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type-safe-programming.tex
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\documentclass{beamer}
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% \usepackage{codehigh}
% \let\fverb\fakeverb
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% Korean-English, e.g., \alt{메모리 재활용}{garbage collection}
% \alt*을 통해 찾아보기에 추가, \alt는 단순히 한영 병기
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% C++ logo
\usepackage{relsize}
\def\cplus{\raisebox{.4ex}{\relsize{-3}{\texttt{+}}}}
\def\C++{C\nolinebreak\hspace{-.081em}\cplus\nolinebreak\hspace{-0.02em}\cplus}
\usetikzlibrary{calc,decorations.pathmorphing,shapes}
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\usepackage{hyperref}
%% Default info
\title{타입으로 안전하게 프로그래밍하기}
\institute{SNULife 개발팀}
\date{2024년 1월 13일}
\author{이재호}
\begin{document}
\maketitle
\begin{frame}[c, fragile]
\frametitle{목차}
\tableofcontents[subsectionstyle=hide]
\end{frame}
\section{타입 시스템이란?}
\begin{frame}[c, fragile]
\frametitle{``자유로운'' 언어의 숙명}
\only<1>{\includegraphics[width=\linewidth]{img/es-lint.png}}
\begin{onlyenv}<2>
\begin{quote}
This rule aims to detect some cases where the use of optional chaining doesn’t prevent runtime errors.
In particular, it flags optional chaining expressions in positions where short-circuiting to \texttt{undefined} causes throwing a TypeError afterward.
\end{quote}%
\begin{jscode}
/*eslint no-unsafe-optional-chaining: "error"*/
(obj?.foo)();
(obj?.foo).bar;
(foo?.()).bar;
(foo?.()).bar();
\end{jscode}
\end{onlyenv}
\only<3>{\includegraphics[width=\linewidth]{img/pylint.png}}
\end{frame}
\begin{frame}[c, fragile]
\frametitle{동적 타입과 정적 타입}
동적 타입 시스템은 사실 ``타입 시스템''이 아닌 실행 중 타입 태그 검사
{\footnotesize\begin{description}
\item[동적 타입 언어] \Alt*{파이썬}{Python}, \Alt*{자바스크립트}{JavaScript}, \Alt*{스킴}{Scheme}, \Alt*{라켓}{Racket}, \Alt*{루아}{Lua}, \Alt*{루비}{Ruby}, \Alt*{펄}{Perl}, \Alt*{라쿠}{Raku}, \ldots
\item[정적 타입 언어] \Alt*{스위프트}{Swift}, \Alt*{코틀린}{Kotlin}, \Alt*{러스트}{Rust}, \Alt*{오캐멀}{OCaml}, \Alt*{하스켈}{Haskell}, \Alt*{리스크립트}{ReScript}, \Alt*{엚}{Elm}, C, \C++, \Alt*{자바}{Java}, \ldots
\end{description}}
\begin{onlyenv}<2>
\begin{pycode}
Python 3.11.4 (main, Jul 4 2023, 19:38:47) [Clang 14.0.3 (clang-1403.0.22.14.1)] on darwin
Type "help", "copyright", "credits" or "license" for more information.
>>> def poison():
... "hello" + 42
...
>>> poison()
Traceback (most recent call last):
File "<stdin>", line 1, in <module>
File "<stdin>", line 2, in poison
TypeError: can only concatenate str (not "int") to str
\end{pycode}
\end{onlyenv}
\begin{onlyenv}<3>
\begin{swiftcode}
Welcome to Apple Swift version 5.9.2 (swiftlang-5.9.2.2.56 clang-1500.1.0.2.5).
Type :help for assistance.
1> func poison() {
2. "hello" + 42
3. }
error: repl.swift:2:13: error: binary operator '+' cannot be applied to operands of type 'String' and 'Int'
"hello" + 42
~~~~~~~ ^ ~~
repl.swift:2:13: note: overloads for '+' exist with these partially matching parameter lists: (Int, Int), (String, String)
"hello" + 42
^
\end{swiftcode}
\end{onlyenv}
\begin{onlyenv}<4>
\begin{ocamlcode}
OCaml version 5.1.0
Enter #help;; for help.
