Do array elements count as a common initial sequence? - c++

Sort of related to my previous question:
Do elements of arrays count as a common initial sequence?
struct arr4 { int arr[4]; };
struct arr2 { int arr[2]; };
union U
{
arr4 _arr4;
arr2 _arr2;
};
U u;
u._arr4.arr[0] = 0; //write to active
u._arr2.arr[0]; //read from inactive
According to this cppreference page:
In a standard-layout union with an active member of non-union class type T1, it is permitted to read a non-static data member m of another union member of non-union class type T2 provided m is part of the common initial sequence of T1 and T2....
Would this be legal, or would it also be illegal type punning?

C++11 says (9.2):
If a standard-layout union contains two or more standard-layout structs that share a common initial sequence,
and if the standard-layout union object currently contains one of these standard-layout structs, it is permitted
to inspect the common initial part of any of them. Two standard-layout structs share a common initial
sequence if corresponding members have layout-compatible types and either neither member is a bit-field or
both are bit-fields with the same width for a sequence of one or more initial members.
As to whether arrays of different size form a valid common initial sequence, 3.9 says:
If two types T1 and T2 are the same type, then T1 and T2 are layout-compatible types
These arrays are not the same type, so this doesn't apply. There is no special further exception for arrays, so the arrays may not be layout-compatible and do not form a common initial sequence.
In practice, though, I know of a compiler (GCC) which:
ignores the "common initial sequence" rule, and
allows type punning anyway, but only when accesses are "via the union type" (as in your example), in which case the "common initial sequence" rule is obeyed indirectly (because a "common initial sequence" implies a common initial layout on the architectures the compiler supports).
I suspect many other compilers take a similar approach. In your example, where you type-pun via the union object, such compilers will give you the expected result - reading from the inactive member should give you value written via the inactive member.

The C Standard would allow an implementation to vary the placement of an array object within a structure based upon the number of elements. Among other things, there may be some circumstances where it may be useful to word-align a byte array which would occupy exactly one word, but not to word-align arrays of other sizes. For example, on a system with 8-bit char and 32-bit words, processing a structure such as:
struct foo {
char header;
char dat[4];
};
in a manner that word-aligns dat may allow an access to dat[i] to be processed by loading a word and shifting it right by a 0, 8, 16, or 24 bits, but such advantages might not be applicable had the structure instead been:
struct foo {
char header;
char dat[5];
};
The Standard was clearly not intended to forbid implementations from laying out structures in such ways, on platforms where doing so would be useful. On the other hand, when the Standard was written, compilers which would place arrays within a structure at offsets that were unaffected by the arrays' sizes would unanimously behave as though array elements that were present in two structures were part of the same Common Initial Sequence, and nothing in the published Rationale for the Standard suggests any intention to discourage such implementations from continuing to behave in such fashion. Code which relied upon such treatment would have been "non-portable", but correct on all implementations which followed common struct layout practices.

Related

ARM Neon: How to convert from uint8x16_t to uint8x8x2_t?

