1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
|
# Using Scheme as a metaphor for template metaprogramming
Back in January I looked at compile time computation in C++ based on handling lists in a _functional fashion_ using a mixture between templates, generic lambda expressions and `constexpr` functions. The conclusion of the [appropriate article] was that the inherent restrictions of the approach taken in [ConstList], namely the missing guarantee on compile time evaluation, inability to make return types depend on actual values and lambda expressions being unable to be declared as constant make viewing types as values and templates as functions the superior approach. This article describes how this approach works out in practice.
While [ConstList] turned out to be of limited use in actually performing compile time computations its list manipulation and query functionality was already inspired by how lists are handled in _LISP_ respectively its more minimalistic dialect _Scheme_, especially by the functionality described in the latter's [SRFI-1].
When I started developing a new library porting this basic concept to the _type as value and templates as functions_ approach called [TypeAsValue] it quickly turned out that a _Scheme_ like paradigm maps quite well to template metaprogramming. This was initially very surprising as I did not expect that C++ templates would actually feel like a - admittedly rather verbose - functional programming language if used in a certain way.
~~~
// (define sum
// (fold +
// 0
// (iota 5 2 2))) => 30
using sum = tav::Fold<
tav::Add,
tav::Int<0>,
tav::Iota<
tav::Size<5>,
tav::Int<2>,
tav::Int<2>
>
>;
~~~
{:.language-cpp}
As we can see compile time computations expressed using this approach are more or less direct mappings of their _Scheme_ equivalent if we overlook the need to explicitly declare types as well as the different syntax used for defining bindings.
While [TypeAsValue] started out as a direct reimplementation of my previous attempt I am happy to say that the conclusions drawn concerning the superiority of a stricly template metaprogramming based implementation held true and enabled the implementation of equivalents for large parts of the _Scheme_ list library. This includes actual content dependent list manipulations such as `filter`, which were impossible to implement in [ConstList], in addition to e.g. a compile time implementation of _Quick Sort_.
## Types as values
The desire to express values in terms of types restricts the set of usable types to _integral types_ as only those types may be used as template parameters. According to the standard[^0] this includes all _integer types_ i.e. all non-floating-point types such as `bool`, `char` and `int`. In case of [TypeAsValue] all values are expressed as specializations of `std::integral_constant` that wrapped in template aliases to simplify their declaration.
~~~
using answer = tav::Int<42>; // std::integral_constant<int, 42>
using letter = tav::Char<'A'>; // std::integral_constant<char, 'A'>
using truth = tav::Boolean<true>; // std::integral_constant<bool, true>
~~~
{:.language-cpp}
This need to explicitly declare all types because deduction during template resolution is not feasible marks one of the instances where the _Scheme metaphor_ does not hold true. Luckily this is not a bad thing as the goal is after all not to develop a exact replica of _Scheme_ in terms of template metaprogramming but to enable compile time computations in a _Scheme like_ fashion. In this context not disregarding the C++ type system is a advantage, especially since it should be possible to enable type deduction where required using an `Any` like [`std::integral_constant`] constructor.
Obviously expressing single values as types is not enough, we also require at least a equivalent for _Scheme_'s fundamental pair type, on top of which more complex structures such as lists and trees may be built.
~~~
template <
typename CAR,
typename CDR
>
struct Pair : detail::pair_tag {
typedef CAR car;
typedef CDR cdr;
typedef Pair<CAR, CDR> type;
};
~~~
{:.language-cpp}
As we can see expressing a pair type in terms of a template type is very straight forward. Note that the recursive `type` definition will be discussed further in the next section on _templates as functions_. Each `Pair` specialization derives from `detail::pair_tag` to simplify verification of values as pairs in `tav::IsPair`. The naming of the parameters as `CAR` and `CDR` is a reference to pair types being constructed using `tav::Cons` analogously to _Scheme_, where the pair `(1 . 2)` may be constructed using `(cons 1 2)`.
To summarize the type concept employed in [TypeAsValue] we can say that all actual values are stored in `std::integral_constant` specializations that enable extraction in a template context via their constant `value` member. Those types are then aggregated into structures using the `tav::Pair` template. This means that we can easily provide _functions_ to work on these _values_ in the form of template types and their parameters as will be discussed in the following section.
## Templates as functions
~~~
template <
typename X,
typename Y
>
using Multiply = std::integral_constant<
decltype(X::value * Y::value),
X::value * Y::value
>;
~~~
{:.language-cpp}
As we can see basic functionality such as a function respectively template to multiply a number by another is easily implemented in terms of a alias for a value type specialization, including automatic result type deduction using `decltype`. This also applies to higher order functionality which can be expressed only using other templates provided by the library such as the `Every` list query.