# let poison () = "hello" + 42;;
Error: This expression has type string but an expression was expected of type
int
\end{ocamlcode}
\end{onlyenv}
\begin{onlyenv}<5>
\begin{jscode}
Welcome to Node.js v20.10.0.
Type ".help" for more information.
> function poison() {
... "hello" + 42
... }
undefined
> poison()
undefined
> "hello" + 42
'hello42'
\end{jscode}
???
\end{onlyenv}
\only<6>{\begin{center}\includegraphics[width=0.27\linewidth]{img/js-meme.png}\end{center}}
\end{frame}
\begin{frame}[c, fragile]
\frametitle{타입 시스템}
타입 시스템은 \textbf{실행 전}에 프로그램이 \textbf{잘} 실행될 수 있는지를 \textbf{겉모습}으로 검사하는 방법
\begin{description}
\item[실행 전에]<2-> 오류를 \Alt*{안전하게}{sound}\ 잡아냄
\item[잘]<3-> 만족해야 하는 성질을 타입 시스템에 녹여낼 수 있고, 실행 중에 어긋나는 일이 없음
\item[겉모습]<4->, 즉 문법적으로 검사를 할 수 있음
\end{description}
\end{frame}
\section{타입 시스템의 장점}
\subsection{\Alt*{안전성}{Soundness}}
\begin{frame}[c, fragile]
\frametitle{잘 타입된 프로그램은 잘못되지 않는다}
% TODO: better quote
``Well-type programs cannot go wrong'' \smaller{--- Robin Milner, in \textit{A Theory of Type Polymorphism in Programming} (1978)}
(\Alt*{안전한}{Sound}) 타입 시스템을 갖춘 언어에서는 실행 중 타입 에러가 발생하지 않는다!
\end{frame}
\begin{frame}[c, fragile]
\frametitle{안전한 타입 시스템이란?}
안전성은 실행 중에 ``잘'' 만족해야 하는 성질이 어긋나지 않는 것을 의미
\begin{itemize}
\item 그 성질은 언어마다 다름
\item 통상적인 타입 시스템은 실행 과정에서 계산된 값이 실제로 타입된 것과 일치하는지를 의미
\end{itemize}
\pause 안전한 타입 시스템이 중요한 이유
\begin{itemize}
\item 타입을 통해 원하는 성질을 표현할 수 있다면, 실제로 값이 그 성질을 만족한다고 신뢰할 수 있음
\pause\begin{itemize}
\item 리스트가 비어있지 않다는 성질을 \textbf{보장}하는 타입?
\end{itemize}
\pause\item 이러한 성질이 충분히 강력하다면, 테스트나 다른 프로그램 검증 기법 없이도 확신할 수 있음
\pause\begin{itemize}
\item 이것의 극단은 \Alt*{콕}{Coq}, \Alt*{아그다}{Agda}, \Alt*{린}{Lean}\ 등과 같은 \Alt*{값에 기댄 타입 언어}{dependently typed language}
\end{itemize}
\end{itemize}
\end{frame}
\note[itemize]{
\item 비어있지 않는 리스트 타입은 뒤에 일반화된 합과 곱 타입을 사용해 보여줄 예정
}
\begin{frame}[c, fragile]
\frametitle{\Alt*{조금씩 타입 붙이기}{Gradual typing}}
동적인 언어에 타입 시스템을 도입하는 방법
\begin{itemize}
\item 안전성에 방점을 두지 않음
\item 버그를 잡는데 도움
\end{itemize}
\pause\begin{description}
\item[자바스크립트] 타입스크립트, \Alt*{플로우}{Flow}
\item[파이썬] \Alt*{마이파이}{MyPy}, \Alt*{파이어}{Pyre}, \Alt*{파이라이트}{Pyright}, \Alt*{파이타이프}{Pytype}
\item[루비] \Alt*{소르베}{Sorbet}
\item[라켓] \Alt*{타입 라켓}{Typed Racket}
\item[라쿠] 언어 자체에 내장
\end{description}
\end{frame}
\begin{frame}[c, fragile]
\frametitle{안전하지 않은 타입 시스템: 타입스크립트}
타입스크립트는 안전하지 않은 타입 시스템
\begin{center}
\includegraphics[width=0.6\linewidth]{img/ts-unsound.png}
\end{center}
\begin{tscode}
Welcome to Node.js v20.10.0.
