C++模板Turing-complete？

Question

我被告知C++中的模板系统在编译时是图灵完备的.这篇文章以及维基百科都提到了这一点.

你能提供一个利用这个属性的计算的重要例子吗？

这个事实在实践中有用吗？

Answer 1

我用C++ 11做了一台图灵机.C++ 11添加的功能确实对图灵机而言并不重要.它只是使用可变参数模板提供任意长度的规则列表,而不是使用反向宏元编程:).条件的名称用于在stdout上输出图表.我已删除该代码以保持样本简短.

#include <iostream>

template<bool C, typename A, typename B>
struct Conditional {
    typedef A type;
};

template<typename A, typename B>
struct Conditional<false, A, B> {
    typedef B type;
};

template<typename...>
struct ParameterPack;

template<bool C, typename = void>
struct EnableIf { };

template<typename Type>
struct EnableIf<true, Type> {
    typedef Type type;
};

template<typename T>
struct Identity {
    typedef T type;
};

// define a type list 
template<typename...>
struct TypeList;

template<typename T, typename... TT>
struct TypeList<T, TT...>  {
    typedef T type;
    typedef TypeList<TT...> tail;
};

template<>
struct TypeList<> {

};

template<typename List>
struct GetSize;

template<typename... Items>
struct GetSize<TypeList<Items...>> {
    enum { value = sizeof...(Items) };
};

template<typename... T>
struct ConcatList;

template<typename... First, typename... Second, typename... Tail>
struct ConcatList<TypeList<First...>, TypeList<Second...>, Tail...> {
    typedef typename ConcatList<TypeList<First..., Second...>, 
                                Tail...>::type type;
};

template<typename T>
struct ConcatList<T> {
    typedef T type;
};

template<typename NewItem, typename List>
struct AppendItem;

template<typename NewItem, typename...Items>
struct AppendItem<NewItem, TypeList<Items...>> {
    typedef TypeList<Items..., NewItem> type;
};

template<typename NewItem, typename List>
struct PrependItem;

template<typename NewItem, typename...Items>
struct PrependItem<NewItem, TypeList<Items...>> {
    typedef TypeList<NewItem, Items...> type;
};

template<typename List, int N, typename = void>
struct GetItem {
    static_assert(N > 0, "index cannot be negative");
    static_assert(GetSize<List>::value > 0, "index too high");
    typedef typename GetItem<typename List::tail, N-1>::type type;
};

template<typename List>
struct GetItem<List, 0> {
    static_assert(GetSize<List>::value > 0, "index too high");
    typedef typename List::type type;
};

template<typename List, template<typename, typename...> class Matcher, typename... Keys>
struct FindItem {
    static_assert(GetSize<List>::value > 0, "Could not match any item.");
    typedef typename List::type current_type;
    typedef typename Conditional<Matcher<current_type, Keys...>::value, 
                                 Identity<current_type>, // found!
                                 FindItem<typename List::tail, Matcher, Keys...>>
        ::type::type type;
};

template<typename List, int I, typename NewItem>
struct ReplaceItem {
    static_assert(I > 0, "index cannot be negative");
    static_assert(GetSize<List>::value > 0, "index too high");
    typedef typename PrependItem<typename List::type, 
                             typename ReplaceItem<typename List::tail, I-1,
                                                  NewItem>::type>
        ::type type;
};

template<typename NewItem, typename Type, typename... T>
struct ReplaceItem<TypeList<Type, T...>, 0, NewItem> {
    typedef TypeList<NewItem, T...> type;
};

enum Direction {
    Left = -1,
    Right = 1
};

template<typename OldState, typename Input, typename NewState, 
         typename Output, Direction Move>
struct Rule {
    typedef OldState old_state;
    typedef Input input;
    typedef NewState new_state;
    typedef Output output;
    static Direction const direction = Move;
};

template<typename A, typename B>
struct IsSame {
    enum { value = false }; 
};

template<typename A>
struct IsSame<A, A> {
    enum { value = true };
};

template<typename Input, typename State, int Position>
struct Configuration {
    typedef Input input;
    typedef State state;
    enum { position = Position };
};

template<int A, int B>
struct Max {
    enum { value = A > B ? A : B };
};

template<int n>
struct State {
    enum { value = n };
    static char const * name;
};

template<int n>
char const* State<n>::name = "unnamed";

struct QAccept {
    enum { value = -1 };
    static char const* name;
};

struct QReject {
    enum { value = -2 };
    static char const* name; 
};

