While the overload table can handle many definitions with the same function (or operator) name, it must be able to tell them apart by the argument types (the result type is usually not available when resolving an overloaded call, so it cannot be used to distinguish overloaded definitions). Moreover some implicit conversions can be applied to arguments, making some overloaded argument types incompatible (for instance having separate definitions for ‘vec’ and ‘[int]’ arguments is not possible, as arguments of these types can be implicitly converted into one another). However if a new definition exactly matches the argument types of a previous definition, the new definition is taken to replace rather than to overload the previous definition. The reply that axis gives to a definition indicates when this happens, for instance:

set inner_class (RootDatum rd, string ict)=
let lt = Lie_type (rd), basis=^coroot_radical(rd)
in inner_class(rd, involution(lt, basis, ict))


Redefined inner_class: (RootDatum,string->InnerClass)


showing that a previous (in this case built-in) definition of ‘inner_class’ has been superseded and has become unavailable (the user-defined version has replaced it; as it happens, it is almost equivalent to the built-in version).

Here is an example where an attempt to add an overloaded definition for a different but closely related argument type is refused (attempt to overload the variant defined above with ‘ratvec’ second argument with a similar instance with a ‘[rat]’ second argument:

set inner_class (LieType lt,[rat] gen,string ict)=
inner_class(lt, [ratvec: gen],ict)


gives the somewhat elaborate error message:

Cannot overload inner_class': previous type (LieType,ratvec,string)
is too close to (LieType,[rat],string),
as either type converts to the other, definition aborted.


The “broadness” phrase relates to the fact that one can for instance overload with types ‘int’ and ‘rat’, even though the former can be implicitly converted to the latter: since the opposite is not possible, ‘rat’ is considered the broader of the two types, and overload resolution will proceed by trying the broader type only if no matching for the narrower type is possible. But between ‘ratvec’ and ‘[rat]’ conversions in either direction are possible, so axis cannot decide which overload would have to be tried first, whence its refusal of the overloaded definition. Note that the full argument types here are ‘(LieType,ratvec,string)’ respectively ‘(LieType,[rat],string)’, which are not directly convertible (just their second components are), but ambiguity may still arise in function calls; axis is conservative and prefers to refuse potentially ambiguous overloads when the ambiguity is introduced, rather than to wait for a usage that actually exposes the ambiguity (at which point it may have become hard to hunt down the offending definitions).

If one wants to replace the old definition so that one with a close but distinct type can be accepted, a command ‘forget’ can be used; here:

atlas> forget inner_class@(LieType,ratvec,string)
Definition of inner_class@(LieType,ratvec,string) forgotten


after which the definition attempted and refused above will be accepted.

Functions can also be unnamed values, which is particularly useful if the are to be passed as arguments to other functions. So supposing g was defined by:

atlas> set g ((int->int) f) = vec: [f(0),f(f(1)),f(5)]
Defined g: ((int->int)->vec)


a function that takes as parameter another function, of type ‘(int->int)’. As argument of ‘g’ one can provide an anonymous function, for instance one that adds 1 to an integer, which can be denoted by ‘(int n) : n+1’, or to make the return type explicit ‘(int n) int: n+1’ (the syntax is a parenthesized parameter list, an optional return type followed by a colon, and an expression giving the function body). One can now call g with this argument:

atlas> g((int n):n+1)
Value: [ 1, 3, 6 ]


One can also call ‘g’ with a named function ‘f’. However since defining a named function just introduces an overload for the function name, a bare ‘f’ does not suffice; one needs to specify the argument type for ‘f’ in the call of ‘g’:

atlas> set f (int n) = 3*n+1
Defined f: (int->int)
atlas> g(f)
Error during analysis of expression at...
Undefined identifier f
Type check failed
atlas> g(f@int)
Value: [  1, 13, 16 ]


Note that ‘f’ was called “Undefined” even though a function ‘f’ is defined: in the call ‘g(f)’ the name ‘f’ is not called, and bindings of ‘f’ in the overload table are not considered. Using ‘set’ with a value of function type always enters in into the overload table, even when it is for instance written:

atlas> set f = (int n): 3*n+1
Redefined f: (int->int)


To get a global variable whose value happens to have function type, a special syntax is provided, namely ‘name : function’ instead of ‘set name = function”:

atlas>  f : (int n): 2*n+3
Identifier f: (int->int)
atlas> g(f)
Value: [  3, 13, 13 ]


A global identifier defined with a function value in this way can coexist with one or more overloaded instances for the same name. However, the presence of overloaded instances make the function variable unavailable for use in direct function calls, though it can still be used to pass the function value, assign it to local variables, or assign a new function (of the same type) to it.

atlas> f(10) { overloaded instance }
Value: 31
atlas> let ff=f in ff(10) { pass global variable to local one, then call it }
Value: 23
atlas> f := ((int n) int: n*n) { syntax requires parentheses here }
Value: Function defined at <standard input>:...
(n): *@(int,int)(n,n)
atlas> g(f) { call g with new value of f }
Value: [  0,  1, 25 ]


The ‘name : value’ syntax is also available for values of non-function type, in which case it is strictly equivalent to ‘set name = value’.

Although the design of axis aims at always giving variables a value at their introduction, deducing their type from that value and avoiding the possibility of uninitialized variables, it does allow introducing a global variable without specifying an initial value, but only a type. To this end there is the variation of the ‘name : value’ syntax consisting of writing ‘name : type’ instead. For instance the function variable ‘f’ above could have been introduced by:

atlas> f: (int->int)
Declaring identifier 'f': (int->int)


after which it has a type but no value, whence the following gives no type error but a runtime error:

atlas> g(f)
Runtime error:
Taking value of uninitialized variable 'f'
Evaluation aborted.


(it would be however possible to use ‘f’ in unevaluated function definitions at this point). Before doing anything useful with ‘f’, it must be assigned to:

atlas> f := ((int n): 2*n+3)
Value: Function defined at <standard input>:...
(n): +@(int,int)(*@(int,int)(2,n),3)
atlas> g(f)
Value: [  3, 13, 13 ]
`

(Note that after the assignment to ‘f’ its function body is printed in an internal representation, in which overload resolution has been made explicit by writing operators prefix and followed by @argument_types; if there had been any implicit conversion, it would also have been made visible by a somewhat cryptic code. If a command is an assignment, the value assigned is printed, as it is for any command that is an expression returning a value; this printing can however be suppressed by adding ‘;()’ at the end of the command.)