Type ".help" for more information.
> const lst = [3, 1, 4];
undefined
> const x = lst[3]; // static type is number but runtime type is undefined.
undefined
> console.log(x.toFixed());
Uncaught TypeError: Cannot read properties of undefined (reading 'toFixed')
\end{tscode}
\end{frame}
\note[itemize]{
\item 타입 시스템을 염두하지 않고 만든 언어에 안전한 타입 시스템을 도입하는 것은 어려움
\item 이러한 언어에서는 타입 검증이 어려운 상용구를 지향하기도 함
}
\begin{frame}[c, fragile]
\frametitle{안전하게 조금씩 타입 붙이기}
타입스크립트는 안전하지 않은 타입 시스템
\begin{center}
\includegraphics[width=0.6\linewidth]{img/ts-unsound.png}
\end{center}
\begin{tscode}
Welcome to Node.js v20.10.0.
Type ".help" for more information.
> const lst = [3, 1, 4];
undefined
> const x = lst[3]; // static type is number but runtime type is undefined.
undefined
> console.log(x.toFixed());
Uncaught TypeError: Cannot read properties of undefined (reading 'toFixed')
\end{tscode}
\end{frame}
\begin{frame}[c, fragile]
\frametitle{안전하지 않은 타입 시스템: 수동 메모리 관리}
수동 메모리 관리를 지원하는 언어는 안전한 타입 시스템을 만들기 굉장히 어려움
\begin{ccode}
int toxic(void)
{
int *x = malloc(sizeof(int));
*x = 42;
int *y = x;
free(x);
float *z = malloc(sizeof(float));
*z = 3.141592f;
return *y + 1;
}
\end{ccode}
\begin{onlyenv}<2>
\Alt*{메모리 재활용}{Garbage collection}은 필수적(이었음)
\begin{itemize}
\item 러스트는 소유권과 빌려주기의 개념을 사용해 문제를 우회
\end{itemize}
\end{onlyenv}
\end{frame}
\subsection{성능 향상}
\begin{frame}[c, fragile]
\frametitle{미루지 말고 미리미리}
정적 타입 시스템은 실행 전에 타입을 검사하기 때문에,
\begin{itemize}
\item 실행 중에 타입 검사를 할 필요가 없고
\item 효율적으로 값을 메모리에 담을 수 있음
\end{itemize}
\pause\begin{pycode}
Python 3.11.4 (main, Jul 4 2023, 19:38:47) [Clang 14.0.3 (clang-1403.0.22.14.1)] on darwin
Type "help", "copyright", "credits" or "license" for more information.