I recently discovered about the vreinterpret{q}_dsttype_srctype casting operator. However this doesn't seem to support conversion in the data type described at this link (bottom of the page):
Some intrinsics use an array of vector types of the form:
<type><size>x<number of lanes>x<length of array>_t
These types are treated as ordinary C structures containing a single
element named val.
An example structure definition is:
struct int16x4x2_t
{
int16x4_t val[2];
};
Do you know how to convert from uint8x16_t to uint8x8x2_t?
Note that that the problem cannot be reliably addressed using union (reading from inactive members leads to undefined behaviour Edit: That's only the case for C++, while it turns out that C allows type punning), nor by using pointers to cast (breaks the strict aliasing rule).
It's completely legal in C++ to type pun via pointer casting, as long as you're only doing it to char*. This, not coincidentally, is what memcpy is defined as working on (technically unsigned char* which is good enough).
Kindly observe the following passage:
For any object (other than a base-class subobject) of trivially
copyable type T, whether or not the object holds a valid value of type
T, the underlying bytes (1.7) making up the object can be copied into
an array of char or unsigned char.
42 If the content of the array of char or unsigned char is copied back
into the object, the object shall subsequently hold its original
value. [Example:
#define N sizeof(T)
char buf[N];
T obj;
// obj initialized to its original value
std::memcpy(buf, &obj, N);
// between these two calls to std::memcpy,
// obj might be modified
std::memcpy(&obj, buf, N);
// at this point, each subobject of obj of scalar type
// holds its original value
— end example ]
Put simply, copying like this is the intended function of std::memcpy. As long as the types you're dealing with meet the necessary triviality requirements, it's totally legit.
Strict aliasing does not include char* or unsigned char*- you are free to alias any type with these.
Note that for unsigned ints specifically, you have some very explicit leeway here. The C++ Standard requires that they meet the requirements of the C Standard. The C Standard mandates the format. The only way that trap representations or anything like that can be involved is if your implementation has any padding bits, but ARM does not have any- 8bit bytes, 8bit and 16bit integers. So for unsigned integers on implementations with zero padding bits, any byte is a valid unsigned integer.
For unsigned integer types other than unsigned char, the bits
of the object representation shall be divided into two groups:
value bits and padding bits (there need not be any of the
latter). If there are N value bits, each bit shall represent
a different power of 2 between 1 and 2N−1, so that objects
of that type shall be capable of representing values from 0
to 2N−1 using a pure binary representation; this shall be
known as the value representation. The values of any padding bits are
unspecified.
Based on your comments, it seems you want to perform a bona fide conversion -- that is, to produce a distinct, new, separate value of a different type. This is a very different thing than a reinterpretation, such as the lead-in to your question suggests you wanted. In particular, you posit variables declared like this:
uint8x16_t a;
uint8x8x2_t b;
// code to set the value of a ...
and you want to know how to set the value of b so that it is in some sense equivalent to the value of a.
Speaking to the C language:
The strict aliasing rule (C2011 6.5/7) says,
An object shall have its stored value accessed only by an lvalue
expression that has one of the following types:
a type compatible with the effective type of the object, [...]
an aggregate or union type that includes one of the aforementioned types among its members [...], or
a character type.
(Emphasis added. Other enumerated options involve differently-qualified and differently-signed versions of the of the effective type of the object or compatible types; these are not relevant here.)
Note that these provisions never interfere with accessing a's value, including the member value, via variable a, and similarly for b. But don't overlook overlook the usage of the term "effective type" -- this is where things can get bolluxed up under slightly different circumstances. More on that later.
