Regular types in C++

Regular types
Ilio Catallo - info@iliocatallo.it

Outline
• Fundamental opera/ons
• Regular types
• Semi-regular & totally ordered types
• Making vector_int regular
• Bibliography

Commonali(es among types
What is in common between chars and doubles1
?
char c1 = 'a', c2 = 'b';
double d1 = 1.3, d2 = 2.7;
1
For the remainder of this presenta1on, we will limit ourselves to non-singular double values, i.e., we will exclude
NaN from the set of possible values

Commonali(es among types
For a start, both chars and doubles and can be swapped
std::swap(c1, c2);
std::swap(d1, d2);

swap implementa*on for doubles
Let us look at the implementa0on of swap as if it was wri4en just
for doubles
void swap(double& x, double& y) {
double tmp = x;
x = y;
y = tmp;
}

Requirements on doubles
Which requirements on doubles are needed for swap to work?
double tmp = x;
x = y;
y = tmp;
}

First, it should be possible to copy construct a double
double tmp = x; // tmp is copy constructed from x
x = y;
y = tmp;
}

Second, it should be possible to assign a double
double tmp = x;
x = y; // x is assigned y
y = tmp; // y is assigned tmp
}

Given that double values are in fact both assignable and copy
construc1ble, we can run swap on them
std::swap(d1, d2);

swap implementa*on for chars
Let us now turn our a,en-on to a version of swap tailored for
chars
void swap(char& x, char& y) {
char tmp = x;
x = y;
y = tmp;
}

Requirements on chars
As before, we no,ce that char values need to be assignable and
copy construc1ble
void swap(char& x, char& y) {
char tmp = x;
x = y;
y = tmp;
}

Requirements on chars
As before, since char values are in fact assignable and copy
construc1ble, it is possible to swap them
std::swap(c1, c2);

Swapping built-in types
It is immediate to see that our ﬁnding generalizes to all built-in
types
std::swap(b1, b2); // b1 and b2 are bools
std::swap(f1, f2); // f1 and f2 are floats
std::swap(i1, i2); // i1 and i2 are ints

Common opera*ons
We conclude that assignment and copy construc0on are
opera/ons common to all built-in types

Common opera*ons
That is, for any built-in type T the following is always possible:
T a; a = b; // assignment
T c = d; // copy construction

Common opera*ons
In light of this, we can write a uniform implementa.on of swap for
all built-in types
void swap(T& x, T& y) { // T is a built-in type
T tmp = x;
x = y;
y = tmp;
}

Do built-in types share any addi3onal opera3ons?

Fundamental opera.ons
As a ma&er of fact, we observe that there is a core set of
opera4ons which are deﬁned for all built-in types

Fundamental opera.ons
We call such opera-ons the fundamental opera.ons
┌────────────────────────┬──────────┐
│ Default construction │ T a; │
├────────────────────────┼──────────┤
│ Copy construction │ T a = b; │
├────────────────────────┼──────────┤
│ Destruction │ ~T(a); │
├────────────────────────┼──────────┤
│ Assignment │ a = b; │
├────────────────────────┼──────────┤
│ Equality │ a == b; │
├────────────────────────┼──────────┤
│ Inequality │ a != b; │
├────────────────────────┼──────────┤
│ Ordering │ a < b; │
└────────────────────────┴──────────┘

Operators for UDTs
The C++ programming language allows the use of built-in type
operator syntax for user-deﬁned types (UDTs)

Example: the point2
UDT
struct point {
point(double x, double y): x(x), y(y) {}
double x;
double y;
};
2
Remember that there is no diﬀerence between a struct and a class, up to default members' visibility.
Conven=onally, we use structs when our UDTs do not need a private sec=on

Example: the point UDT
Intui&vely, it makes sense to deﬁne operator == as equality:
bool operator==(point const& p1, point const& p2) {
return p1.x == p2.x &&
p1.y == p2.y;
}

Overload seman-cs
However, C++ does not dictate any seman&c rules on operator
overloading

Overload seman-cs
That is, developers are free to deﬁne == as:
return p1.x * p2.x >
p1.y * p2.y;
}

Seman&cs mismatch
This could lead to a mismatch between the expected and the
actual seman4cs of an operator

Seman&cs mismatch
This could lead to a mismatch between the expected and the
actual seman4cs of an operator
( °□°

Seman&cs mismatch
"I admire the commi.ee dedica/on to programming freedom, but I
claim that such freedom is an illusion. Finding the laws that govern
so=ware components gives us freedom to write complex programs the
same way that ﬁnding the laws of physics allows us to construct
complex mechanical and electrical systems."
(A. Stepanov)

