We now observe that all casts, except those between $\nat$ and $\dyn$ +and between $\dyn \ra \dyn$ and $\dyn$, are admissible. +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 $\dyn \ra \dyn$. We claim that in $\extlcmm$, all of the casts +present in $\extlcm$ are derivable. +It will suffice to verify that casts for function type are derivable. +This holds becasue function casts are constructed inductively from the cast +for their domain and codomain. The base case is one of the casts inolving $\nat$ +or $\dyn \ra \dyn$ which are present in $\extlcmm$ as injections and case inspections. + + +The resulting calculus now lacks transitivity of type precision, heterogeneous +transitivity of term precision, and arbitrary casts. +In this setting, rather than type precision, it makes more sense to speak of +arbitrary monotone relations on types, which we denote by $A \rel A'$. +We have relations on value types, as well as on computation types. + +\begin{align*} + &\text{Value Relations } R := \nat \alt \dyn \alt (R \ra R) \alt\, \dyn\, \\ + &\text{Computation Relations } S := \Lift R \\ + &\text{Value Relation Contexts } \Gamma^{\rel} := \cdot \alt \Gamma^{\rel} , A^{\rel} (x_l : A_l , x_r : A_r)\\ + &\text{Computation Relation Contexts } \Delta^{\rel} := \cdot \alt \hole{B^{\rel}} \alt + \Delta^{\rel} , A^{\rel} (x_l : A_l , x_r : A_r) \\ +\end{align*} + +% TODO rules for relations + + +\subsection{The Step-Sensitive Lambda Calculus} + +% \textbf{TODO: Subject to change!} + +From here, we define an \emph{step-sensitive} (also called \emph{intensional}) +GSTLC. As mentioned, this language makes the intensional stepping behavior of programs +explicit in the syntax. We do this by adding a syntactic ``later'' type and a +syntactic $\theta$ that maps terms of type later $A$ to terms of type $A$. + +In the step-sensitive 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 step-sensitive syntax to step-insensitive syntax +by induction on the step-sentive types and terms. +The basic idea is that the syntactic type $\later A$ erases to $A$, +and $\nxt$ and $\theta$ erase to the identity. + + + +% \begin{align*} +% &\text{Terms } M, N := \err_A, \, \dots \, \theta_s (\tilde{M}) \\ +% \end{align*} + +% \begin{mathpar} +% \inferrule*[] +% { \later_t (\hasty \Gamma {M_t} A) } +% { \hasty \Gamma {\theta_s M} {A} } +% \end{mathpar} + +% \begin{mathpar} +% \inferrule*[] +% { \later_t (\itmprec \gamlt {M_t} {N_t} d) } +% { \itmprec \gamlt {\theta_s M} {\theta_s N} d } +% \end{mathpar} + +% Recall that $\later_t$ is a dependent form of $\later$ where the argument is allowed +% to mention $t$. In particular, here we apply the tick $t$ to the later-terms $M$ and $N$ to get +% ``now"-terms $M_t$ and $N_t$. + + +\subsection{Quotienting by Syntactic Bisimilarity} + diff --git a/paper-new/technical-background.tex b/paper-new/technical-background.tex new file mode 100644 index 0000000..8a97604 --- /dev/null +++ b/paper-new/technical-background.tex @@ -0,0 +1,286 @@ +\section{Technical Background}\label{sec:technical-background} + +\subsection{Gradual Typing} + +% Cast calculi +In a gradually-typed language, the mixing of static and dynamic code is seamless, in that +the dynamically typed parts are checked at runtime. This type checking occurs at the elimination +forms of the language (e.g., pattern matching, field reference, etc.). +Gradual languages are generally elaborated to a \emph{cast calculus}, in which the dynamic +type checking is made explicit through the insertion of \emph{type casts}. + +% Up and down casts +In a cast calculus, there is a relation $\ltdyn$ on types such that $A \ltdyn