Using a union
C certainly permits you to perform a conversion via an intermediate union, or you could rely on b being a union member in the first place so as to remove the "intermediate" part:
union {
uint8x16_t x1;
uint8x8_2_t x2;
} temp;
temp.x1 = a;
b = temp.x2;
Using a typecast pointer (to produce UB)
However, although it's not so uncommon to see it, C does not permit you to type-pun via a pointer:
// UNDEFINED BEHAVIOR - strict-aliasing violation
b = *(uint8x8x2_t *)&a;
// DON'T DO THAT
There, you are accessing the value of a, whose effective type is uint8x16_t, via an lvalue of type uint8x8x2_t. Note that it is not the cast that is forbidden, nor even, I'd argue, the dereferencing -- it is reading the dereferenced value so as to apply the side effect of the = operator.
Using memcpy()
Now, what about memcpy()? This is where it gets interesting. C permits the stored values of a and b to be accessed via lvalues of character type, and although its arguments are declared to have type void *, this is the only plausible interpretation of how memcpy() works. Certainly its description characterizes it as copying characters. There is therefore nothing wrong with performing a
memcpy(&b, &a, sizeof a);
Having done so, you may freely access the value of b via variable b, as already mentioned. There are aspects of doing so that could be problematic in a more general context, but there's no UB here.
However, contrast this with the superficially similar situation in which you want to put the converted value into dynamically-allocated space:
uint8x8x2_t *c = malloc(sizeof(*c));
memcpy(c, &a, sizeof a);
What could be wrong with that? Nothing is wrong with it, as far as it goes, but here you have UB if you afterward you try to access the value of *c. Why? because the memory to which c points does not have a declared type, therefore its effective type is the effective type of whatever was last stored in it (if that has an effective type), including if that value was copied into it via memcpy() (C2011 6.5/6). As a result, the object to which c points has effective type uint8x16_t after the copy, whereas the expression *c has type uint8x8x2_t; the strict aliasing rule says that accessing that object via that lvalue produces UB.
So there are a bunch of gotchas here. This reflects C++.
First you can convert trivially copyable data to char* or unsigned char* or c++17 std::byte*, then copy it from one location to another. The result is defined behavior. The values of the bytes are unspecified.
If you do this from a value of one one type to another via something like memcpy, this can result in undefined behaviour upon access of the target type unless the target type has valid values for all byte representations, or if the layout of the two types is specified by your compiler.
There is the possibility of "trap representations" in the target type -- byte combinations that result in machine exceptions or something similar if interpreted as a value of that type. Imagine a system that doesn't use IEEE floats and where doing math on NaN or INF or the like causes a segfault.
There are also alignment concerns.
In C, I believe that type punning via unions is legal, with similar qualifications.
Finally, note that under a strict reading of the c++ standard, foo* pf = (foo*)malloc(sizeof(foo)); is not a pointer to a foo even if foo was plain old data. You must create an object before interacting with it, and the only way to create an object outside of automatic storage is via new or placement new. This means you must have data of the target type before you memcpy into it.
Do you know how to convert from uint8x16_t to uint8x8x2_t?
uint8x16_t input = { 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15 };
uint8x8x2_t output = { vget_low_u8(input), vget_high_u8(input) };
One must understand that with neon intrinsics, uint8x16_t represents a 16-byte register; while uint8x8x2_t represents two adjacent 8-byte registers. For ARMv7 these may be the same thing (q0 == {d0, d1}) but for ARMv8 the register layout is different. It's necessary to get (extract) the low 8 bytes and the high 8 bytes of the single 16-byte register using two functions. The clang compiler will determine which instruction(s) are necessary based on the context.