Built-in types as a theore1cal founda1on
The opera)ons common to all built-in types are among the most
well-understood and pervasively used

Built-in types as a theore1cal founda1on
This is because, over the years, the development of built-in types
has led to consistent deﬁni9ons that match
• the programmer intui-on, and
• the underlying mathema-cal understanding

Idea: deriving the seman.cs of user-deﬁned types from that of
built-in types

Preserving opera-on seman-cs
The idea is that when a code fragment has a certain meaning for
built-in types, it should preserve the same meaning for UDTs

Regular types
To this end, we introduce the concept of a regular type as a type
behaving like the built-in types

Regular types
We call such types regular, since their use guarantees regularity of
behavior and – therefore – interoperability

std::vector is regular
For instance, std::vector<T> is regular whenever T is regular

std::vector is regular
As with built-in types, std::vector<T> is thus copy
construc+ble and assignable for each regular type T
std::vector<int> w = v; // copy construction
std::vector<int> u; u = w; // assignment

std::vector is swappable
Hence, we can swap3
two vectors of any regular type T:
std::vector<int> v = {7, 3, 5};
std::vector<int> w = {10, 11, 4};
std::swap(v, w);
3
In reality, for the sake of performance, the Standard Library provides a specialized version std::swap for the
speciﬁc case of std::vector. Although less performing, the general purpose version would however equally work

How can we assure our UDTs to be regular?

What does it mean to be regular?
Ul#mately, we need to inves#gate how several of the built-in
operators should behave when applied to user-deﬁned types

This amounts to understanding how to apply the fundamental
opera5ons to UDTs
┌────────────────────────┬──────────┐
│ Default construction │ T a; │
├────────────────────────┼──────────┤
│ Copy construction │ T a = b; │
├────────────────────────┼──────────┤
│ Destruction │ ~T(a); │
├────────────────────────┼──────────┤
│ Assignment │ a = b; │
├────────────────────────┼──────────┤
│ Equality │ a == b; │
├────────────────────────┼──────────┤
│ Inequality │ a != b; │
├────────────────────────┼──────────┤
└────────────────────────┴──────────┘

First, we no,ce that the fundamental opera,ons can be divided
into three groups
┌────────────────────────┬──────────┐
│ │ T a; │
│ ├──────────┤
│ │ T a = b; │
│ Copying and assignment ├──────────┤
│ │ ~T(a); │
│ ├──────────┤
│ │ a = b; │
├────────────────────────┼──────────┤
│ │ a == b; │
│ Equality ├──────────┤
│ │ a != b; │
├────────────────────────┼──────────┤
└────────────────────────┴──────────┘

Let us now inves,gate in greater detail the proper%es of each such
group
┌────────────────────────┬──────────┐
│ │ T a; │
│ ├──────────┤
│ │ T a = b; │
│ Copying and assignment ├──────────┤
│ │ ~T(a); │
│ ├──────────┤
│ │ a = b; │
├────────────────────────┼──────────┤
│ │ a == b; │
│ Equality ├──────────┤
│ │ a != b; │
├────────────────────────┼──────────┤
└────────────────────────┴──────────┘

Copy and assignment are interchangeable
We know that built-in types allow wri4ng interchangeably:
T a = b; // copy construction

Copy and assignment are interchangeable
The ra'onale behind this is that it allows natural code such as:
T a;
if (something) a = b;
else a = c;

Default construc.on
Observe that we need default construc.on to make assignment
and copy construc7on interchangeable
T a = b; T a; a = b; T a;

Default construc.on
Note that we require a default constructed regular type to be
assignable and destruc.ble

Par$al construc$on
That is, we require default ini2aliza2on only to par$ally form an
object

Preserving equality
The next step is to note that for all built-in types copy construc5on
and assignment are equality-preserving opera5ons
T a = b; // copy construction
a == b; // true!

Preserving equality
The next step is to note that for all built-in types copy construc5on
and assignment are equality-preserving opera5ons
a == b; // true!

