## gp2c types and the description system## By Bill Allombert |

The main feature GP2C adds above GP is the use of types. Types give a semantic to PARI objects, so that GP2C can generate code that use specialized (hence faster) PARI functions instead of generic ones. Please read the section ’Advanced use of GP2C’ in the GP2C manual for how to use the GP2C types and examples.

Such types are used in conjunctions with so-called *descriptions* which
are stored in the field ’Description:’ of the file pari.desc and provide
the actual C code to use depending of the types of the arguments. They are
described in Section 2.

Abstractly, a GP2C type is a set of pairs (A,B) where A is a mathematical
object and B its computer representation. Two different types can contain
the same mathematical object, with a different computer representation.
For example the type bool is the set {(true,1L), (false,0L)},
*i.e.* true and false with true coded by the C-long integer 1 and
false coded by the C-long integer 0; the type negbool is
the set {(true,0L), (false,1L)} which is the same set of mathematical objects,
but now true is coded by the C-long integer 0 and false by the C-long
integer 1.

For each GP2C type, there exists a C type Ct such that for all pairs (A,B) belonging to the type, the C type of B is Ct. This C type is specified by the description _typedef.

The GP2C types are preordered. Abstractly, we say that t_{1}≺
t_{2} if and only if there is a map f such that (A,B)↦ (A,f(B))
defines a one-to-one mapping from t_{1} to t_{2}.
Practically we restrict the relation ≺ to make it a partial order
such that any two types have an upper bound.
This partial order is defined by the chains in the description
_type_preorder. It can be printed by running `gp2c -t`

.

The process of converting a mathematical object from one type to another is called casting. The casting methods known to GP2C are given by the _cast description.

In this section, we list the types known to PARI/GP. The current list is available in the description _typedef.

- small
- Small integers represented by C long integers.
- int
- Multi-precision integers represented by t_INT GENs.
- real
- Multi-precision floating point real numbers represented by t_REAL GENs.
- mp
- Multi-precision numbers. Union of the types int and real.
- vecsmall
- Vectors of small integers represented by t_VECSMALL GENs.
- vec
- Vectors and matrices of PARI objects, represented by t_VEC, t_COL or t_MAT GENs.
- var
- Polynomial variables represented by their variable number which is a C long integer. This is associated to the prototype code ’n’.
- pol
- Polynomials represented by t_POL GENs.
- genstr
- Strings represented by t_STR GENs.
- list
- GP lists represented by t_LIST GENs.
- gen
- Generic PARI objects represented by GENs.

- void
- This type is a set with only one element called void. This is the return type of functions not returning anything. GP allows to cast it to 0.
- bool
- Boolean values represented as C long integers, where 1 is true and 0 is false.
- negbool
- Negated boolean values represented as C long integers, where 0 is true and 1 is false.
- lg
- Vector lengths represented by the lg macro output, i.e. a C long integer equal to the actual length plus one.
- str
- C strings represented by C const char * strings.
- typ
- GEN types represented by C long integers, as returned by the typ() macro.

The following types are mainly for internal use inside GP2C.

- empty
- This type is the empty set. No individual object can be of this type but a set of objects can. In fact this is a default type for an unspecified set of objects.
- small_int
- Small integers represented by C int integers. This is only available for compatibility with PARI function returning int (instead of long).
- bptr
- Byte pointer. This is used for the primepointer global variable.
- func_GG
- function with protype GEN f(GEN,GEN). Used by forvec.
- pari_sp
- This is the stack pointer type pari_sp.

These types are mainly defined to allow the use of inline member functions.

- nf
- Number field as returned by nfinit.
- bnf
- Big number field as returned by bnfinit.
- bnr
- Ray class field as returned by bnrinit.
- ell
- Elliptic curve as returned by ellinit.
- bell
- Big elliptic curve as returned by ellinit.
- clgp
- Class group as returned by the ’clgp’ member function.
- prid
- Prime ideal as returned by idealprimedec.
- gal
- Galois group as returned by galoisinit.

