Changeset 720eec9 for doc/theses


Ignore:
Timestamp:
Oct 25, 2024, 4:09:56 PM (3 weeks ago)
Author:
Michael Brooks <mlbrooks@…>
Branches:
master
Children:
33474e6
Parents:
d031f7f
Message:

Thesis, array, proofread intro sections.

Location:
doc/theses/mike_brooks_MMath
Files:
4 edited

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  • doc/theses/mike_brooks_MMath/array.tex

    rd031f7f r720eec9  
    44
    55\section{Introduction}
    6 
    7 Arrays in C are possible the single most misunderstood and incorrectly used features in the language, resulting in the largest proportion of runtime errors and security violations.
    8 This chapter describes the new \CFA language and library features that introduce a length-checked array-type to the \CFA standard library~\cite{Cforall}.
    9 
    10 Specifically, a new \CFA array is declared:
     6\label{s:ArrayIntro}
     7
     8Arrays in C are possibly the single most misunderstood and incorrectly used feature in the language, resulting in the largest proportion of runtime errors and security violations.
     9This chapter describes the new \CFA language and library features that introduce a length-checked array type to the \CFA standard library~\cite{Cforall}.
     10
     11Specifically, a new \CFA array is declared by instantiating the generic @array@ type,
     12much like instantiating any other standard-library generic type (such as @dlist@),
     13though using a new style of generic parameter.
    1114\begin{cfa}
    1215@array( float, 99 )@ x;                                 $\C[2.75in]{// x contains 99 floats}$
    1316\end{cfa}
    14 using generic type @array@ with arguments @float@ and @99@.
    15 A function @f@ is declared with an @array@ parameter of length @42@.
     17Here, the arguments to the @array@ type are @float@ (element type) and @99@ (length).
     18When this type is used as a function parameter, the type-system requires that a call's argument matches, down to the length.
    1619\begin{cfa}
    1720void f( @array( float, 42 )@ & p ) {}   $\C{// p accepts 42 floats}$
    … …  
    2124Applying untyped:  Name: f ... to:  Name: x
    2225\end{cfa}
    23 The call @f( x )@ is invalid because the @array@ lengths @99@ and @42@ do not match.
    24 
    25 Next, function @g@ introduces a @forall@ prefix on type parameter @T@ and arbitrary \emph{dimension parameter} @N@, the new feature that represents a count of elements managed by the type system.
     26Here, the function @f@'s parameter @p@ is declared with length 42.
     27The call @f( x )@, with the argument being the previously-declared object, is invalid, because the @array@ lengths @99@ and @42@ do not match.
     28
     29A function declaration can be polymorphic over these @array@ arguments by using the @forall@ declaration prefix.
     30This function @g@'s takes arbitrary type parameter @T@ (familiar) and \emph{dimension parameter} @N@ (new).
     31A dimension paramter represents a to-be-determined count of elements, managed by the type system.
    2632\begin{cfa}
    2733forall( T, @[N]@ )
    … …  
    3541\end{cfa}
    3642The call @g( x, 0 )@ is valid because @g@ accepts any length of array, where the type system infers @float@ for @T@ and length @99@ for @N@.
    37 Inferring values for @T@ and @N@ is implicit without programmer involvement.
    38 Furthermore, the runtime subscript @x[0]@ (parameter @i@ is @0@) in @g@ is valid because @0@ is in the dimension range $[0,99)$ of argument @x@.
    39 The call @g( x, 1000 )@ is also valid;
    40 however, the runtime subscript @x[1000]@ is invalid (generates a subscript error) because @1000@ is outside the dimension range $[0,99)$ of argument @x@.
    41 
    42 The generic @array@ type is similar to the C array type, which \CFA inherits from C.
    43 Its runtime characteristics are often identical, and some features are available in both.
    44 For example, assume a caller can instantiates @N@ with 42 in the following (details to follow).
     43Inferring values for @T@ and @N@ is implicit, without programmer involvement.
     44Furthermore, in this case, the runtime subscript @x[0]@ (parameter @i@ being @0@) in @g@ is valid because 0 is in the dimension range $[0,99)$ of argument @x@.
     45The call @g( x, 1000 )@ is also accepted through compile time;
     46however, this case's subscript, @x[1000]@, generates an error, because @1000@ is outside the dimension range $[0,99)$ of argument @x@.
     47
     48The generic @array@ type is comparable to the C array type, which \CFA inherits from C.
