CIVL IR v2.0

Language principles

• CIVL-IR is a subset of CIVL-C. A CIVL-IR program is a CIVL-C program, and has the same semantics.
• CIVL-IR should be a small language. Whenever there is a way to express a new construct using other existing constructs, the preferred approach is to not add the new construct to the language.
• CIVL-IR is to be as close as possible to a language representation of a CIVL "Model". This is the lowest-level representation of a CIVL program, one appropriate for efficient (symbolic) execution and/or model checking.
• The basic representation of a function is as a "program graph"---a directed graph in which the nodes are locations and edges are atomic statements.
• Expressions are side-effect free.
• There are no automatic conversions. All conversions must be by explicit casts or other functions. Operations such as numeric addition (add) require that both operands have the exact same type.
• In general, symbols can be used before they are declared, as long as they are in scope. For example, a function f can call g, even if g is defined after f in the same scope. There is no need for a "prototype". Similarly, a type definition can refer to a type defined later, or can refer to itself in its own definition.
• Many of the core operations are implemented as functions in the standard library. This is similar to C. The header files for the standard library are an integral part of the language.
• Evaluation of expressions and execution of statements should always result in a value or state. There is no concept of exception or error. Hence anything that should be checked should be checked by explicit assertions.

Questions

1. Do variables have default initial values?
   • An object variable may or may not include an explicit initializer. If it does not include an initializer, the variable is "undefined". Even in this case, the variable has some unspecified value of its type. Details are below.
2. How do you initialize a variable?
   • In the declaration. For example $int n=$new($int); will assign n an arbitrary integer, while $int n=0; will assign the integer 0 to n.
3. How is an array allocated?
   • An array variable a is declared with a decl such as T a[] = $new_array(n, T); which assigns to a a new array value for an array of length n of elements of type T in which all entries are undefined. (Alternatively, T a[] = $new(T[n]) will create an array of arbitrary defined values.) For heap-allocation, T * p = $undef; followed by the statement p = $alloc<T>(&heap, n); will add a new object to the heap and return a pointer to the first element. An $alloc-ed object can be deallocated with $free(p);.
4. Is there an "array-pointer pun", as in C?
   • No, if you want a pointer to element 0 of an array, you have to explicitly write something like &a[0].
5. How to you translate between sequences and arrays?
   • There are functions in the standard library to do that.
6. Can you make types values? (reification)
   • For now, no. There are several statements and expressions which do take a type name as an argument, but there is no way for one to define new expressions or functions that do that.
7. How do you monitor reads and writes?
   • There are functions in the standard library (mem.h) for this.
8. Is there a type for $state ?
   • Coming soon.
9. How do you model malloc when the element type is not known?
   • For now, you can't. [Note to self: consider creating a type like $byte for this purpose?]
10. How do you iterate over a domain?
    • Coming soon.
11. Bitvectors?
    • Coming soon.

Grammar

program: typedef* decl* function-definition+ ;
typedef:
  type-param-list?
  ( struct ID '{' decl* '}'
  | union ID '{' decl* '}'
  | 'typedef' type-specifier declarator
  )
  ';'
  ;
decl: function-decl | object-decl ;
function-decl: type-param-list? function-kind function-qualifier*  type-specifier declarator contract-clause* ';' ;
function-kind:
    '$abstract_f'  /* abstract function; only for function decls */
  | '$system_f' '<' STRING ',' STRING '>'  /* function is defined in system code elsewhere */
  ;
function-qualifier:
    '$pure_f'  /* function is a mathematical function of its parameters */
  | '$state_f'  /* function is a mathematical function of the state */
  | '$atomic_f'  /* function invocations take place atomically */
  ;
object-decl: type-specifier declarator ('=' expr)? ';' ;
function-definition: type-param-list? function-qualifier* type-specifier declarator contract-clause* block ;
block: '{' typedef* decl* function-definition* statement* '}' ;
statement: block | simpleStmt | chooseStmt ;
simpleStmt: label? dependency? guard? primitiveStmt gotoStmt? ;
chooseStmt: label? '$choose' '{' simpleStmt* '}' ;
label: ID ':' ;
guard: '$when' '(' expr ')' ;
dependency: '$depends_on' '(' expression-list? ')'
expression-list: expr (',' expr)* ;
gotoStmt: 'goto' ID ;
type-param-list: '<' ID (',' ID)* '>' ;
INT: ... /* integer constant */
ID: ... /* identifier */
STRING: ... /* string literal in double quotes */