Preserving equality
• Copy construc-ng means to make a copy that is equal to the
original
• Assignment copies the right-hand side object to the le;-hand
side object, leaving their values equal

Dependency on equality
Since we need the ability to test for equality, we have that copy
construc7on and assignment depend on equality

A reﬁned deﬁni)on of regularity
In other words, regular types possess the equality operator and
equality-preserving copy construc4on and assignment

Deﬁning equality
Although equality is conceptually central, we do not yet have a
sa.sfactory deﬁni.on of equality of two objects of a regular type

Equality for built-in types
On built-in types, the equality rela3on is normally deﬁned as
bitwise equality

Equality for UDTs
Star%ng from this, we can devise a natural deﬁni,on of equality for
user-deﬁned types

Equality for UDTs
Namely, we can deﬁne equality of user-deﬁned types from the
equality of its parts

Equality for UDTs
However, in order to build the equality operator from the equality
operator of its parts, we must iden%fy its parts

Example: the point type
struct point {
point(double x, double y): x(x), y(y) {}
double x;
double y;
};

Example: operator== for points
return p1.x == p2.x &&
p1.y == p2.y;
}

Remote parts
Complex objects are o&en constructed out of mul0ple simpler
objects, connected by pointers

Remote parts
In such cases, we say that the given complex object has remote
parts

Example: the person type
struct mail_address {...};
class person {
public:
person(...) {...}
private:
std::string name;
std::string surname;
mail_address* address;
};

1st
a&empt: operator== for persons
friend
bool operator==(person const& p1, person const& p2) {
return p1.name == p2.name &&
p1.surname == p2.surname;
}

2nd
friend
p1.surname == p2.surname &&
p1.address == p2.address;
}

3rd
friend
p1.surname == p2.surname &&
*p1.address == *p2.address;
}

Comparing remote parts
For objects with remote parts, the equality operator must compare
the corresponding remote parts, rather than the pointers to them

Comparing remote parts
We do not want objects with equal remote parts to compare
unequal just because the remote parts were in diﬀerent memory
loca3ons

Deﬁni&on of equality
We can therefore devise the following deﬁni4on of equality:
Two objects are equal if their corresponding parts are equal (applied
recursively), including remote parts (but not comparing their addresses),
excluding inessen>al components, and excluding components which
iden>fy related objects

Equality comes with inequality
It is not enough to deﬁne equality, we also need to deﬁne
inequality to match it

Unequal vs. not equal
We need to do so in order to preserve the ability to write
interchangeably:
a != b; // unequal
!(a == b); // not equal

Unequal vs. not equal
That is, the statements "two things are unequal to each other" and
"two 6ngs are not equal to each other" should be equivalent
a != b; // unequal
!(a == b); // not equal

Inequality operator
Fortunately, this is very simple:
// T is some user-defined type
bool operator!=(T const& x, T const& y) {
return !(x == y);
}

Equality and inequality
C++ does not enforce that equality and inequality should be
defined together, so remember to define inequality every 7me
equality is defined

The importance of ordering
We need to have ordering if we want to ﬁnd things quickly, as
ordering enables sor$ng, and therefore binary search

Ordering
For deﬁning ordering on a user-deﬁned type we need to set in
place the four rela%onal operators <, >, <=, >=

Rela%onal operators
However, we need to implement only one rela(onal operator,
which can then be used to derive the deﬁni8on on the remaining
three

Less-than operator
We choose < as the primary operator, since when we refer to
ordering we usually mean ascending ordering

Less-than operator seman.cs
Intui&vely, since we are overloading the less-than operator <, the
ordering criterion we implement must behave like <

How to make our overload behave the way that < behaves?

Strictness
First, let us no-ce this natural property of < when used as an
arithme-c operator:
5 ≮ 5 12 ≮ 12 etc.

Strictness
Hence, also our ordering rela0on of choice must be strict, that is:
a ≮ a for all a in T

Is this deﬁni+on strict?
bool operator<(T const& x, T const& y) {
return true;
}

Is this deﬁni+on strict?
return true; // according to this 5 < 5,
// which violates strictness
}

Transi'vity
Second, it is immediate to see that the less-than operator is
transi've:
4 < 7 and 7 < 12 implies 4 < 12

Transi'vity
Hence, also our ordering rela0on of choice must be transi've
// maintain transitivity
}

Trichotomy law
Finally, the less-than operator < obeys the trichotomy law:
For every pair of elements, exactly one of the following holds:
a < b, b < a or a == b

Trichotomy law
Note that the trichotomy law requires the deﬁni5on of < to be
consistent with the deﬁni5on of == so that:
!(a < b) && !(b < a) is equivalent to a == b

Equality is central
Hence, as with assignment and copy, also ordering depends on
equality
!(a < b) && !(b < a) is equivalent to a == b

Is this a well-formed overload?
bool operator<(point const& p1, point const& p2) {
return p1.x < p2.x;
}