     49Their runtime characteristics are often identical, and some features are available in both.
     50For example, assume a caller instantiates @N@ with 42 (discussion about how to follow) in:
    4551\begin{cfa}
    4652forall( [N] )
    … …  
    5157\end{cfa}
    5258Both of the locally-declared array variables, @x1@ and @x2@, have 42 elements, each element being a @float@.
    53 The two variables have identical size and layout; they both encapsulate 42-float, stack \vs heap allocations with no additional ``bookkeeping'' allocations or headers.
     59The two variables have identical size and layout; they both encapsulate 42-float stack allocations, with no additional ``bookkeeping'' allocations or headers.
    5460Providing this explicit generic approach requires a significant extension to the \CFA type system to support a full-feature, safe, efficient (space and time) array-type, which forms the foundation for more complex array forms in \CFA.
    5561
    5662Admittedly, the @array@ library type (type for @x2@) is syntactically different from its C counterpart.
    57 A future goal (TODO xref) is to provide a built-in array type with syntax approaching C's (type for @x1@);
    58 then, the library @array@ type can be removed giving \CFA a largely uniform array type.
    59 At present, the C syntax @array@ is only partially supported, so the generic @array@ is used exclusively in the discussion;
     63A future goal (TODO xref) is to provide the new features upon a built-in type whose syntax approaches C's (declaration style of @x1@).
     64Then, the library @array@ type could be removed, giving \CFA a largely uniform array type.
     65At present, the C-syntax array gets partial support for the new features, so the generic @array@ is used exclusively when introducing features;
    6066feature support and C compatibility are revisited in Section ? TODO.
    6167
    6268Offering the @array@ type, as a distinct alternative to the C array, is consistent with \CFA's goal of backwards compatibility, \ie virtually all existing C (@gcc@) programs can be compiled by \CFA with only a small number of changes, similar to \CC (@g++@).
    6369However, a few compatibility-breaking changes to the behaviour of the C array are necessary, both as an implementation convenience and to fix C's lax treatment of arrays.
    64 Hence, the @array@ type is an opportunity to start from a clean slate and show a cohesive selection of features, making it unnecessary to deal with every inherited complexity introduced by the C array TODO xref.
     70Hence, the @array@ type is an opportunity to start from a clean slate and show a cohesive selection of features, making it unnecessary to deal with every inherited complexity of the C array.
     71
     72In all discussion following, ``C array'' means the types like that of @x@ and ``\CFA array'' means the standard-library @array@ type (instantiations), like the type of @x2@.
    6573
    6674My contributions in this chapter are:
    … …  
    8189
    8290
     91
     92General dependent typing allows the type system to encode arbitrary predicates (e.g. behavioural specifications for functions),
     93which is an anti-goal for my work.
     94Firstly, this application is strongly associated with pure functional languages,
     95where a characterization of the return value (giving it a precise type, generally dependent upon the parameters)
     96is a sufficient postcondition.
     97In an imperative language like C and \CFA, it is also necessary to discuss side effects,
     98for which an even heavier formalism, like separation logic, is required.
     99Secondly, TODO: bash Rust.
     100TODO: cite the crap out of these claims.
     101
     102
     103
    83104\section{Features added}
    84105
    85 This section presents motivating examples for the new array type, demonstrating the syntax and semantics of the generic @array@.
    86 As stated, the core capability of the new array is tracking all dimensions in the type system, where dynamic dimensions are represented using type variables.
    87 
    88 The definition of type variables preceding object declarations makes the array dimension lexically referenceable where it is needed.
    89 For example, a declaration can share one length, @N@, among a pair of parameters and the return.
     106This section shows more about using the \CFA array and dimension parameters, demonstrating their syntax and semantics by way of motivating examples.
     107As stated, the core capability of the new array is tracking all dimensions within the type system, where dynamic dimensions are represented using type variables.