Notes

• A typedef can be used to define parameterized types. The type parameters are listed between angular brackets preceding the typedef. When the identifier is later used, it must be used with angular brackets and actual type names to replace the type parameters.
• A declaration declares a variable to have either an object type or a function type. (An object type is any type that is not a function type.)
• The optional type-param-list in a declaration may be applied only to a declaration of function type. It declares a "generic" function. When the function is called (or spawned), the call must include the angular brackets with a list of type names that replace the type parameters used in the declaration or definition of the function.
• A variable of function type represents either an abstract or a system function. The declaration of such a variable must use either the qualifier $abstract_f or $system_f. Moreover, those two qualifiers can only be used in a declaration of function type.
• An abstract function represents a new uninterpreted pure function.
• A system function has no definition in the program, but is instead defined elsewhere (for example, in C or Java code). Such a function will always be executed atomically. The first string specifies a path (e.g., Java package) to the library containing the function, the second is the name of the library. These two Strings should be enough to tell CIVL where to find the system definition of the function.
• $atomic_f can only be used in a function definition. It indicates that a defined function is to behave atomically, i.e., every call to such a function will be executed as if in atomic region. (An abstract or system function must necessarily be atomic.)
• $pure_f can be applied to a system function or defined function only. (An abstract function is necessarily pure.) The use of $pure_f indicates that the function has no side-effects, and the value returned in a mathematical function of the arguments. If the function is not actually pure, the behavior is undefined.
• $state_f can be applied to a system function or defined function only. It declares that the function has no side-effects, and the value returned is a mathematical function of the state in which the call occurred; i.e., the value returned may depend on the arguments in the call, the values of global variables, or any other component of the state. But, if called twice from the same state, it will always return the same value. If the function does not actually have this property, the behavior is undefined. Any $pure_f function is necessarily a $state_f, so at most one of these two qualifiers can occur in a declaration or definition.
• Contract clauses can occur only with system functions and defined functions (not with abstract functions, not with variables).
• A program must contain a function definition for a function named main. (Signature?)
• The expr in a guard must have type $bool.

Contracts

contract-clause:
    '$assigns' '(' expression-list? ')'   /* frame condition */
  | '$requires' '(' expr ')'  /* precondition */
  | '$ensures' '(' expr ')'  /* postcondition */
  | '$depends_on' '(' expression-list? ')'  /* dependency specification */
  | '$when '(' ID ')'  /* guard clause */
  ;

Notes

• A guard clause can occur only in the declaration of a system function. The identifier is the name of the function that will be used as the guard in all calls to the system function. The guard function must have the same input signature as the system function, but must return $bool. Any call to the system function will block unless the guard function returns $true on the arguments used in the call. The execution of the system function and the return of "true" from the guard function will occur atomically, i.e., no state change will occur between the return of true and the call of the system function. The guard function must be a $state_f function; in particular, it must be side-effect free. The guard function may be a system function or a defined function; it cannot be abstract. If omitted, the guard is understood to be "true", i.e., a call to the system function will never block.
• The expressions in a dependency clause must have pointer type. This declares that the following transition depends only on the objects pointed to.

Types

type-specifier:
    ID type-args?  /* type parameter or (possibly parameterized) typedef use */
  | '$int'  /* mathematical integers */
  | '$bool'  /* boolean type ($true and $false, unrelated to integers) */
  | '$char'  /* character type (Unicode characters, unrelated to integers) */
  | '$real'   /* mathematical reals */
  | '$float32'  /* IEEE 32-bit floating-point numbers */
  | '$float64'  /* IEEE 64-bit floating-point number */
  | '$herbrand' '<' type-name '>'  /* Herbrand type of non-Herbrand numeric type T */
  | '$proc'  /* process type */
  | '$heap'  /* heap type, for dynamic allocation */
  | '$mem'  /* set of memory locations */
  | 'struct' ID  type-param-list?  /* structure type */
  | 'union' ID  type-param-list?  /* union type */
  | 'void'   /* use as pointer element-type and return type of a function */
  | '$seq' '<' type-name '>'  /* sequence of T */
  | '$set' '<' type-name '>'  /* set of T */
  | '$map' '<' type-name ',' type-name '>'  /* partial function from T1 to T2 */
  | '$rel' type-args  /* relation: set of n-tuples with specified component types */
  ;
declarator: '*'* direct-declarator ;
direct-declarator:
    ID  /* variable being declared */
  | direct-declarator '[' ']'  /* array of ... */
  | direct-declarator  '(' parameter-type-list? ')'  /* function consuming ... and returning ... */
  | '(' declarator ')'
  ;
abstract-declarator: '*'* abstract-direct-declarator ;
abstract-direct-declarator:
  | abstract-direct-declarator '[' expr? ']'
  | abstract-direct-declarator  '(' parameter-type-list? ')'
  | '(' abstract-declarator ')'
  ;
type-name: type-specifier abstract-declarator ;
type-args: '<' type-name (',' type-name)* '>';
parameter-type-list: type-specifier declarator (',' type-specifier declarator)* ;