>>> 42 + "foo"
Traceback (most recent call last):
File "<stdin>", line 1, in <module>
TypeError: unsupported operand type(s) for +: 'int' and 'str'
\end{pycode}
\end{frame}
\subsection{관계 표현}
\begin{frame}[c, fragile]
\frametitle{간증: \Alt*{서로 맞물려 돌아가는}{mutually recursive}\ 구조}
\begin{onlyenv}<1>
\tiny\url{https://easyword.kr}
\vskip-1.4em\begin{center}
\includegraphics[width=0.6\linewidth]{img/easyword.png}
\end{center}
\end{onlyenv}
\begin{onlyenv}<2>
\mbox{\small\url{https://github.com/Zeta611/eko/blob/main/src/Comment.res}}\footnote{영구 링크: \href{https://github.com/Zeta611/eko/blob/9d140e8b578030b99a0965719be7aa8fd7f18899/src/Comment.res}{\texttt{https://github.com/Zeta611/eko/...}}}
\vskip-1em
\begin{ocamlcode}
// Firestore comment entity
@deriving({abstract: light})
type t = {
@optional id: string,
content: string,
user: string,
timestamp: Firebase.Timestamp.t,
parent: string,
}
type rec node = {
comment: t,
mutable parent: option<node>,
mutable children: list<node>,
}
let rec countDescendents = children => {
switch children {
| list{} => 0
| list{{children}, ...tl} => 1 + countDescendents(children) + countDescendents(tl)
}
}
\end{ocamlcode}
\end{onlyenv}
\end{frame}
\begin{frame}[c, fragile]
\frametitle{맛보기: 여러 종류의 다형성}
\scriptsize\begin{description}
\item[매개변수 다형성] \texttt{CollapsibleView}가 받아들이는 \texttt{State}, \texttt{Control}, \texttt{Collapsible} 타입
\item[타입 적응 다형성] \texttt{State} 타입은 \texttt{Togglable}의 모양(스위프트에서는 \Alt*{프로토콜}{protocol}이라고 부름)을, \texttt{Control}과 \texttt{Collapsible} 타입은 \texttt{View}의 모양을 가짐
\begin{itemize}
\item \texttt{CollapsibleView}는 \texttt{View}의 모양을 가짐
\end{itemize}
\item[아래 타입 다형성] 아래 코드에는 드러나지 않았지만, \texttt{Togglable} 타입은 \texttt{Equatable}의 아래 타입
\end{description}
\href{https://github.com/snulife/snulife-gp-ios-2/blob/cded38ba6f728bf5429520424a8326ac4cc23a1b/GP/Views/CollapsibleView.swift#L11}{\texttt{CollapsibleView.swift}}
\begin{swiftcode}
struct CollapsibleView<State: Togglable, Control: View, Collapsible: View>: View {
@Binding var globalState: State
let state: State
let control: () -> Control
let collapsible: () -> Collapsible
@FocusState private var foo: Bool
private let val = false
var body: some View { ... }
init( ... ) { ... }
}
\end{swiftcode}
\end{frame}
\section{타입 시스템과 다형성}
\subsection{간단한 타입 시스템}
\begin{frame}[c, fragile]
\frametitle{\Alt*{람다 계산법}{Lambda Calculus}}
람다 계산법은 계산의 원리를 연구하기 위해 만들어진 수학적 모델 (Alonzo Church, 1920대)
\begin{itemize}
\item 그런데 프로그래밍 언어를 설명하는데 효과적이라는 것이 밝혀짐! (Peter Landin, 1960대)
\end{itemize}
\pause\begin{pycode}
Python 3.11.4 (main, Jul 4 2023, 19:38:47) [Clang 14.0.3 (clang-1403.0.22.14.1)] on darwin
Type "help", "copyright", "credits" or "license" for more information.
>>> fact = lambda n: 1 if n == 0 else n * fact(n - 1)
>>> fact(10)
3628800
\end{pycode}
\pause
\begin{center}
fact = $\lambda$ n. if n = 0 then 1 else n * fact(n - 1)
\end{center}
\end{frame}
\begin{frame}[c, fragile]
\frametitle{람다 계산법의 \Alt*{문법 구조}{syntax}와 \Alt*{실행 의미 구조}{operational semantics}}