`std::complex<T>[n]` and `T[n*2]` type aliasing

Since C++11 std::complex<T>[n] is guaranteed to be aliasable as T[n*2], with well defined values. Which is exactly what one would expect for any mainstream architecture. Is this guarantee achievable with standard C++ for my own types, say struct vec3 { float x, y, z; } or is it only possible with special support from the compiler?
TL;DR: The compiler must inspect reinterpret_casts and figure out that (standard library) specializations of std::complex are involved. We cannot conformably mimic the semantics.
I think it's fairly clear that treating three distinct members as array elements is not going to work, since pointer arithmetic on pointers to them is extremely restricted (e.g. adding 1 yields a pointer past-the-end).
So let's assume vec3 contained an array of three ints instead.
Even then, the underlying reinterpret_cast<int*>(&v) you implicitly need (where v is a vec3) does not leave you with a pointer to the first element. See the exhaustive requirements on pointer-interconvertibility:
Two objects a and b are pointer-interconvertible if:
they are the same object, or
one is a standard-layout union object and the other is a non-static data member of that object ([class.union]), or
one is a standard-layout class object and the other is the first non-static data member of that object, or, if the object has no
non-static data members, the first base class subobject of that object
([class.mem]), or
there exists an object c such that a and c are pointer-interconvertible, and c and b are
pointer-interconvertible.
If two objects are pointer-interconvertible, then they have the same
address, and it is possible to obtain a pointer to one from a pointer
to the other via a reinterpret_­cast. [ Note: An array object
and its first element are not pointer-interconvertible, even though
they have the same address.  — end note ]
That's quite unequivocal; while we can get a pointer to the array (being the first member), and while pointer-interconvertibility is transitive, we cannot obtain a pointer to its first element.
And finally, even if you managed to obtain a pointer to the first element of your member array, if you had an array of vec3s, you cannot traverse all the member arrays using simple pointer increments, since we get pointers past-the-end of the arrays in between. launder doesn't solve this problem either, because the objects that the pointers are associated with don't share any storage (cf [ptr.launder] for specifics).
It's only possible with special support from the compiler, mostly.
Unions don't get you there because the common approach actually has undefined behaviour, although there are exceptions for layout-compatible initial sequences, and you may inspect an object through an unsigned char* as a special case. That's it, though.
Interestingly, unless we assume a broad and useless meaning of "below", the standard is technically contradictory in this regard:
[C++14: 5.2.10/1]: [..] Conversions that can be performed explicitly using reinterpret_cast are listed below. No other conversion can be performed explicitly using reinterpret_cast.
The case for complex<T> is then not mentioned. Finally the rule you're referring to is introduced much, much later, in [C++14: 26.4/4].
I think it would work for a single vec3 if your type contained float x[3] instead, and you ensure sizeof(vec3) == 3*sizeof(float) && is_standard_layout_v<vec3>. Given those conditions, the standard guarantees that the first member is at zero offset so the address of the first float is the address of the object, and you can perform array arithmetic to get the other elements in the array:
struct vec3 { float x[3]; } v = { };
float* x = reinterpret_cast<float*>(&v); // points to first float
assert(x == v.x);
assert(&x[0] == &v.x[0]);
assert(&x[1] == &v.x[1]);
assert(&x[2] == &v.x[2]);
What you can't do is treat an array of vec3 as an array of floats three times the length. Array arithmetic on the array inside each vec3 won't allow you to access the array inside the next vec3. CWG 2182 is relevant here.