A C-Type is just a level of indirection toward real C types. C-types are defined by the descriptions _decl_base and _decl_ext. Each type belongs to a C-type as specified by the description _typedef.

The description system is a way to describe the PARI application programming interface in a way understandable by both the GP2C compiler and human beings. The present document is mostly addressed to this second category. We start by a simple example:

The description of the GP function sqr is

(int):int sqri($1) (mp):mp gsqr($1) (gen):gen gsqr($1)

Each line is called a rule, which in this case consists of three
parts. Let us consider the first one: the parts `(int)`

, `:int`

and
`sqri($1)`

are respectively called the pattern, type,
and action part.

When GP2C compiles sqr(1), it computes the types of the arguments
(here 1 is of type small) and matches them against the patterns
from top to bottom. The “best” rule is used; in case of a tie, the topmost
rule wins. Here, all three rules apply, and the first rule wins. Since the type
of this rule is int, GP2C sets the type of the expression
sqr(1) to int. The action part is `sqri($1)`

, so GP2C
generates the C code sqri($1) where $1 is replaced by the
code of the argument 1 cast to the pattern type (int). The result
is the C code sqri(gen_1).

Now a more complex example: the description of the GP function exp is

(real):real mpexp($1) (mp):mp:prec gexp($1, prec) (gen):gen:prec gexp($1, prec)

When GP2C compiles exp(1), it looks for the "best" rules.
The first rule cannot be used, because there is no way to cast a
small to a real, so it uses the second rule.
This time the result will be
of type mp. The extra part `:prec`

is called a mode. The
mode ‘prec’ states that the action part will use the special ‘prec’
variable that holds the current real precision. This is obvious from
the action part code, but GP2C do not parse the action part so it
needs this mode. Note that the last rule is also valid and has the
same action part so would generate the exact same code. However, the
type of the expression would be less precise.

The description of the GP function matdet is

(gen, ?0):gen det($1) (gen, 1):gen det2($1) (gen, #small):gen $"incorrect flag in matdet" (gen, small):gen det0($1, $2)

We see several new pattern atoms:

- 1 matches a literal 1, e.g. matdet(M,1) would match the second rule.
- ?0 matches an optional literal 0: matdet(M), matdet(M,0) and matdet(M,) all match the first rule.
- #small matches an unspecified literal small.

Finally, we also see a new action $"…", which causes GP2C to display the error message and abort.

We now give a formal definition of descriptions.

A description is a line-separated list of rules.

A rule is a line of the form

(pattern):type:modelist action

Only the pattern part is mandatory, though most rules also include an action and a type.

A pattern is a comma-separated list of pattern atoms.

The type of a rule is a standard GP2C type.

A modelist is a colon-separated list of modes.

An action is a string (normally a piece of C code) that can include replacement strings. Replacement strings start by a $ and are substituted according to the replacement rules.

A pattern atom is one of the following, where type is any GP2C type, n any small integer, "str" any character string and ctype any C-type. A pattern atom can match an object.

- type. This matches any object of type comparable to type.
- n. This matches a constant small integer value equal to n.
- ?n. This matches an optional small value which defaults to n.
- ?type. This matches an optional type value with standard default value.
- "str". This matches a constant character string equal to str.
- &type. This matches a reference (the GP &x construction) to an object of type equal or less than type referencing the same data type.
- nothing. This matches a missing argument.
- #type. This matches a constant value of type type.
- ... This matches any number of arguments matching the previous atom. This must be the last atom of the pattern. This allows to implement functions taking an unlimited number of arguments.
- C!ctype. This matches any object of C-type ctype.
- @type. This matches a variable of type type. This is mainly used for expressions that evaluate their arguments several times.
- *type. This matches an lvalue of type type. This is used in constructions that modify their arguments.