\begin{minipage}[t]{0.49\linewidth}
문법 구조
\begin{center}
\begin{bnf}
t : 항 ::=
| x : 변수
| \lambda x.t : 함수
| t\ t : 적용
;;
v : 값 ::=
| \lambda x.t : 함수 값
\end{bnf}
\end{center}
\end{minipage}
\begin{minipage}[t]{0.49\linewidth}
실행 의미 구조\hfill\fbox{$t \rightsquigarrow t'$}
\begin{center}
\begin{InfRule}{E-App1}
\hypo{t_1 \rightsquigarrow t_1'}
\infer1{t_1\ t_2 \rightsquigarrow t_1'\ t_2}
\end{InfRule}
\begin{InfRule}{E-App2}
\hypo{t_2 \rightsquigarrow t_2'}
\infer1{v_1\ t_2 \rightsquigarrow v_1\ t_2'}
\end{InfRule}
\begin{InfRule}{E-AppAbs}
\infer0{(\lambda x.t_{12})\ v_2 \rightsquigarrow [x \mapsto v_2]t_{12}}
\end{InfRule}
\end{center}
\end{minipage}
\pause\[\begin{tblr}{rl}
& (\lambda x.\lambda y.x)\ \underline{((\lambda x.x)\ a)}\ b \\
\xrsquigarrow{\textsc{E-App2, E-AppAbs}} & \underline{(\lambda x.\lambda y.x)\ a}\ b \\
\xrsquigarrow{\textsc{E-App1, E-AppAbs}} & \underline{(\lambda y.a)\ b} \\
\xrsquigarrow{\textsc{E-App1}} & a
\end{tblr}\]
\end{frame}
\begin{frame}[c, fragile]
\frametitle{\Alt*{간단한 타입을 가진 람다 계산법}{Simply Typed Lambda Calculus, STLC}}
\begin{minipage}[t]{0.49\linewidth}
문법 구조
\begin{center}
\def\typecolon{{:}}
\begin{bnf}
t : 항 ::=
| x : 변수
| \lambda x \typecolon \tau.t : 함수
| t\ t : 적용
;;
v : 값 ::=
| \lambda x \typecolon \tau .t : 함수 값
;;
\tau : 타입 ::=
| \tau \to \tau : 함수 타입
;;
\Gamma : 환경 ::=
| \varnothing : 빈 환경
| \Gamma, x \mathrel{\typecolon} \tau : 변수 타입 정의
\end{bnf}
\end{center}
\end{minipage}
\begin{minipage}[t]{0.49\linewidth}
타입 규칙\hfill\fbox{$\Gamma \vdash t : \tau$}
\begin{center}
\begin{InfRule}{T-Var}
\hypo{x : \tau \in \Gamma}
\infer1{\Gamma \vdash x : \tau}
\end{InfRule}
\begin{InfRule}{T-Abs}
\hypo{\Gamma, x : \tau_1 \vdash t_2 : \tau_2}
\infer1{\Gamma \vdash \lambda x {:} \tau_1.t_2 : \tau_1 \to \tau_2}
\end{InfRule}
\begin{InfRule}{T-App}
\hypo{\Gamma \vdash t_1 : \tau_{11} \to \tau_{12}}
\hypo{\Gamma \vdash t_2 : \tau_{11}}
\infer{1,1}{\Gamma \vdash t_1\ t_2 : \tau_{12}}
\end{InfRule}
\end{center}
\end{minipage}
\end{frame}
\begin{frame}[c, fragile]
\frametitle{타입 규칙 예시}
새로운 불리언 항 \texttt{true}, \texttt{false}를 추가하고, 불리언 타입 \texttt{Bool}을 추가:
\begin{center}
\begin{InfRule}{T-True}
\infer0{\Gamma \vdash \texttt{true} : \texttt{Bool}}
\end{InfRule}
\begin{InfRule}{T-False}
\infer0{\Gamma \vdash \texttt{false} : \texttt{Bool}}
\end{InfRule}
\end{center}
\begin{prooftree*}
\hypo{\texttt{x} : \texttt{Bool} \in \texttt{x} : \texttt{Bool}}
\infer1[T-Var]{\texttt{x} : \texttt{Bool} \vdash \texttt{x} : \texttt{Bool}}
\infer1[T-Abs]{\vdash \lambda \texttt{x}{:}\texttt{Bool}.\texttt{x} : \texttt{Bool} \to \texttt{Bool}}
\infer0[T-True]{\vdash \texttt{true} : \texttt{Bool}}
\infer2[T-App]{\vdash (\lambda \texttt{x}{:}\texttt{Bool}.\texttt{x})\ \texttt{true} : \texttt{Bool}}
\end{prooftree*}
\end{frame}
\subsection{\Alt*{매개변수 다형성}{Parametric polymorphism}}
\begin{frame}[c, fragile]
\frametitle{매개변수 다형성의 동기}
인터넷에서 데이터를 가져오는 함수 \verb/fetch/를 만들어보자!
\begin{itemize}
\pause\item 그런데 데이터의 타입마다 다른 함수를 만들어야 하나?