Are members of a POD-struct or standard layout type guaranteed to be aligned according to their alignment requirements?

Given a POD-struct (in C++03) or a standard layout type (in C++11), with all members having a fundamental alignment requirement, is it true that every member is guaranteed to be aligned according to its alignment requirement?
In other words, for all members m_k in { m0 ... mn } of standard layout type S,
struct S {
T0 m0;
T1 m1;
...
TN mn;
};
is the following expression guaranteed to evaluate to true?
(offsetof(S,m_k) % alignof(decltype(S::m_k))) == 0
Please give answers for both C++03 and C++11 and cite the relevant parts of the standard. Supporting evidence from C standards would also be helpful.
My reading of the C++03 standard (ISO/IEC 14882:2003(E)) is that it is silent regarding the alignment of members within a POD-struct, except for the first member. The relevant paragraphs are:
In the language of the specification, an object is a "region of storage":
1.8 The C + + object model [intro.object]
1.8/1 The constructs in a C + + program create, destroy, refer to, access, and manipulate objects. An object is a region of storage. ...
Objects are allocated according to their alignment requirement:
3.9 Types [basic.types]
3.9/5 Object types have alignment requirements (3.9.1, 3.9.2). The alignment of a complete object type is an implementation-defined integer value representing a number of bytes; an object is allocated at an address that meets the alignment requirements of its object type.
Fundamental types have alignment requirements:
3.9.1 Fundamental types [basic.fundamental]
3.9.1/3 For each of the signed integer types, there exists a corresponding (but different) unsigned integer type: "unsigned char", "unsigned short int", "unsigned int", and "unsigned long int," each of which occupies the same amount of storage and has the same alignment requirements (3.9) as the corresponding signed integer type;...
Padding may occur due to "implementation alignment requirements":
9.2 Class members [class.mem]
9.2/12 Nonstatic data members of a (non-union) class declared without an intervening access-specifier are allocated so that later members have higher addresses within a class object. The order of allocation of nonstatic
data members separated by an access-specifier is unspecified (11.1). Implementation alignment requirements might cause two adjacent members not to be allocated immediately after each other; so might requirements for space for managing virtual functions (10.3) and virtual base classes (10.1).
Does the word "allocated" in 9.2/12 have the same meaning as "allocated" in 3.9/5? Most uses of "allocated" in the spec refer to dynamic storage allocation, not struct-internal layout. The use of may in 9.2/12 seems to imply that the alignment requirements of 3.9/5 and 3.9.1/3 might not be strictly required for struct members.
The first member of a POD-struct will be aligned according to the alignment requirement of the struct:
9.2/17 A pointer to a POD-struct object, suitably converted using a reinterpret_cast, points to its initial member (or if that member is a bit-field, then to the unit in which it resides) and vice versa. [Note: There might therefore be unnamed padding within a POD-struct object, but not at its beginning, as necessary to achieve appropriate alignment. ]
[Emphasis added in all of the above quotes.]
Each element of a POD struct is itself an object, and objects can only be allocated in accordance with the alignment requirements for those objects. The alignment requirements may change, though, due to the fact that something is a sub-object of another object ([basic.align]/1, 2:
1 Object types have alignment requirements (3.9.1, 3.9.2) which place restrictions on the addresses at which an object of that type may be allocated. An alignment is an implementation-defined integer value representing the number of bytes between successive addresses at which a given object can be allocated. An object type imposes an alignment requirement on every object of that type; stricter alignment can be requested using the alignment specifier (7.6.2).
2 A fundamental alignment is represented by an alignment less than or equal to the greatest alignment supported by the implementation in all contexts, which is equal to alignof(std::max_align_t) (18.2). The alignment required for a type might be different when it is used as the type of a complete object and when it is used as the type of a subobject. [Example:
struct B { long double d; };
struct D : virtual B { char c; }
When D is the type of a complete object, it will have a subobject of type B, so it must be aligned appropriately for a long double. If D appears as a subobject of another object that also has B as a virtual base class, the
B subobject might be part of a different subobject, reducing the alignment requirements on the D subobject.—end example ] The result of the alignof operator reflects the alignment requirement of the type in the
complete-object case.
[emphasis added]
Although the examples refer to a sub-object via inheritance, the normative wording just refers to sub-objects in general, so I believe the same rules apply, so on one hand you can assume that each sub-object is aligned so that it can be accessed. On the other hand, no you can't necessarily assume that will be the same alignment that alignof gives you.
[The reference is from N4296, but I believe the same applies to all recent versions. C++98/03, of course, didn't have alignof at all, but I believe the same basic principle applies--members will be aligned so they can be used, but that alignment requirement isn't necessarily the same as when they're used as independent objects.]
There are two distinct alignment requirements for each type. One corresponds to complete objects, the other one to subobjects. [basic.align]/2:
The alignment required for a type might be different when it is used
as the type of a complete object and when it is used as the type of a
subobject. [ Example:
struct B { long double d; };
struct D : virtual B { char c; };
When D is the type of a complete object, it will have a subobject of
type B, so it must be aligned appropriately for a long double. If
D appears as a subobject of another object that also has B as a
virtual base class, the B subobject might be part of a different
subobject, reducing the alignment requirements on the D subobject.
—end example ] The result of the alignof operator reflects the alignment requirement of the type in the complete-object case.
I can't think of any specific example for member subobjects - and there assuredly is none! -, but the above paragraph concedes the possibility that a member subobject has weaker alignment than alignof will yield, which would fail your condition.

Is sizeof(T) == sizeof(int)?