#define DEF_STATE(ID, NAME) \
    typedef State<ID> NAME ; \
    NAME :: name = #NAME ;

template<int n>
struct Input {
    enum { value = n };
    static char const * name;

    template<int... I>
    struct Generate {
        typedef TypeList<Input<I>...> type;
    };
};

template<int n>
char const* Input<n>::name = "unnamed";

typedef Input<-1> InputBlank;

#define DEF_INPUT(ID, NAME) \
    typedef Input<ID> NAME ; \
    NAME :: name = #NAME ;

template<typename Config, typename Transitions, typename = void> 
struct Controller {
    typedef Config config;
    enum { position = config::position };

    typedef typename Conditional<
        static_cast<int>(GetSize<typename config::input>::value) 
            <= static_cast<int>(position),
        AppendItem<InputBlank, typename config::input>,
        Identity<typename config::input>>::type::type input;
    typedef typename config::state state;

    typedef typename GetItem<input, position>::type cell;

    template<typename Item, typename State, typename Cell>
    struct Matcher {
        typedef typename Item::old_state checking_state;
        typedef typename Item::input checking_input;
        enum { value = IsSame<State, checking_state>::value && 
                       IsSame<Cell,  checking_input>::value
        };
    };
    typedef typename FindItem<Transitions, Matcher, state, cell>::type rule;

    typedef typename ReplaceItem<input, position, typename rule::output>::type new_input;
    typedef typename rule::new_state new_state;
    typedef Configuration<new_input, 
                          new_state, 
                          Max<position + rule::direction, 0>::value> new_config;

    typedef Controller<new_config, Transitions> next_step;
    typedef typename next_step::end_config end_config;
    typedef typename next_step::end_input end_input;
    typedef typename next_step::end_state end_state;
    enum { end_position = next_step::position };
};

template<typename Input, typename State, int Position, typename Transitions>
struct Controller<Configuration<Input, State, Position>, Transitions, 
                  typename EnableIf<IsSame<State, QAccept>::value || 
                                    IsSame<State, QReject>::value>::type> {
    typedef Configuration<Input, State, Position> config;
    enum { position = config::position };
    typedef typename Conditional<
        static_cast<int>(GetSize<typename config::input>::value) 
            <= static_cast<int>(position),
        AppendItem<InputBlank, typename config::input>,
        Identity<typename config::input>>::type::type input;
    typedef typename config::state state;

    typedef config end_config;
    typedef input end_input;
    typedef state end_state;
    enum { end_position = position };
};

template<typename Input, typename Transitions, typename StartState>
struct TuringMachine {
    typedef Input input;
    typedef Transitions transitions;
    typedef StartState start_state;

    typedef Controller<Configuration<Input, StartState, 0>, Transitions> controller;
    typedef typename controller::end_config end_config;
    typedef typename controller::end_input end_input;
    typedef typename controller::end_state end_state;
    enum { end_position = controller::end_position };
};

#include <ostream>

template<>
char const* Input<-1>::name = "_";

char const* QAccept::name = "qaccept";
char const* QReject::name = "qreject";

int main() {
    DEF_INPUT(1, x);
    DEF_INPUT(2, x_mark);
    DEF_INPUT(3, split);

    DEF_STATE(0, start);
    DEF_STATE(1, find_blank);
    DEF_STATE(2, go_back);

    /* syntax:  State, Input, NewState, Output, Move */
    typedef TypeList< 
        Rule<start, x, find_blank, x_mark, Right>,
        Rule<find_blank, x, find_blank, x, Right>,
        Rule<find_blank, split, find_blank, split, Right>,
        Rule<find_blank, InputBlank, go_back, x, Left>,
        Rule<go_back, x, go_back, x, Left>,
        Rule<go_back, split, go_back, split, Left>,
        Rule<go_back, x_mark, start, x, Right>,
        Rule<start, split, QAccept, split, Left>> rules;

    /* syntax: initial input, rules, start state */
    typedef TuringMachine<TypeList<x, x, x, x, split>, rules, start> double_it;
    static_assert(IsSame<double_it::end_input, 
                         TypeList<x, x, x, x, split, x, x, x, x>>::value, 
                "Hmm... This is borky!");
}

Answer 2

例

#include <iostream>

template <int N> struct Factorial
{
    enum { val = Factorial<N-1>::val * N };
};

template<>
struct Factorial<0>
{
    enum { val = 1 };
};

int main()
{
    // Note this value is generated at compile time.
    // Also note that most compilers have a limit on the depth of the recursion available.
    std::cout << Factorial<4>::val << "\n";
}

这有点好玩但不太实用.