\end{itemize}
\pause 타입에 상관없이 동작하는 \Alt*{일반적인}{generic} 함수를 만들고 싶다!
\begin{swiftcode}
func fetch<T>(
_ type: T.Type,
router: Router,
accessToken: String? = nil
) async throws -> T where T: Decodable {
logger.debug("fetch(\(type), router: \(router.path), accessToken: \(accessToken ?? "nil")")
let request = try generateRequest(router: router, accessToken: accessToken)
let (data, response) = try await session.data(for: request)
... }
\end{swiftcode}
대충:
\[
\texttt{fetch} : \forall \texttt{T}. \texttt{T.Type} \to \texttt{Router} \to \texttt{String?} \to \texttt{T}
\]
\end{frame}
\begin{frame}[c, fragile]
\frametitle{시스템 F}
\scriptsize
\begin{minipage}[t]{0.49\linewidth}
문법 구조
\begin{center}
\def\typecolon{{:}}
\begin{bnf}
t : 항 ::=
| x : 변수
| \lambda x \typecolon \tau.t : 함수
| t\ t : 적용
| \Lambda X.t : 타입 함수
| t [\tau] : 타입 적용
;;
v : 값 ::=
| \lambda x \typecolon \tau .t : 함수 값
| \Lambda X.t : 타입 함수 값
;;
\tau : 타입 ::=
| X : 타입 변수
| \tau \to \tau : 함수 타입
| \forall X.\tau : 보편 타입
;;
\Gamma : 환경 ::=
| \varnothing : 빈 환경
| \Gamma, x \mathrel{\typecolon} \tau : 변수 타입 정의
| \Gamma, X : 타입 변수 정의
\end{bnf}
\end{center}
\end{minipage}
\begin{onlyenv}<1>
\begin{minipage}[t]{0.49\linewidth}
실행 의미 구조\hfill\fbox{$t \rightsquigarrow t'$}
\begin{center}
\begin{InfRule}{E-App1}
\hypo{t_1 \rightsquigarrow t_1'}
\infer1{t_1\ t_2 \rightsquigarrow t_1'\ t_2}
\end{InfRule}
\begin{InfRule}{E-App2}
\hypo{t_2 \rightsquigarrow t_2'}
\infer1{v_1\ t_2 \rightsquigarrow v_1\ t_2'}
\end{InfRule}
\begin{InfRule}{E-AppAbs}
\infer0{(\lambda x.t_{12})\ v_2 \rightsquigarrow [x \mapsto v_2]t_{12}}
\end{InfRule}
\begin{InfRule}{E-TApp}
\hypo{t_1 \rightsquigarrow t_1'}
\infer1{t_1[\tau_2] \rightsquigarrow t_1'[\tau_2]}
\end{InfRule}
\begin{InfRule}{E-TAppTAbs}
\infer0{(\Lambda X.t_{12})[\tau_2] \rightsquigarrow [X \mapsto \tau_2]t_{12}}
\end{InfRule}
\end{center}
\end{minipage}
\end{onlyenv}
\begin{onlyenv}<2>
\begin{minipage}[t]{0.49\linewidth}
타입 규칙\hfill\fbox{$\Gamma \vdash t : \tau$}
\begin{center}
\begin{InfRule}{T-Var}
\hypo{x : \tau \in \Gamma}
\infer1{\Gamma \vdash x : \tau}
\end{InfRule}
\begin{InfRule}{T-Abs}
\hypo{\Gamma, x : \tau_1 \vdash t_2 : \tau_2}
\infer1{\Gamma \vdash \lambda x{:}\tau_1.t_2 : \tau_1 \to \tau_2}
\end{InfRule}
\begin{InfRule}{T-App}
\hypo{\Gamma \vdash t_1 : \tau_{11} \to \tau_{12}}
\hypo{\Gamma \vdash t_2 : \tau_{11}}
\infer{1,1}{\Gamma \vdash t_1\ t_2 : \tau_{12}}
\end{InfRule}
\begin{InfRule}{T-TAbs}
\hypo{\Gamma, X \vdash t_2 : \tau_2}