I've been poring over the draft standard and can't seem to find what I'm looking for.
If I have a standard-layout type
struct T {
unsigned handle;
};
Then I know that reinterpret_cast<unsigned*>(&t) == &t.handle for some T t;
The goal is to create some vector<T> v and pass &v[0] to a C function that expects a pointer to an array of unsigned integers.
So, does the standard define sizeof(T) == sizeof(unsigned) and does that imply that an array of T would have the same layout as an array of unsigned?
While this question addresses a very similar topic, I'm asking about the specific case where both the data member and the class are standard layout, and the data member is a fundamental type.
I've read some paragraphs that seem to hint that maybe it might be true, but nothing that hits the nail on the head. For example:
§ 9.2.17
Two standard-layout struct (Clause 9) types are layout-compatible if
they have the same number of non-static data members and corresponding
non-static data members (in declaration order) have layout-compatible
types
This isn't quite what I'm looking for, I don't think.
You essentially are asking, given:
struct T {
U handle;
};
whether it's guaranteed that sizeof(T) == sizeof(U). No, it is not.
Section 9.2/17 of the ISO C++03 standard says:
A pointer to a POD-struct object, suitably converted using a
reinterpret_cast, points to its initial member (or if that member is a
bit-field, then to the unit in which it resides) and vice versa.
Suppose you have an array of struct T. The vice versa part means that the address of any of the T::handle members must also be a valid address of a struct T. Now, suppose that these members are of type char and that your claim is true. This would mean that struct T would be allowed to have an unaligned address, which seems rather unlikely. The standard usually tries to not tie the hands of implementations in such a way. For your claim to be true, the standard would have to require that struct T be allowed to have unaligned addresses. And it would have to be allowed for all structures, because struct T could be a forward-declared, opaque type.
Furthermore, section 9.2/17 goes on to state:
[Note: There might therefore be unnamed padding within a POD-struct object, but not at its beginning, as necessary to achieve appropriate alignment.]
Which, taken a different way, means that there is no guarantee that there will never be padding.
I am used to Borland environments and for them:
T is a struct in your case so sizeof(T) is size of struct
that depends on #pragma pack and align setting of your compiler
so sometimes it can be greater than sizeof(unsigned) !!!
for the same reason if you have 4Byte struct (uint32) and 16Byte allign
struct T { uint32 u; };
then T a[100] is not the same as uint32 a[100];
because T is uint32 + 12 Byte empty space !!!
RETRACTION: The argument is erroneous. The proof of Lemma 2 relies on a hidden premise that the alignment of an aggregate type is determined strictly by the alignments of its member types. As Dyp points out in the commentary, that premise is not supported by the standard. It is therefore admissible for struct { Foo f } to have a more strict alignment requirement that Foo.
I'll play devil's advocate here, since no one else seems to be willing. I will argue that standard C++ (I'll refer to N3797 herein) guarantees that sizeof(T) == sizeof(U) when T is a standard layout class (9/7) with default alignment having a single default-aligned non-static data member U, e.g,
struct T { // or class, or union
U u;
};
It's well-established that:
sizeof(T) >= sizeof(U)
offsetof(T, u) == 0 (9.2/19)
U must be a standard layout type for T to be a standard layout class
u has a representation consisting of exactly sizeof(U) contiguous bytes of memory (1.8/5)
Together these facts imply that the first sizeof(U) bytes of the representation of T are occupied by the representation of u. If sizeof(T) > sizeof(U), the excess bytes must then be tail padding: unused padding bytes inserted after the representation of u in T.
The argument is, in short:
The standard details the circumstances in which an implementation may add padding to a standard-layout class,
None of those cirumstances applies in this particular instance, and hence
A conforming implementation may not add padding.
Potential Sources of Padding
Under what circumstances does the standard allow an implementation to add such padding to the representation of a standard layout class? When it's necessary for alignment: per 3.11/1, "An alignment is an implementation-defined integer value representing the number of bytes between successive addresses at which a given object can be allocated." There are two mentions of adding padding, both for alignment reasons:
5.3.3 Sizeof [expr.sizeof]/2 states "When applied to a reference or a reference type, the result is the size of the referenced type. When applied
to a class, the result is the number of bytes in an object of that class including any padding required for placing objects of that type in an array. The size of a most derived class shall be greater than zero (1.8). The result of applying sizeof to a base class subobject is the size of the base class type.77 When applied to an array, the result is the total number of bytes in the array. This implies that the size of an array of n elements is n times the size of an element."
9.2 Class members [class.mem]/13 states "Implementation alignment requirements might cause two adjacent members not to be allocated immediately after each other; so might requirements for space for managing virtual functions (10.3) and virtual base classes (10.1)."
(Notably the C++ standard does not contain a blanket statement allowing implementations to insert padding in structures as in the C standards, e.g., N1570 (C11-ish) §6.7.2.1/15 "There may be unnamed padding within a structure object, but not at its beginning." and /17 "There may be unnamed padding at the end of a structure or union.")
Clearly the text of 9.2 doesn't apply to our problem, since (a) T has only one member and thus no "adjacent members", and (b) T is standard layout and hence has no virtual functions or virtual base classes (per 9/7). Demonstrating that 5.3.3/2 doesn't allow padding in our problem is more challenging.
Some Prerequisites
Lemma 1: For any type W with default alignment, alignof(W) divides sizeof(W): By 5.3.3/2, the size of an array of n elements of type W is exactly n times sizeof(W) (i.e., there is no "external" padding between array elements). The addresses of consecutive array elements are then sizeof(W) bytes apart. By the definition of alignment, it must then be that alignof(W) divides sizeof(W).
Lemma 2: The alignment alignof(W) of a default-aligned standard layout class W with only default-aligned data members is the least common multiple LCM(W) of the alignments of the data members (or 1 if there are none): Given an address at which an object of W can be allocated, the address LCM(W) bytes away must also be appropriately aligned: the difference between the addresses of member subobjects would also be LCM(W) bytes, and the alignment of each such member subobject divides LCM(W). Per the definition of alignment in 3.11/1, we have that alignof(W) divides LCM(W). Any whole number of bytes n < LCM(W) must not be divisible by the alignment of some member v of W, so an address that is only n bytes away from an address at which an object of W can be allocated is consequently not appropriately aligned for an object of W, i.e., alignof(W) >= LCM(W). Given that alignof(W) divides LCM(W) and alignof(W) >= LCM(W), we have alignof(W) == LCM(W).
Conclusion
Application of this lemma to the original problem has the immediate consequence that alignof(T) == alignof(U). So how much padding might be "required for placing objects of that type in an array"? None. Since alignof(T) == alignof(U) by the second lemma, and alignof(U) divides sizeof(U) by the first, it must be that alignof(T) divides sizeof(U) so zero bytes of padding are required to place objects of type T in an array.
Since all possible sources of padding bytes have been eliminated, an implementation may not add padding to T and we have sizeof(T) == sizeof(U) as required.