Notes

• Sequences, sets, maps, relations, and $mem objects are immutable. An assignment using objects of this type creates a new copy of the object, just as with primitive types like $int.
• Arrays are similar to sequences. The main differences are as follows:
  • An object (i.e., a variable or component of an object) of array type is initialized once, then will never be assigned to again. Hence there cannot be a statement of the form a=... where a has array type.
  • After initialization, an object of array type can appear only as the first (left) argument to the subscript operator [] or as the argument to the $length operator.
  • Arrays are mutable. As in C, the left hand size of an assignment may have the form a[i], where a is an object of array type. Sequences are immutable.
  • Elements of an array are addressable, i.e., one can form a pointer such as &a[i]. This is not possible with sequences, sets, maps, or relations---there is no way to have a pointer to any component of such a type.
  • A function may neither consume nor return an object of array type. There is no such restriction for sequences.
• The difference between the function type and map type:
  • a function cannot return a function; nor can a function consume an argument of functional type
  • a function may return a pointer to a function and may consume an argument of type pointer to function...
  • a function pointer is formed by &f, where f is the name of a function
  • there is no pointer arithmetic on function pointers
  • a map has a finite number of entries
  • a map cannot be executed in a call or spawn; nor can functions on maps have any side effects
  • a function can return a map and can consume an argument of map type
• A type name in a type-args list may be void. In this case, the type name designates the type whose domain is the union, over all types T of the type name obtained by substituting T for void. For example, the domain of $seq<void> is the union over all types T of $seq<T>. The sequence {1,2,3} is in the domain of $seq<$int> and of $seq<void>. The sequence {1,3.14} is not in $seq<void> since there is no type T containing $int and $real. Hence $seq<void> is a super type of all $seq<T>, and an expression of type $seq<T> can be used where one of type $seq<void> is required. It is similar to the relation between void* and T*.
  • A "bundle" type is not needed. A bundle is just a $seq<void>.
  • A message in an MPI communication might be represented as an element of type $seq<$seq<void>>. That is, a sequence, each element of which is a sequence of elements of the same type. An example of a message would be {{1,2,3}, {3.14,2.7},{'a','b','c'}}.
  • A general C "object" might be represented as a structure

    struct $object {
      $int size; // number of bytes
      $map<$int, $seq<void>> data; // offset (in bytes), contiguous values
    };
    

• A *pure type name* is one in which the optional length expression for an array type never occurs.
• A *complete type name* is defined recursively as follows: a struct or union type is complete if and only if all the member types are complete; an array type is complete if and only if the element type is complete and the length is specified; all other types are complete.

Statements

An lvalue is an expression that represents an object, as in C. It is either a variable, an array element expression, a struct or union field expression, or dereference expression.

primitiveStmt:
    ';'  /* noop */
  | lvalue '=' expr ';'  /* assignment */
  | (lvalue '=')? expr type-args? '(' expression-list? ')' ';'  /* function call */
  | lvalue '=' '$spawn' expr type-args? '(' expression-list? ')' ';'  /* process creation */
  | '$wait' expr ';'  /* wait until p terminates */
  | 'return' expr? ';'  /* return from function call */
  | '$atomic_enter' ';'  /* enter atomic region */
  | '$atomic_exit' ';'  /* exit atomic region */
  | '$yield' ';'  /* release atomic lock */
  | '$assert' expr (',' expr)* ';'  /* assertion with optional error message */
  | '$assume' expr ';'  /* assumption */
  ;

Notes

• For function calls and spawns, the first expression shall have a functional type. That function must be a system or defined function (not an abstract function).
• The first expression following $alloc has type $heap*. It is a pointer to the heap that will be modified by allocating the new memory. The second expression has type $int and is the number of elements being allocated. This is followed by the element type. The function returns a pointer to the first element of an array, similar to C's malloc. It is deallocated using function$free.