     108
     109By declaring type variables at the front of object declarations, an array dimension is lexically referenceable where it is needed.
     110For example, a declaration can share one length, @N@, among a pair of parameters and the return,
     111meaning that it requires both input arrays to be of the same length, and guarantees that the result with be of that length as well.
    90112\lstinput{10-17}{hello-array.cfa}
    91 Here, the function @f@ does a pointwise comparison, checking if each pair of numbers is within half a percent of each other, returning the answers in a newly allocated @bool@ array.
    92 The dynamic allocation of the @ret@ array by @alloc@ uses the parameterized dimension information in its implicit @_Alignof@ and @sizeof@ determinations, and casting the return type.
    93 \begin{cfa}
     113This function @f@ does a pointwise comparison of its two input arrays, checking if each pair of numbers is within half a percent of each other, returning the answers in a newly allocated @bool@ array.
     114The dynamic allocation of the @ret@ array by preexisting @alloc@ uses the parameterized dimension information implicitly within its @sizeof@ determination, and casts the return type.
     115Note that alloc only sees one whole type for its @T@ (which is @f@'s @array(bool, N)@); this type's size is a computation based on @N@.
     116\begin{cfa}
     117// simplification
    94118static inline forall( T & | sized(T) )
    95 T * alloc( size_t dim ) {
    96         if ( _Alignof(T) <= libAlign() ) return (T *)aalloc( dim, sizeof(T) ); // calloc without zero fill
    97         else return (T *)amemalign( _Alignof(T), dim, sizeof(T) ); // array memalign
    98 }
    99 \end{cfa}
    100 Here, the type system deduces from the left-hand side of the assignment the type @array(bool, N)@, and forwards it as the type variable @T@ for the generic @alloc@ function, making it available in its body.
    101 Hence, preexisting \CFA behaviour is leveraged here, both in the return-type polymorphism, and the @sized(T)@-aware standard-library @alloc@ routine.
     119T * alloc() {
     120        return (T *)malloc( sizeof(T) );
     121}
     122\end{cfa}
    102123This example illustrates how the new @array@ type plugs into existing \CFA behaviour by implementing necessary @sized@ assertions needed by other types.
    103 (@sized@ implies a concrete \vs abstract type with a compile-time size.)
     124(@sized@ implies a concrete \vs abstract type with a runtime-available size, exposed as @sizeof@.)
    104125As a result, there is significant programming safety by making the size accessible and implicit, compared with C's @calloc@ and non-array supporting @memalign@, which take an explicit length parameter not managed by the type system.
    105126
    … …  
    111132\end{figure}
    112133
    113 \VRef[Figure]{f:fHarness} shows a harness that uses function @f@ to illustrate how dynamic values are fed into the @array@ type.
     134\VRef[Figure]{f:fHarness} shows a harness that uses function @f@, illustrating how dynamic values are fed into the @array@ type.
    114135Here, the dimension of arrays @x@, @y@, and @result@ is specified from a command-line value, @dim@, and these arrays are allocated on the stack.
    115136Then the @x@ array is initialized with decreasing values, and the @y@ array with amounts offset by constant @0.005@, giving relative differences within tolerance initially and diverging for later values.
    … …  
    119140The example shows @dim@ adapting into a type-system managed length at the declarations of @x@, @y@, and @result@, @N@ adapting in the same way at @f@'s loop bound, and a pass-thru use of @dim@ at @f@'s declaration of @ret@.
    120141Except for the lifetime-management issue of @result@, \ie explicit @free@, this program has eliminated both the syntactic and semantic problems associated with C arrays and their usage.
    121 These benefits cannot be underestimated.
     142The result is a significant improvement in safety and usability.
    122143
    123144In general, the @forall( ..., [N] )@ participates in the user-relevant declaration of the name @N@, which becomes usable in parameter/return declarations and within a function.
    124145The syntactic form is chosen to parallel other @forall@ forms:
    125146\begin{cfa}
    126 forall( @[N]@ ) ...     $\C[1.5in]{// array kind}$
    127 forall( T & ) ...       $\C{// reference kind (dtype)}$
    128 forall( T ) ...         $\C{// value kind (otype)}\CRT$
     147forall( @[N]@ ) ...     $\C[1.5in]{// dimension}$