B$ means that +$A$ is a \emph{more precise} type than $B$. +There a dynamic type $\dyn$ with the property that $A \ltdyn\, \dyn$ for all $A$. +% +If $A \ltdyn B$, a term $M$ of type $A$ may be \emph{up}casted to $B$, written $\up A B M$, +and a term $N$ of type $B$ may be \emph{down}casted to $A$, written $\dn A B N$. +Upcasts always succeed, while downcasts may fail at runtime. +% +% Syntactic term precision +We also have a notion of \emph{syntactic term precision}. +If $A \ltdyn B$, and $M$ and $N$ are terms of type $A$ and $B$ respectively, we write +$M \ltdyn N : A \ltdyn B$ to mean that $M$ is more precise than $N$, i.e., $M$ and $N$ +behave the same except that $M$ may error more. + +% Modeling the dynamic type as a recursive sum type? +% Observational equivalence and approximation? + +% synthetic guarded domain theory, denotational semantics therein + +\subsection{Difficulties in Prior Semantics} + % Difficulties in prior semantics + + In this work, we compare our approach to proving graduality to the approach + introduced by New and Ahmed \cite{new-ahmed2018} which constructs a step-indexed + logical relations model and shows that this model is sound with respect to their + notion of contextual error approximation. + + Because the dynamic type is modeled as a non-well-founded + recursive type, their logical relation needs to be paramterized by natural numbers + to restore well-foundedness. This technique is known as a \emph{step-indexed logical relation}. + Reasoning about step-indexed logical relations + can be tedious and error-prone, and there are some very subtle aspects that must + be taken into account in the proofs. Figure \ref{TODO} shows an example of a step-indexed logical + relation for the gradually-typed lambda calculus. + + In particular, the prior approach of New and Ahmed requires two separate logical + relations for terms, one in which the steps of the left-hand term are counted, + and another in which the steps of the right-hand term are counted. + Then two terms $M$ and $N$ are related in the ``combined'' logical relation if they are + related in both of the one-sided logical relations. Having two separate logical relations + complicates the statement of the lemmas used to prove graduality, becasue any statement that + involves a term stepping needs to take into account whether we are counting steps on the left + or the right. Some of the differences can be abstracted over, but difficulties arise for properties %/results + as fundamental and seemingly straightforward as transitivty. + + Specifically, for transitivity, we would like to say that if $M$ is related to $N$ at + index $i$ and $N$ is related to $P$ at index $i$, then $M$ is related to $P$ at $i$. + But this does not actually hold: we requrie that one of the two pairs of terms + be related ``at infinity'', i.e., that they are related at $i$ for all $i \in \mathbb{N}$. + Which pair is required to satisfy this depends on which logical relation we are considering, + (i.e., is it counting steps on the left or on the right), + and so any argument that uses transitivity needs to consider two cases, one + where $M$ and $N$ must be shown to be related for all $i$, and another where $N$ and $P$ must + be related for all $i$. % This may not even be possible to show in some scenarios! + + % These complications introduced by step-indexing lead one to wonder whether there is a + % way of proving graduality without relying on tedious arguments involving natural numbers. + % An alternative approach, which we investigate in this paper, is provided by + % \emph{synthetic guarded domain theory}, as discussed