\infer1{\Gamma \vdash \Lambda X.t_2 : \forall X.\tau_2}
\end{InfRule}
\begin{InfRule}{T-TApp}
\hypo{\Gamma \vdash t_1 : \forall X. \tau_{12}}
\infer1{\Gamma \vdash t_1[\tau_2] : [X \mapsto \tau_2]\tau_{12}}
\end{InfRule}
\end{center}
\end{minipage}
\end{onlyenv}
\end{frame}
\begin{frame}[c, fragile]
\frametitle{시스템 F의 예시}
\begin{lstlisting}
double = $\Lambda$X. $\lambda$f: X $\to$ X. $\lambda$x: X. f (f x)
> double : $\forall$X. (X $\to$ X) $\to$ X $\to$ X
doubleInt = double [Int]
> doubleInt : (Int $\to$ Int) $\to$ Int $\to$ Int
doubleIntArrowInt = double [Int $\to$ Int]
> doubleIntArrowInt : ((Int $\to$ Int) $\to$ Int $\to$ Int)
$\to$ (Int $\to$ Int) $\to$ Int $\to$ Int
quadruple = $\Lambda$X. double [X $\to$ X] (double [X])
> quadruple : $\forall$X. (X $\to$ X) $\to$ X $\to$ X
\end{lstlisting}
\end{frame}
\begin{frame}[c, fragile]
\frametitle{\Alt*{let 다형성}{let-polymorphism}}
\tiny
\begin{ocamlcode}
OCaml version 5.1.0
Enter #help;; for help.
# let double f a = f (f a);;
val double : ('a -> 'a) -> 'a -> 'a = <fun>
# double ((+) 1) 1;;
- : int = 3
# double ((+.) 1.) 1.;;
- : float = 3.
# let quadruple = double double;;
val quadruple : ('_weak1 -> '_weak1) -> '_weak1 -> '_weak1 = <fun>
# quadruple ((+) 1) 1;;
- : int = 5
# quadruple ((+.) 1.) 1.;;
Error: This expression has type float -> float
but an expression was expected of type int -> int
Type float is not compatible with type int
# let quadruple a = double double a;;
val quadruple : ('a -> 'a) -> 'a -> 'a = <fun>
# quadruple ((+) 1) 1;;
- : int = 5
# quadruple ((+.) 1.) 1.;;
- : float = 5.
\end{ocamlcode}
\end{frame}
\subsection{\Alt*{아래 타입 다형성}{Subtype polymorphism}}
\begin{frame}[c, fragile]
\frametitle{\texttt{Self}의 기묘함}
\begin{swiftcode}
Welcome to Apple Swift version 5.9.2 (swiftlang-5.9.2.2.56 clang-1500.1.0.2.5).
Type :help for assistance.
1> class A {
2. let x: Int
3. func isEqual(to other: Self) -> Bool {
4. self.x == other.x
5. }
6. init(x: Int) {
7. self.x = x
8. }
9. }
error: repl.swift:3:28: error: covariant 'Self' or 'Self?' can only appear as the type of a property, subscript or method result; did you mean 'A'?
func isEqual(to other: Self) -> Bool {
^~~~
A
\end{swiftcode}
\pause \verb/A/를 상속하는 클래스의 \verb/isEqual(to:)/가 이상하다!
\end{frame}
\begin{frame}[c, fragile]
\frametitle{\Alt*{맞춰 변하기}{covariant}, \Alt*{거슬러 변하기}{contravariant}, \Alt*{안 변하기}{invariant}}
같은 타입의 다른 값과 같은지 비교하는 함수를 만들자:
\begin{ocamlcode}
OCaml version 5.1.0
Enter #help;; for help.