     148forall( T & ) ...       $\C{// opaque datatype (formerly, "dtype")}$
     149forall( T ) ...         $\C{// value datatype (formerly, "otype")}\CRT$
    129150\end{cfa}
    130151% The notation @array(thing, N)@ is a single-dimensional case, giving a generic type instance.
    … …  
    132153\begin{itemize}
    133154\item
    134 @[N]@ within a forall declares the type variable @N@ to be a managed length.
    135 \item
    136 The type of @N@ within code is @size_t@.
    137 \item
    138 The value of @N@ within code is the acquired length derived from the usage site, \ie generic declaration or function call.
    139 \item
    140 @array( thing, N0, N1, ... )@ is a multi-dimensional type wrapping $\prod_i N_i$ adjacent occurrences of @thing@ objects.
     155@[N]@ within a @forall@ declares the type variable @N@ to be a managed length.
     156\item
     157@N@ can be used an expression of type @size_t@ within the declared function body.
     158\item
     159The value of an @N@-expression is the acquired length, derived from the usage site, \ie generic declaration or function call.
     160\item
     161@array( thing, N0, N1, ... )@ is a multi-dimensional type wrapping $\prod_i N_i$ adjacent occurrences of @thing@-typed objects.
    141162\end{itemize}
    142163
    … …  
    144165\begin{enumerate}[leftmargin=*]
    145166\item
    146 The \CC template @N@ is a compile-time value, while the \CFA @N@ is a runtime value.
     167The \CC template @N@ can only be compile-time value, while the \CFA @N@ may be a runtime value.
     168% agreed, though already said
     169\item
     170\CC does not allow a template function to be nested, while \CFA lests its polymorphic functions to be nested.
     171% why is this important?
    147172\item
    148173The \CC template @N@ must be passed explicitly at the call, unless @N@ has a default value, even when \CC can deduct the type of @T@.
    149174The \CFA @N@ is part of the array type and passed implicitly at the call.
     175% fixed by comparing to std::array
     176% mycode/arrr/thesis-examples/check-peter/cs-cpp.cpp, v2
    150177\item
    151178\CC cannot have an array of references, but can have an array of pointers.
    … …  
    153180In the \CC example, the arrays fall back on C arrays, which have a duality with references with respect to automatic dereferencing.
    154181The \CFA array is a contiguous object with an address, which can be stored as a reference or pointer.
     182% not really about forall-N vs template-N
     183% any better CFA support is how Rob left references, not what Mike did to arrays
     184% https://stackoverflow.com/questions/1164266/why-are-arrays-of-references-illegal
     185% https://stackoverflow.com/questions/922360/why-cant-i-make-a-vector-of-references
    155186\item
    156187C/\CC arrays cannot be copied, while \CFA arrays can be copied, making them a first-class object (although array copy is often avoided for efficiency).
     188% fixed by comparing to std::array
     189% mycode/arrr/thesis-examples/check-peter/cs-cpp.cpp, v10
    157190\end{enumerate}
     191TODO: settle Mike's concerns with this comparison (perhaps, remove)
    158192
    159193\begin{figure}
    … …  
    166200}
    167201int main() {
     202
    168203        int ret[10], x[10];
    169204        for ( int i = 0; i < 10; i += 1 ) x[i] = i;
    … …  
    177212\begin{cfa}
    178213int main() {
    179         @forall( T, [N] )@   // nested function
     214        @forall( T, [N] )@              // nested function
    180215        void copy( array( T, @N@ ) & ret, array( T, @N@ ) & x ) {
    181                 for ( i; 10 ) ret[i] = x[i];
    182         }
    183 
    184         array( int, 10 ) ret, x;
    185         for ( i; 10 ) x[i] = i;
     216                for ( i; N ) ret[i] = x[i];
     217        }
     218
     219        const int n = promptForLength();
     220        array( int, n ) ret, x;
     221        for ( i; n ) x[i] = i;
    186222        @copy( ret,  x );@
    187         for ( i; 10 )
     223        for ( i; n )
    188224                sout | ret[i] | nonl;
    189225        sout | nl;
    … …  
    191227\end{cfa}
    192228\end{tabular}
    193 \caption{\CC \lstinline[language=C++]{template} \vs \CFA generic type }
     229\caption{\lstinline{N}-style paramters, for \CC template \vs \CFA generic type }
    194230\label{f:TemplateVsGenericType}
    195231\end{figure}
    196232
    197 Continuing the discussion of \VRef[Figure]{f:fHarness}, the example has @f@ expecting two arrays of the same length.