Expressions

expr:
  | ID  /* variable or function identifier */
  | '$true'
  | '$false'
  | INT
  | REAL
  | FLOAT
  | CHAR
  | STRING
  | 'NULL'
  | '$proc_null'
  | '(' type-name ')' expr  /* cast */
  | '(' type-name ')' '{' expression-list? '}'  /* concrete array, struct, $seq, $set, $mem */
  | '(' type-name ')' '{' expr-pair-list? '}'  /* concrete map */
  | '(' type-name ')' '$lambda' '(' '$int' identifier-list ')' expr  /* array literal, aka array lambda */
  | expr '+' expr  /* numeric or pointer addition */
  | expr '-' expr  /* numeric or pointer subtraction */
  | expr '/' expr  /* division */
  | expr '%' expr  /* modulo */
  | expr '&&' expr  /* logical and */
  | expr '||' expr  /* logical or */
  | expr '==>' expr  /* logical implication */
  | expr '==' expr  /* equality */
  | '!' expr  /* logical not */
  | expr '<' expr  /* less than */
  | expr '<=' expr  /* less than or equal to */
  | expr '[' expr ']'  /* array element access */
  | expr '.' ID  /* field access */
  | '(' expr ')'
  | '-' expr  /* negative */
  | '*' expr  /* pointer dereference */
  | '&' lvalue  /* address-of */
  | expr type-args? '(' expression-list? ')'  /* application of abstract function (and perhaps other functions?) */
  | '$new' '(' type-name ')'  /* returns a new arbitrary value of the given type */
  | '$new_array' '(' expr ',' type-name ')'  /* allocation of array of undefined values*/
  | '$forall' '(' decl expr? ')' expr  /* universal quantification */
  | '$exists' '(' decl expr? ')' expr  /* existential quantification */
  | expr '?' expr ':' expr  /* if-then-else expression */
  | '$length' '(' expr ')'  /* length of array */
  | '$summable' '(' expr ',' expr ')'  /* is pointer addition p+i defined? */
  | '$subtractable' '(' expr ',' 'expr ')'  /* is pointer subtraction p-q defined? */
  | '$castable' '(' expr ',' type-name ')'  /* can expr be cast to type T? */
  | '$defined' '(' lvalue ')'  /* has the specified memory location been assigned? */
  | '$valid '(' expr ')'  /* does a pointer point to an object of its base type? */
  ;
expr-pair-list: expr-pair (',' expr-pair)* ;
expr-pair: '{' expr ',' expr '}' ;

Notes

• For operations on float types the rounding mode used is round-to-nearest.
• Comparisons likes == and < return a $bool (not an int, like C).
• Every object holds a value of its declared type, always. There is no notion of "trap representation", as in C.
• Evaluation of an expression always results in a value of the type of that expression. There are no "exceptions".
• For some operations, the result of applying the operation on certain inputs in unspecified. We say those applications are "unsafe". The result of an unsafe operation is unspecified, but will result in some value of the appropriate type. If an application is not unsafe, it is "safe".
• For every operation, there is a way to determine if performing that operation is safe. Hence an assertion on a certain expression can be checked before performing the operation. Examples:
  • a+b, for the $int, $real, or float types: always safe, assuming a and b are defined. Will will assume all objects are defined from now on.
  • a/b: b!=0
  • p+i (pointer arithmetic): $summable(p, i)
  • a[i]: 0<=i && i<$length(a)
  • p-q (pointer subtraction): $subtractable(p,q)
  • (T)e (cast): $castable(e, T)
  • *p: $valid(p) --- this means it is safe to write to *p. (It may not be safe to read *p because the object pointed to by p may not yet have been initialized.)
• Explanation of $defined:
  • A bit is associated to each object. We say an object is "defined" if the bit is true, else it is undefined.
  • An object is initially undefined. It becomes defined by assignment. If the right hand side of the assignment involves undefined objects, it is unspecified whether the left hand side will be defined.
  • Objects of pointer type can change from defined to undefined. This happens when the object into which they point is deallocated or goes out of scope. Likewise, an object of $proc type becomes undefined when a call to $wait on the process returns.
  • An object of array type becomes defined when assigned a $new_array. However, its elements are undefined.
  • An object of array type also becomes defined when assigned a $new value. In that case, the elements are also defined.
  • A pointer becomes defined when assigned the result of an $alloc. Each element in the newly allocated object is undefined.
  • Example: to determine whether it is safe to read *(p+i), assert $summable(p, i) && $valid(p+i) && $defined(*(p+i)). You must check (1) you can sum p and i, (2) p+i points to an object of the base type, so dereference is safe, and (3) the object is defined (has been assigned a value). If you aren't sure p and i have been defined, you can add $defined(p) && $defined(i) to the assertion. It may be more efficient to use temporary variables and break this up into multiple assertions.