The best rule is determined as follows:

- The result of matching a pattern atom against some GP code is either ’reject’ or ’match’.
- There are three matching levels: ’partial’, ’normal’ and ’perfect’.
- A pattern matches if all the atoms match.
- A rule matches if its pattern matches.
- The best rule is the matching rule with the higher number of normal and perfect matches. In case of a tie, the highest number of perfect matches wins. If there is still a tie, the topmost rule wins.

When matching the pattern atoms type and ?type, the matching level is determined as follows:

- a perfect match occurs when the type of the object is exactly type,
- a normal match when the type is less than type,
- a partial match when the type is bigger than type.
- Rejection happens when the types are uncomparable.

Other pattern atoms always result in a reject or a perfect match.

Modes are used in descriptions to give more information to GP2C about the action part. They are usually useless to human beings that are smart enough to understand the action part. The current list of modes is:

- prec
- The action uses the prec variable.
- parens
- The action does not have top precedence. GP2C will put it between parentheses when needed (see $())
- copy
- The action returns data that access memory belonging to other objects. GP2C will generate calls to gcopy() when needed.

The following special sequences can occur in the action part:

- $n. This is replaced by the n-th argument of the function.
- $(n). This is replaced by the n-th argument of the function between parenthesis if it has the parens mode.
- $type:n. This is replaced by the n-th argument of the function cast to type type.
- $(type:n). Combination of $(n) and $type:n.
- $%n. This is replaced by the n-th argument of the function, which must be a constant string, with all % characters doubled and no quotes. This is for use inside format specification.
- $prec: short cut for $prec.
- $bitprec: short cut for $bitprec.
- $"message". Signals an invalid condition. GP2C will abort by printing the error message message.
- ${RPN sequence} The RPN sequence is a space separated list of RPN commands that will be evaluated by the GP2C internal RPN evaluator. If the stack is empty at the end of the evaluation, this is replaced by the empty string, else this is replaced by the integer at the top of the stack. Some RPN commands generate text, in that case it is pasted just before the $ sign.

The commands are evaluated with respect to a stack of integer values, initially empty. The exact list of command supported by a particular GP2C version is the %accepted_command hash in the script scripts/822_desc.pl.in.

- literal integer
- push the integer at the top of the stack.
- :type
- push the type type at the top of the stack.
- add, sub, mul, div, mod
- 2-ary arithmetic operators
- neg
- 1-ary arithmetic operator
- and, or, xor
- 2-ary logical operators
- not
- 1-ary logical operator
- type
- pop an integer n and push the type of the n-th argument.
- value
- pop an integer n and push the value of the n-th argument, provided it is a constant integer.
- code
- pop an integer n and generate the C code for the n-th argument.
- cast
- pop an integer n and a type t and generate the C code for the n-th argument cast to type t.
- parens
- this is a flag requesting code and cast to enclose the C code between parenthesis if the argument has the parens mode.
- str_format, str_raw
- pop an integer n such that the n-th argument is a constant string and display the string without leading and ending ". Furthermore str_format will display the string in a way suitable for inclusion in a format string by quoting meta-characters.
- prec
- display the local precision (in prec format).
- bitprec
- display the local precision in bit.

The following RPN commands are useful with the ... pattern atom to implement functions that take an unlimited number of arguments.

- nbarg
- push the actual number of arguments of the function.
- format_string, format_args
- pop an integer n such that the n-th argument corresponds to a ... pattern atom and generate a format string and a list of arguments, see the description of the GP function print.
- code, cast
- If the integer n corresponds to a ... pattern atom, generate a comma-separated list of C code for the arguments n−1, n, n+1, …, nbarg, by matching each argument against the n−1 pattern atom.
- stdref
- this is a flag requesting code and type to prepend a ’&’ before each arguments.

The following RPN commands are useful to implement functions that take closures as arguments.

- wrapper, cookie
- pop an integer n and generate a call to the wrapper (resp. the cookie associated to the wrapper) for the n-th argument. The wrapper generated depends on the wrapper prototype in the Wrapper field. The cookie is the list of local variables seen by the closure.

This document was translated from L^{A}T_{E}X byH^{E}V^{E}A.