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A.5.2 Random Number Generation

1
   [Facilities for the generation of pseudo-random floating point numbers are provided in the package Numerics.Float_Random; the generic package Numerics.Discrete_Random provides similar facilities for the generation of pseudo-random integers and pseudo-random values of enumeration types. {random number} For brevity, pseudo-random values of any of these types are called random numbers.
2
   Some of the facilities provided are basic to all applications of random numbers. These include a limited private type each of whose objects serves as the generator of a (possibly distinct) sequence of random numbers; a function to obtain the ``next'' random number from a given sequence of random numbers (that is, from its generator); and subprograms to initialize or reinitialize a given generator to a time-dependent state or a state denoted by a single integer.
3
   Other facilities are provided specifically for advanced applications. These include subprograms to save and restore the state of a given generator; a private type whose objects can be used to hold the saved state of a generator; and subprograms to obtain a string representation of a given generator state, or, given such a string representation, the corresponding state.]
3.a
Discussion: These facilities support a variety of requirements ranging from repeatable sequences (for debugging) to unique sequences in each execution of a program.

Static Semantics

4
   The library package Numerics.Float_Random has the following declaration:
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package Ada.Numerics.Float_Random is
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   -- Basic facilities
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   type Generator is limited private;
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   subtype Uniformly_Distributed is Float range 0.0 .. 1.0;
   function Random (Gen : Generator) return Uniformly_Distributed;
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   procedure Reset (Gen       : in Generator;
                    Initiator : in Integer);
   procedure Reset (Gen       : in Generator);
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   -- Advanced facilities
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   type State is private;
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   procedure Save  (Gen        : in  Generator;
                    To_State   : out State);
   procedure Reset (Gen        : in  Generator;
                    From_State : in  State);
13
   Max_Image_Width : constant := implementation-defined integer value;
14
   function Image (Of_State    : State)  return String;
   function Value (Coded_State : String) return State;
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private
   ... -- not specified by the language
end Ada.Numerics.Float_Random;
16
    The generic library package Numerics.Discrete_Random has the following declaration:
17

generic
   type Result_Subtype is (<>);
package Ada.Numerics.Discrete_Random is
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   -- Basic facilities
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   type Generator is limited private;
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   function Random (Gen : Generator) return Result_Subtype;
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   procedure Reset (Gen       : in Generator;
                    Initiator : in Integer);
   procedure Reset (Gen       : in Generator);
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   -- Advanced facilities
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   type State is private;
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   procedure Save  (Gen        : in  Generator;
                    To_State   : out State);
   procedure Reset (Gen        : in  Generator;
                    From_State : in  State);
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   Max_Image_Width : constant := implementation-defined integer value;
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   function Image (Of_State    : State)  return String;
   function Value (Coded_State : String) return State;
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private
   ... -- not specified by the language
end Ada.Numerics.Discrete_Random;
27.a
Implementation defined: The value of Numerics.Float_Random.Max_Image_Width.
27.b
Implementation defined: The value of Numerics.Discrete_Random.Max_Image_Width.
27.c/1
Implementation Note: {8652/0097} The following is a possible implementation of the private part of Numerics.Float_Random each package (assuming the presence of ``with Ada.Finalization;'' as a context clause):
27.d
type State is ...;
type Access_State is access State;
type Generator is new Finalization.Limited_Controlled with
   record
      S : Access_State := new State'(...);
   end record;
procedure Finalize (G : in out Generator);
27.d.1/1
{8652/0097} Unfortunately, Numerics.Discrete_Random.Generator cannot be implemented this way, as Numerics.Discrete_Random can be instantiated at any nesting depth. However, Generator could have a component of a controlled type, as long as that type is declared in some other (non-generic) package. One possible solution would be to implement Numerics.Discrete_Random in terms of Numerics.Float_Random, using a component of Numerics.Float_Random.Generator to implement Numerics.Float_Random.Generator.
27.e
Clearly some level of indirection is required in the implementation of a Generator, since the parameter mode is in for all operations on a Generator. For this reason, Numerics.Float_Random and Numerics.Discrete_Random cannot be declared pure.
28
    An object of the limited private type Generator is associated with a sequence of random numbers. Each generator has a hidden (internal) state, which the operations on generators use to determine the position in the associated sequence. {unspecified [partial]} All generators are implicitly initialized to an unspecified state that does not vary from one program execution to another; they may also be explicitly initialized, or reinitialized, to a time-dependent state, to a previously saved state, or to a state uniquely denoted by an integer value.
28.a
Discussion: The repeatability provided by the implicit initialization may be exploited for testing or debugging purposes.
29
    An object of the private type State can be used to hold the internal state of a generator. Such objects are only needed if the application is designed to save and restore generator states or to examine or manufacture them.
30
    The operations on generators affect the state and therefore the future values of the associated sequence. The semantics of the operations on generators and states are defined below.
31
function Random (Gen : Generator) return Uniformly_Distributed;
function Random (Gen : Generator) return Result_Subtype;
32
Obtains the ``next'' random number from the given generator, relative to its current state, according to an implementation-defined algorithm. The result of the function in Numerics.Float_Random is delivered as a value of the subtype Uniformly_Distributed, which is a subtype of the predefined type Float having a range of 0.0 .. 1.0. The result of the function in an instantiation of Numerics.Discrete_Random is delivered as a value of the generic formal subtype Result_Subtype.
32.a
Implementation defined: The algorithms for random number generation.
32.b
Reason: The requirement for a level of indirection in accessing the internal state of a generator arises from the desire to make Random a function, rather than a procedure.
33
procedure Reset (Gen       : in Generator;
                 Initiator : in Integer);
procedure Reset (Gen       : in Generator);
34
{unspecified [partial]} Sets the state of the specified generator to one that is an unspecified function of the value of the parameter Initiator (or to a time-dependent state, if only a generator parameter is specified). {Time-dependent Reset procedure (of the random number generator)} The latter form of the procedure is known as the time-dependent Reset procedure.
34.a
Implementation Note: The time-dependent Reset procedure can be implemented by mapping the current time and date as determined by the system clock into a state, but other implementations are possible. For example, a white-noise generator or a radioactive source can be used to generate time-dependent states.
35
procedure Save  (Gen        : in  Generator;
                 To_State   : out State);
procedure Reset (Gen        : in  Generator;
                 From_State : in  State);
36
Save obtains the current state of a generator. Reset gives a generator the specified state. A generator that is reset to a state previously obtained by invoking Save is restored to the state it had when Save was invoked.
37
function Image (Of_State    : State)  return String;
function Value (Coded_State : String) return State;
38
Image provides a representation of a state coded (in an implementation-defined way) as a string whose length is bounded by the value of Max_Image_Width. Value is the inverse of Image: Value(Image(S)) = S for each state S that can be obtained from a generator by invoking Save.
38.a
Implementation defined: The string representation of a random number generator's state.