How to implement $defined using symbolic execution

Create a symbolic constant named UNDEF for each type T. These can be created on-the-fly, as they are needed.

A variable without initializer is initialized to the UNDEF of its type. The exception is a struct, which may be initialized to a concrete struct in which each field is UNDEF. (Either way will work, but the latter will probably be more effective.)

For a complete type T, the "undef-initializer" for T is defined as follows: if T is not an array type, the undef-initializer is defined exactly as above: either the symbolic constant UNDEF or a concrete struct of UNDEFs. If T is the complete array type S[n], the undef-initializer is the array-lambda expression with length n and defining expression the undef-initializer for S.

$new_array(n,T) returns the undef-initializer for T[n].

Evaluation of $defined(expr) returns !FORALL UNDEF:T . expr == UNDEF, where T is the type of expr.

Allowed Casts

• $int -> $real
• $float32 -> $real
• $float64 -> $real
• T1* -> T2*. How to check a cast is valid? What happens if it is not? If there are no exceptions, something must be returned. Perhaps undefined value.
  • this is OK if one is a pointer to element 0 of an array, and the other is a pointer to the array
  • if one is a pointer to the first member of a struct or union, and the other is a pointer to the struct or union
  • these rules can be applied recursively. This needs to be fleshed out.
• others?

Standard CIVL Library

civlc.cvh

Work in progress. Refer to existing library.

<T1,T2> struct $pair { T1 left; T2 right; };
<T1,T2> typedef struct $pair<T1,T2> $pair;
<T> $system T * $alloc($heap * h, T obj);  // add an object to a heap
$system $free($void * p);  // remove an object from the heap
$system void $exit();  // terminate process immediately
$system $bool $terminated($proc p);  // has the process terminated?
$system int $choose_int($int n);  // nondeterministic choice of int in 0..n-1

Notes

A heap is an abstraction for a set of objects of possibly different types which can be referenced by pointers. $alloc adds an object to the heap and returns a pointer to that object. The object can then be read or modified using the pointer. Example: C code that mallocs 10 ints

int * p = malloc(10*sizeof(int));

could be modeled as

$int * p = $undef;
$int (*p1)[] = $undef; // pointer to array of $int
p1 = $alloc(&heap, $new_array(10, $int));
p = ($int*)p1; // equivalently, p = &(*p1)[0];

More generally, if the type of objects being malloced is not known, or changes dynamically, one could define a general "object" type...

struct $object {
  $int size; // in bytes
  $map<$int,$seq<void>> data; // byte offset -> element sequence
}
typedef struct $object $object;

math.cvh

Work in progress...

Rounding modes:

• 0 = to nearest
• 1 = upward
• 2 = downward
• 3 = toward zero.

$int $round($real x); // round to nearest integer
$int $floor($real x); // greatest integer less than or equal to
$int $ceil($real x); // least integer greater than or equal to
$float32 $round32($real x, $int mode);  // round to float32 (see modes above)
$float64 $round64($real x, $int mode);  // round to float64 (see modes above)
$real $abs($real x); // absolute value on reals
$int $absi($int x); // absolute value on integers
$float32 $abs32($float32 x); // absolute value on $float32
$float64 $abs64($float64 x); // absolute value on $float64
$real $pow($real x, $int n);  // x to the n-th power.  n>=0
$int $powi($int x, $int n);
$float32 $pow32($float32 x, $int n);
$float64 $pow64($float64 x, $int n);
// do we need herbrand versions of above?  make these expressions?
// trig functions? pi?
<T> T $eval($herbrand<T> x);  // returns value of type T obtained by evaluating all the delayed operations
<T> $herbrand<T> $herbrandize(T x);  // trivial symbolic expression consisting of a single node

• Is it OK to have parameterized functions that only work for certain T? Would it better to expand for each numeric type? E.g., absi, abs, abs32, abs64, ...

mem.cvh

Work in progress...