    198 As stated previous, a compile-time error occurs when attempting to call @f@ with arrays of differing lengths.
    199 % removing leading whitespace
    200 \lstinput[tabsize=1]{52-53}{hello-array.cfa}
    201 \lstinput[tabsize=1,aboveskip=0pt]{62-64}{hello-array.cfa}
    202 C allows casting to assert knowledge not shared with the type system.
    203 \lstinput{70-74}{hello-array.cfa}
    204 
    205 Orthogonally, the new @array@ type works with \CFA's generic types, providing argument safety and the associated implicit communication of array length.
    206 Specifically, \CFA allows aggregate types to be generalized with multiple type parameters, including parameterized element types and lengths.
    207 Doing so gives a refinement of C's ``flexible array member'' pattern, allowing nesting structures with array members anywhere within the structures.
     233Just as the first example in \VRef[Section]{s:ArrayIntro} shows a compile-time rejection of a length mismatch,
     234so are length mismatches stopped when they invlove dimension parameters.
     235While \VRef[Figure]{f:fHarness} shows successfully calling a function @f@ expecting two arrays of the same length,
     236\begin{cfa}
     237array( bool, N ) & f( array( float, N ) &, array( float, N ) & );
     238\end{cfa}
     239a static rejection occurs when attempting to call @f@ with arrays of potentially differing lengths.
     240\lstinput[tabsize=1]{70-74}{hello-array.cfa}
     241When the argument lengths themselves are statically unknown,
     242the static check is conservative and, as always, \CFA's casting lets the programmer use knowledge not shared with the type system.
     243\begin{tabular}{@{\hspace{0.5in}}l@{\hspace{1in}}l@{}}
     244\lstinput{90-97}{hello-array.cfa}
     245&
     246\lstinput{110-117}{hello-array.cfa}
     247\end{tabular}
     248
     249\noindent
     250This static check's full rules are presented in \VRef[Section]{s:ArrayTypingC}.
     251
     252Orthogonally, the \CFA array type works within generic \emph{types}, \ie @forall@-on-@struct@.
     253The same argument safety and the associated implicit communication of array length occurs.
     254Preexisting \CFA allowed aggregate types to be generalized with type parameters, enabling parameterizing for element types.
     255Now, \CFA also allows parameterizing them by length.
     256Doing so gives a refinement of C's ``flexible array member'' pattern[TODO: cite ARM 6.7.2.1 pp18]\cite{arr:gnu-flex-mbr}.
     257While a C flexible array member can only occur at the end of the enclosing structure,
     258\CFA allows length-parameterized array members to be nested at arbitrary locations.
     259This flexibility, in turn, allows for multiple array members.
    208260\lstinput{10-15}{hello-accordion.cfa}
    209 This structure's layout has the starting offset of @studentIds@ varying in generic parameter @C@, and the offset of @preferences@ varying in both generic parameters.
     261This structure's layout has the starting offset of @studentIds@ varying according to the generic parameter @C@, and the offset of @preferences@ varying according to both generic parameters.
    210262For a function that operates on a @School@ structure, the type system handles this memory layout transparently.
    211263\lstinput{40-45}{hello-accordion.cfa}
    212 \VRef[Figure]{f:checkHarness} shows the @School@ harness and results with different array sizes, where multidimensional arrays are discussed next.
     264\VRef[Figure]{f:checkHarness} shows the @School@ harness and results with different array sizes.
     265Note the declaration of the @school@ variable.
     266It is on the stack and its initialization does not use any casting or size arithmetic.
     267Both of these points are impossible with a C flexible array member.
     268When heap allocation is preferred, the original pattern still applies.
     269\begin{cfa}
     270School( classes, students ) * sp = alloc();
     271\end{cfa}
     272This ability to avoid casting and size arithmetic improves safety and usability over C flexible array members.
     273
    213274
    214275\begin{figure}
    … …  
    216277\begin{tabular}{@{}ll@{\hspace{25pt}}l@{}}
    217278\begin{tabular}{@{}p{3.25in}@{}}
    218 \lstinput{60-66}{hello-accordion.cfa}
     279\lstinput{60-64}{hello-accordion.cfa}
    219280\vspace*{-3pt}
    220281\lstinput{73-80}{hello-accordion.cfa}
    … …  
    233294
    234295\section{Typing of C Arrays}
     296\label{s:ArrayTypingC}
    235297
    236298Essential in giving a guarantee of accurate length is the compiler's ability