Dynamic Semantics

39
    {Range_Check [partial]} {check, language-defined (Range_Check)} {Constraint_Error (raised by failure of run-time check)} Instantiation of Numerics.Discrete_Random with a subtype having a null range raises Constraint_Error.
40/1
      This paragraph was deleted.{8652/0050} {Range_Check [partial]} {check, language-defined (Range_Check)} {Constraint_Error (raised by failure of run-time check)} Invoking Value with a string that is not the image of any generator state raises Constraint_Error.

Bounded (Run-Time) Errors

40.1/1
        {8652/0050} It is a bounded error to invoke Value with a string that is not the image of any generator state. {Program_Error (raised by failure of run-time check)} {Constraint_Error (raised by failure of run-time check)} If the error is detected, Constraint_Error or Program_Error is raised. Otherwise, a call to Reset with the resulting state will produce a generator such that calls to Random with this generator will produce a sequence of values of the appropriate subtype, but which might not be random in character. That is, the sequence of values might not fulfill the implementation requirements of this subclause.

Implementation Requirements

41
    A sufficiently long sequence of random numbers obtained by successive calls to Random is approximately uniformly distributed over the range of the result subtype.
42
    The Random function in an instantiation of Numerics.Discrete_Random is guaranteed to yield each value in its result subtype in a finite number of calls, provided that the number of such values does not exceed 2 15.
43
    Other performance requirements for the random number generator, which apply only in implementations conforming to the Numerics Annex, and then only in the ``strict'' mode defined there (see G.2), are given in G.2.5.

Documentation Requirements

44
    No one algorithm for random number generation is best for all applications. To enable the user to determine the suitability of the random number generators for the intended application, the implementation shall describe the algorithm used and shall give its period, if known exactly, or a lower bound on the period, if the exact period is unknown. Periods that are so long that the periodicity is unobservable in practice can be described in such terms, without giving a numerical bound.
45
    The implementation also shall document the minimum time interval between calls to the time-dependent Reset procedure that are guaranteed to initiate different sequences, and it shall document the nature of the strings that Value will accept without raising Constraint_Error.
45.a
Implementation defined: The minimum time interval between calls to the time-dependent Reset procedure that are guaranteed to initiate different random number sequences.