• mem_reach(ptr), where ptr is an expression with a pointer type.
  • This represents the set of all memory units reachable from ptr, including the memory unit pointed to by ptr itself.
• mem_union(mem1,mem2), where mem1 and mem2 are expressions of type Mem.
  • This is the union of the two memory sets.
• mem_isect(mem1,mem1) : set intersection
• mem_comp(mem1) : set complement (everything not in mem1)
• mem_slice(a,dom)
  • where a is an expression of array type and dom is an expression of Domain type.
  • The dimension of the array must match the dimension of the domain. This represents all memory units which are the cells in the array indexed by a tuple in dom.

seq.cvh

<T> $int $seq_length( $seq<T> a ); // length of a
<T> T $seq_get( $seq<T> a, $int i ); // get element i of a
<T> $seq<T> $seq_set( $seq<T> a, $int i, T x ); // seq obtained by replacing element i of a with x
<T> $seq<T> $seq_subseq( $seq<T> a, $int start, $int stop ); // subsequence from start to stop-1
<T> $seq<T> $seq_add( $seq<T> a, T x ); // sequence obtained by adding element x to the end of a
<T> $seq<T> $seq_append( $seq<T> a1, $seq<T> a2 ); //  sequence obtained by concatenating a1 and a2
<T> $seq<T> $seq_remove( $seq<T> a, $int i ); // sequence obtained by removing element at position i from a
<T> $seq<T> $seq_insert( $seq<T> a, $int i, T x ); // sequence obtained by inserting element x at position i in a
<T> void $seq_write( $int n, T * ptr, $seq<T> a );  // write the first n elements of a to ptr, ptr+1, ..., ptr+n-1
<T> $seq<T> $seq_read( $int n, T * ptr ); // form a sequence by reading ptr, ptr+1, ..., ptr+n-1

set.cvh

<T> $int $set_size( $set<T> s ); // number of elements in s
<T> $seq<T> $set_seq( $set<T> s ); // elements of s arranged in a sequence
<T> $bool $set_contains( $set<T> s, T x ); // does s contain x as member?
<T> $bool $set_containsAll( $set<T> s1, $set<T> s2 ); // is s2 a subset of s1?
<T> $set<T> $set_add( $set<T> s, T x ); // result of adding x to s
<T> $set<T> $set_union( $set<T> s1, $set<T> s2 ); // union of s1 and s2
<T> $set<T> $set_intersection( $set<T> s1, $set<T> s2 ); // intersection of s1 and s2
<T> $set<T> $set_difference( $set<T> s1, $set<T> s2 ); // elements of s1 not in s2

map.cvh

<K,V> $set<K> $map_domain( $map<K,V> m ); // returns the domain of the map (the set of keys)
<K,V> $set<V> $map_range( $map<K,V> m ); // returns the range of the map (the set of values)
<K,V> $set<$pair<K,V>> $map_entries( $map<K,V> m ); // key value pairs of m arranged in a sequence
<K,V> $map<K,V> $map_put( $map<K,V> m, K k, V v ); // result of (re)defining k to map to v.
<K,V> $map<K,V> $map_putAll( $map<K,V> m1, $map<K,V> m2); // result of (re)defining k to map to v for every pair (k,v) in m2
<K,V> $map<K,V> $map_remove( $map<K,V> m, K k ); // result of declaring k undefined
<K,V> $map<K,V> $map_removeAll( $map<K,V> m, $set<K> ks ); // result of declaring k undefined for every k in ks
<K,V> $bool $map_contains( $map<K,V> m, K k ); // is k defined in the map?
<K,V> V $map_get( $map<K,V> m, K k ); // get the value defined at k (returns something if no k in domain)
? <K,V> $bool $map_injective( $map<K,V> m ); // does m form an injection?
? <K,V> $map<V,K> $map_inverse( $map<K,V> m ); // returns the inverse map of m. Should check $map_injective first (what do we do when m is not injective?)
? <K,I,V> $map<K,V> $map_compose( $map<I,V> m2, $map<K,I> m1 ); // returns the composition m2.m1

rel.cvh

concurrency.cvh

void $parspawn($proc * proc_array, $domain d, void (*f)($int, ...));  /* parallel spawn returns immediately*/
void $waitall( $int nprocs, $proc * procs );  /* wait for all procs in list to terminate */

• In$parspawn *f must have the function type that consumes n $ints, where n is the dimension of the domain. The function is spawned once for each element of the domain. References to the new processes are stored in the process array. The call to $parspawn returns immediately.