~~~
// (define (every predicate list)
// (fold (lambda (x y) (and x y))
// #t
// (map predicate list)))
template <
template<typename> class Predicate,
typename List
>
using Every = Fold<
And,
Boolean<true>,
Map<Predicate, List>
>;
~~~
{:.language-cpp}
If we ignore the need to explicitly declare the predicate as a _template template parameter_ i.e. as a function this example is very simmilar to its _Scheme_ equivalent. Concerning the function used to fold the list it is actually less verbose than the _Scheme_ version of `Every` as we can directly pass `tav::And` instead of wrapping it in a lambda expression as is required in _Scheme_ [^1].
Sadly the actual implementations of e.g. `tav::Fold` are not as easily expressed. The recursive nature of folding a list or even constructing its underlying pair based structure using the variadic `tav::List` requires partial template specializations which are not allowed for template aliases[^2].
~~~
template <
template<typename, typename> class Function,
typename Initial,
typename Pair
>
struct fold_pair {
typedef Function<
Car<Pair>,
Eval<fold_pair<Function, Initial, Cdr<Pair>>>
> type;
};
template <
template<typename, typename> class Function,
typename Initial
>
struct fold_pair<Function, Initial, void> {
typedef Initial type;
};
~~~
{:.language-cpp}
While the listing of `tav::detail::fold_pair` shows how the partial specialization of its `Pair` parameter allows recursion until the list ends[^3], it also forces us to define it's actual type in terms of a public `type` member `typedef`. Such a member type definition can lead to cluttering the code with `typename` keywords which is why [TypeAsValue] employs a simple `tav::Eval` template alias to hide all `typename *::type` constructs.
~~~
template <
template<typename, typename> class Function,
typename Initial,
typename List
>
using Fold = Eval<detail::fold_pair<Function, Initial, List>>;
~~~
{:.language-cpp}
Hiding member type defintions behind template aliases enables most _higher_ functionality and applications built using it's functions to be written in a reasonably minimalistic - _Scheme_ like - fashion as we can see in e.g. the listing of `Every`. This also explains why `tav::Pair` recursively defines it's own type, as we would otherwise have to be quite careful where to resolve `tav::Eval`.
### Bindings
Not all programs are sensibly expressed as a nested chain of function calls if we want to reuse some values, separate functionality into appropriately named functions and hide complexity via private bindings. While in _Scheme_ one would use `let` and the like for this purpose, such functionality is not easily replicated in the context of template metaprogramming. This is why _TypeAsValue_ uses a alternate and more _C++ like_ solution to this problem by simply making use of the standard object oriented aspects of the language.
~~~
template <
template<typename, typename> class Comparator,
typename Sequence
>
class quick_sort {
private:
using index = Divide<Length<Sequence>, Size<2>>;
using pivot = Nth<index, Sequence>;
using partitions = Partition<
Apply<Comparator, pivot, _0>::template function,
DeleteNth<index, Sequence>
>;
using lhs = Car<partitions>;
using rhs = Cdr<partitions>;
public:
using type = Concatenate<
Eval<quick_sort<Comparator, lhs>>,
List<pivot>,
Eval<quick_sort<Comparator, rhs>>
>;
};
template <template<typename, typename> class Comparator>
struct quick_sort<Comparator, void> {
typedef void type;
};
~~~
{:.language-cpp}
Above we can see the complete listing of the _Quick Sort_ implementation employed by the `tav::Sort` template alias. At first glance this looks rather different from a _Scheme_ program but if we look closer on could argue that it is a equivalent of a `let` binding where the _private_ section of `tav::detail::quick_sort` contains the bound constants and the _public_ section contains the body.
### Partial function application
## Examples
## Summary
[^0]: ISO C++ Standard draft, N3797, § 3.9.1 _Fundamental types_, Section 7
[^1]: `and` needs to be wrapped in anonymous function in _Scheme_ because it is not a function but a macro
[^2]: ISO C++ Standard draft, N3797, § 14.5 _Template declarations_, Section 3
[^3]: _TypeAsValue_ currently represents _Scheme_'s empty list `'()` as `void`
[appropriate article]: /article/a_look_at_compile_time_computation_in_cpp/
[ConstList]: /page/const_list/
[TypeAsValue]: /page/type_as_value/
[SRFI-1]: http://srfi.schemers.org/srfi-1/srfi-1.html
[`std::integral_constant`]: http://en.cppreference.com/w/cpp/types/integral_constant
|