  • doc/theses/mike_brooks_MMath/programs/hello-accordion.cfa

    rd031f7f r720eec9  
    3939
    4040forall( [C], [S] )
    41 void init( @School( C, S ) & classes@, int class, int student, int pref ) with( classes ) {
    42         classIds[class] = class; $\C{// handle dynamic offsets of fields within structure}$
    43         studentIds[student] = student;
    44         preferences[class][student] = pref;
     41void put( @School( C, S ) & s@, int class, int student, int pref ) {
     42        s.classIds[class] = class; $\C{// fields' offsets are dynamic }$
     43        s.studentIds[student] = student;
     44        s.preferences[class][student] = pref;
    4545}
    4646
    … …  
    6262        sin | classes | students;
    6363        @School( classes, students ) school;@
    64         int class, student, preference;
    65         // read data into school calling init
    66         // for each student's class/preferences
    67         try {
    68                 for ( ) {
    69                         sin | class | student | preference;
    70                         init( school, class, student, preference );
    71                 }
    72         } catch( end_of_file * ) {}
     64        // elided: read data into school, calling put
     65        {       int class, student, preference;
     66                // for each student's class/preferences
     67                try {
     68                        for ( ) {
     69                                sin | class | student | preference;
     70                                put( school, class, student, preference );
     71                        }
     72                } catch( end_of_file * ) {}        }
    7373        for ( s; students ) {
    7474                sout | "student" | s | nonl;

  • doc/theses/mike_brooks_MMath/programs/hello-array.cfa

    rd031f7f r720eec9  
    4949*/
    5050
     51
     52
     53
     54
     55
     56
     57
     58
     59
     60
     61
     62
     63
     64#ifdef SHOWERR1
     65
    5166void fred() {
     67
     68
     69
     70        array( float, @10@ ) x;
     71        array( float, @20@ ) y;
     72        f( x, x );              $\C[0.5in]{// ok}$
     73        f( y, y );              $\C{// ok}$
     74        f( x, y );              $\C{// error}\CRT$
     75
     76
     77
     78
     79}
     80
     81
     82
     83
     84
     85
     86
     87
     88
     89
     90forall( [M], [N] )
     91void bad( array(float, M) &x,
     92                array(float, N) &y ) {
     93        f( x, x );              $\C[0.5in]{// ok}$
     94        f( y, y );              $\C{// ok}$
     95
     96        f( x, y );              $\C{// error}\CRT$
     97}
     98
     99#endif
     100
     101
     102
     103
     104
     105
     106
     107
     108
     109
     110forall( [M], [N] )
     111void bad_fixed( array( float, M ) & x,
     112                array( float, N ) & y ) {
     113        f( x, x );              $\C[0.5in]{// ok}$
     114        f( y, y );              $\C{// ok}$
     115        if ( M == N )
     116                f( x, @(array( float, M ) &)@y ); $\C{// ok}$
     117}
     118
     119
     120
     121
     122
     123
     124void fred_ok_only() {
    52125        array( float, @10@ ) x;
    53126        array( float, @20@ ) y;
    … …  
    57130}
    58131
    59 #ifdef SHOWERR1
    60 forall( [M], [N] )
    61 void bad( array(float, M) &x, array(float, N) &y ) {
    62         f( x, x );              $\C[1.5in]{// ok}$
    63         f( y, y );              $\C{// ok}$
    64         f( x, y );              $\C{// error}\CRT$
    65 }
    66 #endif
    67 
    68 
    69132
    70133forall( [M], [N] )
    71 void bad_fixed( array( float, M ) & x, array( float, N ) & y ) {
    72         if ( M == N )
    73                 f( x, @(array( float, M ) &)@y ); $\C{// cast y to matching type}$
     134void bad_ok_only( array(float, M) &x,
     135                array(float, N) &y ) {
     136        f( x, x );
     137        f( y, y );
     138
     139//      f( x, y );
    74140}
     141
    75142
    76143// Local Variables: //

  • doc/theses/mike_brooks_MMath/uw-ethesis.bib

    rd031f7f r720eec9  
    1717    year      = {2018},
    1818}
     19
     20% --------------------------------------------------
     21% C Array facts
     22
     23@misc{arr:gnu-flex-mbr,
     24    title       = {Arrays of Length Zero},
     25    howpublished= {\url{https://gcc.gnu.org/onlinedocs/gcc/Zero-Length.html}},
     26}
     27
    1928
    2029% --------------------------------------------------
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