Are anonymous unions acceptable for aliasing member variables in a struct?

Let's say that I have the following C++ code:
struct something
{
// ...
union { int size, length; };
// ...
};
This would create two members of the struct which access the same value: size and length.
Would treating the two members as complete aliases (i.e. setting the size, then accessing the length and vice/versa) be undefined behaviour? Is there a "better" way to implement this type of behaviour, or is this an acceptable implementation?
Yes, this is allowed and well-defined. According to §3.10 [basic.lval]:
10/ If a program attempts to access the stored value of an object through a glvalue of other than one of the following types the behavior is undefined:
— the dynamic type of the object
[...]
Since here we store an int and read through an int, we access the object through a glvalue of the same dynamic type than the object, thus things are fine.
There even is a special caveat in the Standard for structures that share the same prefix. Or, in standardese, standard-layout types that share a common initial sequence.
§9.2/18 If a standard-layout union contains two or more standard-layout structs that share a common initial sequence, and if the standard-layout union object currently contains one of these standard-layout structs, it is permitted to inspect the common initial part of any of them. Two standard-layout structs share a common initial sequence if corresponding members have layout-compatible types and either neither member is a bit-field or both are bit-fields with the same width for a sequence of one or more initial members.
That is:
struct A { unsigned size; char type; };
struct B { unsigned length; unsigned capacity; };
union { A a; B b; } x;
assert(x.a.size == x.b.length);
EDIT: Given that int is not a struct (nor a class) I am afraid it's actually not formally defined (I certainly could not see anything in the Standard), but should be safe in practice... I've brought the matters to the isocpp forums; you might have found a hole.
EDIT: Following the above mentionned discussion, I have been shown §3.10/10.
It is not undefined behavior. Both of the aliases in the union will be accessing the same location in the memory. See below:
§9.2/18 If a standard-layout union contains two or more
standard-layout structs that share a common initial sequence, and if
the standard-layout union object currently contains one of these
standard-layout structs, it is permitted to inspect the common initial
part of any of them. Two standard-layout structs share a common
initial sequence if corresponding members have layout-compatible types
and either neither member is a bit-field or both are bit-fields with
the same width for a sequence of one or more initial members.
It is undefined if types have different initial sequence.
Values will be same. If you assign 5 to size then length will also be 5.