below. + % Synthetic guarded domain theory allows the resulting logical relation to look almost + % identical to a typical, non-step-indexed logical relation. + +\subsection{Synthetic Guarded Domain Theory} +One way to avoid the tedious reasoning associated with step-indexing is to work +axiomatically inside of a logical system that can reason about non-well-founded recursive +constructions while abstracting away the specific details of step-indexing required +if we were working analytically. +The system that proves useful for this purpose is called \emph{synthetic guarded +domain theory}, or SGDT for short. We provide a brief overview here, but more +details can be found in \cite{birkedal-mogelberg-schwinghammer-stovring2011}. + +SGDT offers a synthetic approach to domain theory that allows for guarded recursion +to be expressed syntactically via a type constructor $\later : \type \to \type$ +(pronounced ``later''). The use of a modality to express guarded recursion +was introduced by Nakano \cite{Nakano2000}. +% +Given a type $A$, the type $\later A$ represents an element of type $A$ +that is available one time step later. There is an operator $\nxt : A \to\, \later A$ +that ``delays'' an element available now to make it available later. +We will use a tilde to denote a term of type $\later A$, e.g., $\tilde{M}$. + +% TODO later is an applicative functor, but not a monad + +There is a \emph{guarded fixpoint} operator + +\[ + \fix : \forall T, (\later T \to T) \to T. +\] + +That is, to construct a term of type $T$, it suffices to assume that we have access to +such a term ``later'' and use that to help us build a term ``now''. +This operator satisfies the axiom that $\fix f = f (\nxt (\fix f))$. +In particular, this axiom applies to propositions $P : \texttt{Prop}$; proving +a statement in this manner is known as $\lob$-induction. + +The operators $\later$, $\next$, and $\fix$ described above can be indexed by objects +called \emph{clocks}. A clock serves as a reference relative to which steps are counted. +For instance, given a clock $k$ and type $T$, the type $\later^k T$ represents a value of type +$T$ one unit of time in the future according to clock $k$. +If we only ever had one clock, then we would not need to bother defining this notion. +However, the notion of \emph{clock quantification} is crucial for encoding coinductive types using guarded +recursion, an idea first introduced by Atkey and McBride \cite{atkey-mcbride2013}. + + +% Clocked Cubical Type Theory +\subsubsection{Ticked Cubical Type Theory} +% TODO motivation for Clocked Cubical Type Theory, e.g., delayed substitutions? + +In Ticked Cubical Type Theory \cite{TODO}, there is an additional sort +called \emph{ticks}. Given a clock $k$, a tick $t : \tick k$ serves +as evidence that one unit of time has passed according to the clock $k$. +The type $\later A$ is represented as a function from ticks of a clock $k$ to $A$. +The type $A$ is allowed to depend on $t$, in which case we write $\later^k_t A$ +to emphasize the dependence. + +% TODO next as a function that ignores its input tick argument? + +The rules for tick abstraction and application are similar to those of dependent +$\Pi$ types. +In particular, if we have a term $M$ of type $\later^k A$, and we +have available in the context a tick $t' : \tick k$, then we can apply the tick to $M$ to get +a term $M[t'] : A[t'/t]$. We will also write tick application as $M_t$. +Conversely, if in a context $\Gamma, t : \tick k$ we have that $M$ has type $A$, +then in context $\Gamma$ we have $\lambda t.M$ has type $\later A$. % TODO dependent version? + +The statements