Implementation Advice

46
    Any storage associated with an object of type Generator should be reclaimed on exit from the scope of the object.
46.a
Ramification: A level of indirection is implicit in the semantics of the operations, given that they all take parameters of mode in. This implies that the full type of Generator probably should be a controlled type, with appropriate finalization to reclaim any heap-allocated storage.
47
    If the generator period is sufficiently long in relation to the number of distinct initiator values, then each possible value of Initiator passed to Reset should initiate a sequence of random numbers that does not, in a practical sense, overlap the sequence initiated by any other value. If this is not possible, then the mapping between initiator values and generator states should be a rapidly varying function of the initiator value.
NOTES
48
14  If two or more tasks are to share the same generator, then the tasks have to synchronize their access to the generator as for any shared variable (see 9.10).
49
15  Within a given implementation, a repeatable random number sequence can be obtained by relying on the implicit initialization of generators or by explicitly initializing a generator with a repeatable initiator value. Different sequences of random numbers can be obtained from a given generator in different program executions by explicitly initializing the generator to a time-dependent state.
50
16  A given implementation of the Random function in Numerics.Float_Random may or may not be capable of delivering the values 0.0 or 1.0. Portable applications should assume that these values, or values sufficiently close to them to behave indistinguishably from them, can occur. If a sequence of random integers from some fixed range is needed, the application should use the Random function in an appropriate instantiation of Numerics.Discrete_Random, rather than transforming the result of the Random function in Numerics.Float_Random. However, some applications with unusual requirements, such as for a sequence of random integers each drawn from a different range, will find it more convenient to transform the result of the floating point Random function. For M >= 1, the expression
51
   Integer(Float(M) * Random(G)) mod M
52
transforms the result of Random(G) to an integer uniformly distributed over the range 0 .. M-1; it is valid even if Random delivers 0.0 or 1.0. Each value of the result range is possible, provided that M is not too large. Exponentially distributed (floating point) random numbers with mean and standard deviation 1.0 can be obtained by the transformation
53
   -Log(Random(G) + Float'Model_Small))
54
where Log comes from Numerics.Elementary_Functions (see A.5.1); in this expression, the addition of Float'Model_Small avoids the exception that would be raised were Log to be given the value zero, without affecting the result (in most implementations) when Random returns a nonzero value.

Examples

55
    Example of a program that plays a simulated dice game:
56
with Ada.Numerics.Discrete_Random;
procedure Dice_Game is
   subtype Die is Integer range 1 .. 6;
   subtype Dice is Integer range 2*Die'First .. 2*Die'Last;
   package Random_Die is new Ada.Numerics.Discrete_Random (Die);
   use Random_Die;
   G : Generator;
   D : Dice;
begin
   Reset (G);  -- Start the generator in a unique state in each run
   loop
      -- Roll a pair of dice; sum and process the results
      D := Random(G) + Random(G);
      ...
   end loop;
end Dice_Game;
57
    Example of a program that simulates coin tosses:
58
with Ada.Numerics.Discrete_Random;
procedure Flip_A_Coin is
   type Coin is (Heads, Tails);
   package Random_Coin is new Ada.Numerics.Discrete_Random (Coin);
   use Random_Coin;
   G : Generator;
begin
   Reset (G);  -- Start the generator in a unique state in each run
   loop
      -- Toss a coin and process the result
      case Random(G) is
          when Heads =>
             ...
          when Tails =>
             ...
      end case;
   ...
   end loop;
end Flip_A_Coin;
59
    Example of a parallel simulation of a physical system, with a separate generator of event probabilities in each task:
60
with Ada.Numerics.Float_Random;
procedure Parallel_Simulation is
   use Ada.Numerics.Float_Random;
   task type Worker is
      entry Initialize_Generator (Initiator : in Integer);
      ...
   end Worker;
   W : array (1 .. 10) of Worker;
   task body Worker is
      G : Generator;
      Probability_Of_Event : Uniformly_Distributed;
   begin
      accept Initialize_Generator (Initiator : in Integer) do
         Reset (G, Initiator);
      end Initialize_Generator;
      loop
         ...
         Probability_Of_Event := Random(G);
         ...
      end loop;
   end Worker;
begin
   -- Initialize the generators in the Worker tasks to different states
   for I in W'Range loop
      W(I).Initialize_Generator (I);
   end loop;
   ... -- Wait for the Worker tasks to terminate
end Parallel_Simulation;
NOTES
61
17  Notes on the last example: Although each Worker task initializes its generator to a different state, those states will be the same in every execution of the program. The generator states can be initialized uniquely in each program execution by instantiating Ada.Numerics.Discrete_Random for the type Integer in the main procedure, resetting the generator obtained from that instance to a time-dependent state, and then using random integers obtained from that generator to initialize the generators in each Worker task.

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