回答问题的第二部分:
这个事实在实践中是否有用？

简答:排序.

长答案:是的,但前提是你是模板守护进程.

使用对其他人来说非常有用的模板元编程(即库)来实现良好的编程真的很难(尽管可行).有助于提升甚至有MPL又名(元编程库).但是,尝试在模板代码中调试编译器错误,您将需要长时间的努力.

但它是用于有用的东西的一个很好的实际例子:

Scott Meyers一直在使用模板工具扩展C++语言(我使用松散的术语).你可以在这里阅读他的工作' 执行代码功能 '

Answer 3

" C++模板图灵完成 "在模板中提供了图灵机的实现......这是非常重要的,并且以非常直接的方式证明了这一点.当然,它也不是很有用!

Answer 4

我的C++有点生疏,所以可能并不完美,但它很接近.

template <int N> struct Factorial
{
    enum { val = Factorial<N-1>::val * N };
};

template <> struct Factorial<0>
{
    enum { val = 1 };
}

const int num = Factorial<10>::val;    // num set to 10! at compile time.

关键是要证明编译器正在完全评估递归定义,直到它得到答案.

Answer 5

举一个非平凡的例子:http://gitorious.org/metatrace,C ++编译时光线跟踪器.

请注意,C++ 0x将以下列形式添加非模板,编译时,图灵完备工具constexpr:

constexpr unsigned int fac (unsigned int u) {
        return (u<=1) ? (1) : (u*fac(u-1));
}

您可以constexpr在需要编译时常量的地方使用-expression,但也可以constexpr使用非const参数调用-functions.

一个很酷的事情是,这将最终启用编译时浮点数学,尽管标准明确指出编译时浮点算术不必匹配运行时浮点算术:

bool f(){
    char array[1+int(1+0.2-0.1-0.1)]; //Must be evaluated during translation
    int  size=1+int(1+0.2-0.1-0.1); //May be evaluated at runtime
    return sizeof(array)==size;
}

f()的值是真还是假是不明确的.

Answer 6

Book Modern C++ Design -由Andrei Alexandrescu 设计的通用编程和设计模式是获得有用且强大的通用编程模式经验的最佳场所.

Answer 7

因子实例实际上没有显示模板是图灵完整的,因为它表明它们支持原始递归.显示模板完成的最简单方法是通过Church-Turing论文,即通过实现图灵机(凌乱和有点无意义)或无类型lambda演算的三个规则(app,abs var).后者更简单,更有趣.

当您了解C++模板允许在编译时进行纯函数式编程时,正在讨论的是一个非常有用的功能,这是一种富有表现力,强大而优雅的形式主义,但如果您没有经验,那么编写起来也非常复杂.还要注意有多少人发现只是获得高度模板化的代码通常需要付出很大的努力:这与(纯)函数语言的情况完全相同,这使得编译更难,但令人惊讶的是产生不需要调试的代码.

Answer 8

我认为它被称为模板元编程.

Answer 9

好吧，这是一个运行 4 状态 2 符号忙海狸的编译时图灵机实现

#include <iostream>

#pragma mark - Tape

constexpr int Blank = -1;

template<int... xs>
class Tape {
public:
    using type = Tape<xs...>;
    constexpr static int length = sizeof...(xs);
};

#pragma mark - Print

template<class T>
void print(T);

template<>
void print(Tape<>) {
    std::cout << std::endl;
}

template<int x, int... xs>
void print(Tape<x, xs...>) {
    if (x == Blank) {
        std::cout << "_ ";
    } else {
        std::cout << x << " ";
    }
    print(Tape<xs...>());
}

#pragma mark - Concatenate

template<class, class>
class Concatenate;

template<int... xs, int... ys>
class Concatenate<Tape<xs...>, Tape<ys...>> {
public:
    using type = Tape<xs..., ys...>;
};

#pragma mark - Invert

template<class>
class Invert;

template<>
class Invert<Tape<>> {
public:
    using type = Tape<>;
};

template<int x, int... xs>
class Invert<Tape<x, xs...>> {
public:
    using type = typename Concatenate<
        typename Invert<Tape<xs...>>::type,
        Tape<x>
    >::type;
};

#pragma mark - Read

template<int, class>
class Read;

template<int n, int x, int... xs>
class Read<n, Tape<x, xs...>> {
public:
    using type = typename std::conditional<
        (n == 0),
        std::integral_constant<int, x>,
        Read<n - 1, Tape<xs...>>
    >::type::type;
};