in this paper have been formalized in a variant of Agda called +Guarded Cubical Agda \cite{TODO}, an implementation of Clocked Cubical Type Theory. + + +% TODO axioms (clock irrelevance, tick irrelevance)? + + +\subsubsection{The Topos of Trees Model} + +The topos of trees model provides a useful intuition for reasoning in SGDT +\cite{birkedal-mogelberg-schwinghammer-stovring2011}. +This section presupposes knowledge of category theory and can be safely skipped without +disrupting the continuity. + +The topos of trees $\calS$ is the presheaf category $\Set^{\omega^o}$. +% +We assume a universe $\calU$ of types, with encodings for operations such +as sum types (written as $\widehat{+}$). There is also an operator +$\laterhat \colon \later \calU \to \calU$ such that +$\El(\laterhat(\nxt A)) =\,\, \later \El(A)$, where $\El$ is the type corresponding +to the code $A$. + +An object $X$ is a family $\{X_i\}$ of sets indexed by natural numbers, +along with restriction maps $r^X_i \colon X_{i+1} \to X_i$ (see Figure \ref{fig:topos-of-trees-object}). +These should be thought of as ``sets changing over time", where $X_i$ is the view of the set if we have $i+1$ +time steps to reason about it. +We can also think of an ongoing computation, with $X_i$ representing the potential results of the computation +after it has run for $i+1$ steps. + +\begin{figure} + % https://q.uiver.app/?q=WzAsNSxbMiwwLCJYXzIiXSxbMCwwLCJYXzAiXSxbMSwwLCJYXzEiXSxbMywwLCJYXzMiXSxbNCwwLCJcXGRvdHMiXSxbMiwxLCJyXlhfMCIsMl0sWzAsMiwicl5YXzEiLDJdLFszLDAsInJeWF8yIiwyXSxbNCwzLCJyXlhfMyIsMl1d +\[\begin{tikzcd}[ampersand replacement=\&] + {X_0} \& {X_1} \& {X_2} \& {X_3} \& \dots + \arrow["{r^X_0}"', from=1-2, to=1-1] + \arrow["{r^X_1}"', from=1-3, to=1-2] + \arrow["{r^X_2}"', from=1-4, to=1-3] + \arrow["{r^X_3}"', from=1-5, to=1-4] +\end{tikzcd}\] + \caption{An example of an object in the topos of trees.} + \label{fig:topos-of-trees-object} +\end{figure} + +A morphism from $\{X_i\}$ to $\{Y_i\}$ is a family of functions $f_i \colon X_i \to Y_i$ +that commute with the restriction maps in the obvious way, that is, +$f_i \circ r^X_i = r^Y_i \circ f_{i+1}$ (see Figure \ref{fig:topos-of-trees-morphism}). + +\begin{figure} +% https://q.uiver.app/?q=WzAsMTAsWzIsMCwiWF8yIl0sWzAsMCwiWF8wIl0sWzEsMCwiWF8xIl0sWzMsMCwiWF8zIl0sWzQsMCwiXFxkb3RzIl0sWzAsMSwiWV8wIl0sWzEsMSwiWV8xIl0sWzIsMSwiWV8yIl0sWzMsMSwiWV8zIl0sWzQsMSwiXFxkb3RzIl0sWzIsMSwicl5YXzAiLDJdLFswLDIsInJeWF8xIiwyXSxbMywwLCJyXlhfMiIsMl0sWzQsMywicl5YXzMiLDJdLFs2LDUsInJeWV8wIiwyXSxbNyw2LCJyXllfMSIsMl0sWzgsNywicl5ZXzIiLDJdLFs5LDgsInJeWV8zIiwyXSxbMSw1LCJmXzAiLDJdLFsyLDYsImZfMSIsMl0sWzAsNywiZl8yIiwyXSxbMyw4LCJmXzMiLDJdXQ== +\[\begin{tikzcd}[ampersand replacement=\&] + {X_0} \& {X_1} \& {X_2} \& {X_3} \& \dots \\ + {Y_0} \& {Y_1} \& {Y_2} \& {Y_3} \& \dots + \arrow["{r^X_0}"', from=1-2, to=1-1] + \arrow["{r^X_1}"', from=1-3, to=1-2] + \arrow["{r^X_2}"', from=1-4, to=1-3] + \arrow["{r^X_3}"', from=1-5, to=1-4] + \arrow["{r^Y_0}"', from=2-2, to=2-1] + \arrow["{r^Y_1}"', from=2-3, to=2-2] + \arrow["{r^Y_2}"', from=2-4, to=2-3] + \arrow["{r^Y_3}"', from=2-5, to=2-4] + \arrow["{f_0}"', from=1-1, to=2-1] + \arrow["{f_1}"', from=1-2, to=2-2] + \arrow["{f_2}"', from=1-3, to=2-3] + \arrow["{f_3}"', from=1-4, to=2-4] +\end{tikzcd}\] + \caption{An example of a morphism in the topos of trees.} + \label{fig:topos-of-trees-morphism} +\end{figure} + + +The type operator $\later$ is defined on an object $X$ by +$(\later X)_0 = 1$ and $(\later X)_{i+1} = X_i$. +The restriction maps are given by $r^\later_0 =\, !$, where $!$ is the +unique map into $1$, and $r^\later_{i+1} = r^X_i$. +The morphism $\nxt^X \colon X \to \later X$ is defined pointwise by +$\nxt^X_0 =\, !