# type x = [ `X ];;
type x = [ `X ]
# type xy = [ `X | `Y ];;
type xy = [ `X | `Y ]
# let x : x = `X;;
val x : x = `X
# let x' = (x :> xy);;
val x' : xy = `X
# let l : x list = [`X; `X];;
val l : x list = [`X; `X]
# let l' = (l :> xy list);;
val l' : xy list = [`X; `X]
# let f : xy -> unit = function `X -> () | `Y -> ();;
val f : xy -> unit = <fun>
# let f' = (f :> x -> unit);;
val f' : x -> unit = <fun>
# let x : x ref = ref `X;;
val x : x ref = {contents = `X}
# let x' = (x :> xy ref);;
Error: Type x ref is not a subtype of xy ref
The first variant type does not allow tag(s) `Y
\end{ocamlcode}
\end{frame}
\begin{frame}[c, fragile]
\frametitle{아래 타입 다형성}
\tiny
\begin{minipage}[t]{0.49\linewidth}
문법 구조
\begin{center}
\def\typecolon{{:}}
\begin{bnf}
t : 항 ::=
| x : 변수
| \lambda x \typecolon \tau.t : 함수
| t\ t : 적용
;;
v : 값 ::=
| \lambda x \typecolon \tau .t : 함수 값
;;
\tau : 타입 ::=
| \top : 최대 타입
| \tau \to \tau : 함수 타입
;;
\Gamma : 환경 ::=
| \varnothing : 빈 환경
| \Gamma, x \mathrel{\typecolon} \tau : 변수 타입 정의
\end{bnf}
\end{center}
실행 의미 구조\hfill\fbox{$t \rightsquigarrow t'$}
\begin{center}
\begin{InfRule}{E-App1}
\hypo{t_1 \rightsquigarrow t_1'}
\infer1{t_1\ t_2 \rightsquigarrow t_1'\ t_2}
\end{InfRule}
\begin{InfRule}{E-App2}
\hypo{t_2 \rightsquigarrow t_2'}
\infer1{v_1\ t_2 \rightsquigarrow v_1\ t_2'}
\end{InfRule}
\begin{InfRule}{E-AppAbs}
\infer0{(\lambda x.t_{12})\ v_2 \rightsquigarrow [x \mapsto v_2]t_{12}}
\end{InfRule}
\end{center}
\end{minipage}
\begin{minipage}[t]{0.49\linewidth}
아래 타입 규칙\hfill\fbox{$\varsigma <: \tau$}
\begin{center}
\begin{InfRule}{S-Refl}
\infer0{\varsigma <: \varsigma}
\end{InfRule}
\begin{InfRule}{S-Trans}
\hypo{\varsigma <: \upsilon}
\hypo{\upsilon <: \tau}
\infer2{\varsigma <: \tau}
\end{InfRule}
\begin{InfRule}{S-Top}
\infer0{\varsigma <: \top}
\end{InfRule}
\begin{InfRule}{S-Arrow}
\hypo{\tau_1 <: \varsigma_1}
\hypo{\varsigma_2 <: \tau_2}
\infer2{\varsigma_1 \to \varsigma_2 <: \tau_1 \to \tau_2}
\end{InfRule}
\end{center}
타입 규칙\hfill\fbox{$\Gamma \vdash t : \tau$}
\begin{center}
\begin{InfRule}{T-Var}
\hypo{x : \tau \in \Gamma}
\infer1{\Gamma \vdash x : \tau}
\end{InfRule}
\begin{InfRule}{T-Abs}
\hypo{\Gamma, x : \tau_1 \vdash t_2 : \tau_2}
\infer1{\Gamma \vdash \lambda x{:}\tau_1.t_2 : \tau_1 \to \tau_2}
\end{InfRule}
\begin{InfRule}{T-App}
\hypo{\Gamma \vdash t_1 : \tau_{11} \to \tau_{12}}
\hypo{\Gamma \vdash t_2 : \tau_{11}}
\infer2{\Gamma \vdash t_1\ t_2 : \tau_{12}}
\end{InfRule}
\begin{InfRule}{T-Sub}
\hypo{\Gamma \vdash t : \varsigma}
\hypo{\varsigma <: \tau}
\infer2{\Gamma \vdash t : \tau}
\end{InfRule}
\end{center}
\end{minipage}
\end{frame}
\subsection{\Alt*{타입 적응 다형성}{Ad-hoc polymorphism}}
\begin{frame}[c, fragile]
\frametitle{타입 적응 다형성}