#pragma mark - N first and N last

template<int, class>
class NLast;

template<int n, int x, int... xs>
class NLast<n, Tape<x, xs...>> {
public:
    using type = typename std::conditional<
        (n == sizeof...(xs)),
        Tape<xs...>,
        NLast<n, Tape<xs...>>
    >::type::type;
};

template<int, class>
class NFirst;

template<int n, int... xs>
class NFirst<n, Tape<xs...>> {
public:
    using type = typename Invert<
        typename NLast<
            n, typename Invert<Tape<xs...>>::type
        >::type
    >::type;
};

#pragma mark - Write

template<int, int, class>
class Write;

template<int pos, int x, int... xs>
class Write<pos, x, Tape<xs...>> {
public:
    using type = typename Concatenate<
        typename Concatenate<
            typename NFirst<pos, Tape<xs...>>::type,
            Tape<x>
        >::type,
        typename NLast<(sizeof...(xs) - pos - 1), Tape<xs...>>::type
    >::type;
};

#pragma mark - Move

template<int, class>
class Hold;

template<int pos, int... xs>
class Hold<pos, Tape<xs...>> {
public:
    constexpr static int position = pos;
    using tape = Tape<xs...>;
};

template<int, class>
class Left;

template<int pos, int... xs>
class Left<pos, Tape<xs...>> {
public:
    constexpr static int position = typename std::conditional<
        (pos > 0),
        std::integral_constant<int, pos - 1>,
        std::integral_constant<int, 0>
    >::type();

    using tape = typename std::conditional<
        (pos > 0),
        Tape<xs...>,
        Tape<Blank, xs...>
    >::type;
};

template<int, class>
class Right;

template<int pos, int... xs>
class Right<pos, Tape<xs...>> {
public:
    constexpr static int position = pos + 1;

    using tape = typename std::conditional<
        (pos < sizeof...(xs) - 1),
        Tape<xs...>,
        Tape<xs..., Blank>
    >::type;
};

#pragma mark - States

template <int>
class Stop {
public:
    constexpr static int write = -1;
    template<int pos, class tape> using move = Hold<pos, tape>;
    template<int x> using next = Stop<x>;
};

#define ADD_STATE(_state_)      \
template<int>                   \
class _state_ { };

#define ADD_RULE(_state_, _read_, _write_, _move_, _next_)          \
template<>                                                          \
class _state_<_read_> {                                             \
public:                                                             \
    constexpr static int write = _write_;                           \
    template<int pos, class tape> using move = _move_<pos, tape>;   \
    template<int x> using next = _next_<x>;                         \
};

#pragma mark - Machine

template<template<int> class, int, class>
class Machine;

template<template<int> class State, int pos, int... xs>
class Machine<State, pos, Tape<xs...>> {
    constexpr static int symbol = typename Read<pos, Tape<xs...>>::type();
    using state = State<symbol>;

    template<int x>
    using nextState = typename State<symbol>::template next<x>;

    using modifiedTape = typename Write<pos, state::write, Tape<xs...>>::type;
    using move = typename state::template move<pos, modifiedTape>;

    constexpr static int nextPos = move::position;
    using nextTape = typename move::tape;

public:
    using step = Machine<nextState, nextPos, nextTape>;
};

#pragma mark - Run

template<class>
class Run;

template<template<int> class State, int pos, int... xs>
class Run<Machine<State, pos, Tape<xs...>>> {
    using step = typename Machine<State, pos, Tape<xs...>>::step;

public:
    using type = typename std::conditional<
        std::is_same<State<0>, Stop<0>>::value,
        Tape<xs...>,
        Run<step>
    >::type::type;
};

ADD_STATE(A);
ADD_STATE(B);
ADD_STATE(C);
ADD_STATE(D);

ADD_RULE(A, Blank, 1, Right, B);
ADD_RULE(A, 1, 1, Left, B);

ADD_RULE(B, Blank, 1, Left, A);
ADD_RULE(B, 1, Blank, Left, C);

ADD_RULE(C, Blank, 1, Right, Stop);
ADD_RULE(C, 1, 1, Left, D);

ADD_RULE(D, Blank, 1, Right, D);
ADD_RULE(D, 1, Blank, Right, A);

using tape = Tape<Blank>;
using machine = Machine<A, 0, tape>;
using result = Run<machine>::type;

int main() {
    print(result());
    return 0;
}

Ideone 验证运行：https ://ideone.com/MvBU3Z

说明：http : //victorkomarov.blogspot.ru/2016/03/compile-time-turing-machine.html

Github 有更多例子：https : //github.com/fnz/CTTM