It respects strictness
return p1.x < p2.x; // (5, 2) ≮ (5, 2)
}

It respects transi,vity
return p1.x < p2.x; // (5,2) < (6, 3) < (10, 5)
}

But it does not obey to the trichotomy law
return p1.x < p2.x; // (5, 3) ≮ (5, 8) and
// (5, 8) ≮ (5, 3) but...
// (5, 8) ≠ (5, 3)
}

if (p1.x < p2.x) return true;
if (p1.x > p2.x) return false;
return p1.y < p2.y;
}

We are imposing a lexicographic ordering over points
return p1.y < p2.y;
}

Lexicographic ordering respects strictness, transi2vity and the
trichotomy law
return p1.y < p2.y;
}

Strict total ordering
We refer to an ordering rela.on that is strict, transi.ve and that
obeys the trichotomy law as a strict total ordering

Remaining operators
As with equality, we must deﬁne manually the remaining rela5onal
operators >, >=, <=

Greater-than operators
bool operator>(T const& x, T const& y) {
return y < x;
}

Greater-or-equal-than operator
bool operator>=(T const& x, T const& y) {
return !(x < y);
}

Less-or-equal-than operator
bool operator<=(T const& x, T const& y) {
return !(y < x);
}

Semi-regular &
totally ordered types

Regularity
Some%mes, although a type looks like a regular type, we cannot
formally guarantee its regularity

complex seems regular
For instance, the complex type we previously wrote feels like a
regular type
complex a; // default construction
complex b = a; // copy construction
complex c; c = b; // assignment
a == b; // equality
a != c; // inequality

complex numbers cannot be ordered
However, we cannot impose a natural order on complex numbers4
complex a(5, 2);
complex b(8, 0);
a < b; // not defined
4
See, e.g., h)p://math.stackexchange.com/a/488015/79684

complex is non-regular
As such, complex is formally a non-regular type
complex a(5, 2);
complex b(8, 0);

complex is non-regular
However, this might be too restric1ve. For this reason, we opt for a
more ﬁne-grained classiﬁca.on

A relaxed no+on of regularity
We relax our deﬁni.on of regular types and consider regular also
types for which it is not possible to devise a natural order
complex a(5, 2);
complex b(8, 0);

Totally ordered types
We instead refer to regular types for which a natural order is
possible as totally ordered types

Semi-regular types
Similarly, we refer to type that supports equality-preserving copy
and assignment without being equality comparable as semi-regular

Semi-regular types
Intui&vely, this means that a semi-regular type is copyable

Type classiﬁca,on
┌────────────────────────┬──────────┐
│ │ T a; │
│ ├──────────┤
│ │ T a = b; │
│ Semi-regular ├──────────┤
│ │ ~T(a); │
│ ├──────────┤
│ │ a = b; │
├────────────────────────┼──────────┤
│ │ a == b; │
│ Regular ├──────────┤
│ │ a != b; │
├────────────────────────┼──────────┤
│ Totally ordered │ a < b; │
└────────────────────────┴──────────┘

vector_int
class vector_int {
public:
vector_int(): sz(0), elem(nullptr) {}
vector_int(std::size_t sz): sz(sz), elem(new int[sz]) {...}
vector_int(vector_int const& v): sz(v.sz), elem(new int[v.sz]) {...}
~vector_int() {...}
std::size_t size() const {...}
int operator[](std::size_t i) const {...}
int& operator[](std::size_t i) {...}
vector_int& operator=(vector_int const& v) {...}
private:
std::size_t sz;
int* elem;
};

Making vector_int regular
In order to make vector_int regular we need to deﬁne
(in)equality and the four rela-on operators

Equality
class vector_int {
public:
...
bool operator==(vector_int const& v) const {
if (sz != v.sz) return false;
for (std::size_t i = 0; i < sz; ++i)
if (elem[i] != v.elem[i]) return false;
return true;
}
};

Less-than operator
class vector_int {
public:
...
bool operator<(vector_int const& v) const {
std::size_t min_size = std::min(sz, v.sz);
std::size_t i = 0;
while (i < min_size && elem[i] == v.elem[i]) ++i;
if (i < min_size)
return elem[i] < v.elem[i];
else
return sz < v.sz;
}
};

Bibliography
• J.C. Dehnert, A. Stepanov, Fundamentals of generic programming
• A. Stepanov, Notes on programming
• A. Stepanov, P. McJones, Elements of programming

Bibliography
• A. Stepanov, D. Rose, From Mathema6cs to Generic
Programming
• A. Stepanov, Eﬃcient Programming with Components (Lecture 1
and Lecture 2)

Regular types in C++

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