$, and $\nxt^X_{i+1} = r^X_i$. It is easily checked that +this satisfies the commutativity conditions required of a morphism in $\calS$. +% +Given a morphism $f \colon \later X \to X$, i.e., a family of functions +$f_i \colon (\later X)_i \to X_i$ that commute with the restrictions in the appropriate way, +we define $\fix(f) \colon 1 \to X$ pointwise +by $\fix(f)_i = f_{i} \circ \dots \circ f_0$. +This can be visualized as a diagram in the category of sets as shown in +Figure \ref{fig:topos-of-trees-fix}. +% Observe that the fixpoint is a \emph{global element} in the topos of trees. +% Global elements allow us to view the entire computation on a global level. + + +% https://q.uiver.app/?q=WzAsOCxbMSwwLCJYXzAiXSxbMiwwLCJYXzEiXSxbMywwLCJYXzIiXSxbMCwwLCIxIl0sWzAsMSwiWF8wIl0sWzEsMSwiWF8xIl0sWzIsMSwiWF8yIl0sWzMsMSwiWF8zIl0sWzMsNCwiZl8wIl0sWzAsNSwiZl8xIl0sWzEsNiwiZl8yIl0sWzIsNywiZl8zIl0sWzAsMywiISIsMl0sWzEsMCwicl5YXzAiLDJdLFsyLDEsInJeWF8xIiwyXSxbNSw0LCJyXlhfMCJdLFs2LDUsInJeWF8xIl0sWzcsNiwicl5YXzIiXV0= +% \[\begin{tikzcd}[ampersand replacement=\&] +% 1 \& {X_0} \& {X_1} \& {X_2} \\ +% {X_0} \& {X_1} \& {X_2} \& {X_3} +% \arrow["{f_0}", from=1-1, to=2-1] +% \arrow["{f_1}", from=1-2, to=2-2] +% \arrow["{f_2}", from=1-3, to=2-3] +% \arrow["{f_3}", from=1-4, to=2-4] +% \arrow["{!}"', from=1-2, to=1-1] +% \arrow["{r^X_0}"', from=1-3, to=1-2] +% \arrow["{r^X_1}"', from=1-4, to=1-3] +% \arrow["{r^X_0}", from=2-2, to=2-1] +% \arrow["{r^X_1}", from=2-3, to=2-2] +% \arrow["{r^X_2}", from=2-4, to=2-3] +% \end{tikzcd}\] + +% \begin{figure} +% % https://q.uiver.app/?q=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 +% \[\begin{tikzcd}[ampersand replacement=\&] +% {\fix(f)_0} \& {\fix(f)_1} \& {\fix(f)_2} \& {\fix(f)_3} \\ +% 1 \& 1 \& 1 \& 1 \\ +% {X_0} \& {X_0} \& {X_0} \& {X_0} \& \cdots \\ +% \& {X_1} \& {X_1} \& {X_1} \\ +% \&\& {X_2} \& {X_2} \\ +% \&\&\& {X_3} +% \arrow["{f_0}", from=2-1, to=3-1] +% \arrow["{f_1}", from=3-2, to=4-2] +% \arrow["{f_0}", from=2-2, to=3-2] +% \arrow["{f_0}", from=2-3, to=3-3] +% \arrow["{f_1}", from=3-3, to=4-3] +% \arrow["{f_2}", from=4-3, to=5-3] +% \arrow["{f_0}", from=2-4, to=3-4] +% \arrow["{f_1}", from=3-4, to=4-4] +% \arrow["{f_2}", from=4-4, to=5-4] +% \arrow["{f_3}", from=5-4, to=6-4] +% \end{tikzcd}\] +% \caption{The first few approximations to the guarded fixpoint of $f$.} +% \label{fig:topos-of-trees-fix-approx} +% \end{figure} + + +\begin{figure} + % https://q.uiver.app/?q=WzAsNixbMywwLCIxIl0sWzAsMiwiWF8wIl0sWzIsMiwiWF8xIl0sWzQsMiwiWF8yIl0sWzYsMiwiWF8zIl0sWzgsMiwiXFxkb3RzIl0sWzIsMSwicl5YXzAiXSxbNCwzLCJyXlhfMiJdLFswLDEsIlxcZml4KGYpXzAiLDFdLFswLDIsIlxcZml4KGYpXzEiLDFdLFswLDMsIlxcZml4KGYpXzIiLDFdLFswLDQsIlxcZml4KGYpXzMiLDFdLFswLDUsIlxcY2RvdHMiLDMseyJzdHlsZSI6eyJib2R5Ijp7Im5hbWUiOiJub25lIn0sImhlYWQiOnsibmFtZSI6Im5vbmUifX19XSxbMywyLCJyXlhfMSJdLFs1LDQsInJeWF8zIl1d + \[\begin{tikzcd}[ampersand replacement=\&,column sep=2.25em] + \&\&\& 1 \\ + \\ + {X_0} \&\& {X_1} \&\& {X_2} \&\& {X_3} \&\& \dots + \arrow["{r^X_0}", from=3-3, to=3-1] + \arrow["{r^X_2}", from=3-7, to=3-5] + \arrow["{\fix(f)_0}"{description}, from=1-4, to=3-1] + \arrow["{\fix(f)_1}"{description}, from=1-4, to=3-3] + \arrow["{\fix(f)_2}"{description}, from=1-4, to=3-5] + \arrow["{\fix(f)_3}"{description}, from=1-4, to=3-7] + \arrow["\cdots"{marking}, draw=none, from=1-4, to=3-9] + \arrow["{r^X_1}", from=3-5, to=3-3] + \arrow["{r^X_3}", from=3-9, to=3-7] + \end{tikzcd}\] + \caption{The guarded fixpoint of $f$.} + \label{fig:topos-of-trees-fix} +\end{figure} + +% TODO global elements? \ No newline at end of file -- GitLab