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version 1.3, 2000/01/23 05:28:33

version 1.11, 2001/08/27 05:39:15

Line 1

%% $OpenXM: OpenXM/doc/OpenXM-specs/cmo-basic1.tex,v 1.1.1.1 2000/01/20 08:52:46 noro Exp $

%% $OpenXM: OpenXM/doc/OpenXM-specs/cmo-basic1.tex,v 1.10 2001/04/10 11:56:29 takayama Exp $

//&jp \section{ 数, 多項式の CMO 表現 }

//&eg \section{ CMOexpressions for numbers and polynomials }

\label{sec:basic1}

/*&C

@../SSkan/plugin/cmotag.h

\begin{verbatim}

Line 24

/*&jp

以下, グループ CMObject/Basic1, CMObject/Tree

以下, グループ CMObject/Basic, CMObject/Tree

および CMObject/DistributedPolynomial

に属する CMObject の形式を説明する.

\noroa{ tagged list を導入すべきか? cf. SSkan/plugin/cmo.txt }

\noindent

{\tt OpenXM/src/ox\_toolkit} にある {\tt bconv} をもちいると

CMO expression を binary format に変換できるので,

これを参考にするといい.

/*&eg

In the sequel, we will explain on the groups

CMObject/Basic1, CMObject/Tree

CMObject/Basic, CMObject/Tree

and CMObject/DistributedPolynomial.

\noindent

The program {\tt bconv} at {\tt OpenXM/src/ox\_toolkit}

translates

CMO expressions into binary formats.

It is convinient to understand the binary formats explained in

this section.

/*&C

\noindent Example:

\begin{verbatim}

bash$ ./bconv

> (CMO_ZZ,123123);

00 00 00 14 00 00 00 01 00 01 e0 f3

\end{verbatim}

/*&jp

\bigbreak

\noindent

Group CMObject/Basic1 requires CMObject/Basic0. \\

Group CMObject/Basic requires CMObject/Primitive. \\

ZZ, QQ, Zero, Rational, Indeterminate,$\in$ CMObject/Basic1. \\

ZZ, QQ, Zero, Rational, Indeterminate,$\in$ CMObject/Basic. \\

\begin{eqnarray*}

\mbox{Zero} &:& ({\tt CMO\_ZERO}) \\

& & \mbox{ --- ユニバーサルなゼロを表す. } \\

\mbox{ZZ} &:& ({\tt CMO\_ZZ},{\sl int32}\, {\rm f}, {\sl byte}\, \mbox{a[1]}, \ldots

{\sl byte}\, \mbox{a[m]} ) \\

{\sl byte}\, \mbox{a[$|$f$|$]} ) \\

&:& \mbox{ --- bignum をあらわす. a[i] についてはあとで説明}\\

\mbox{QQ} &:& ({\tt CMO\_QQ}, {\sl ZZ}\, {\rm a}, {\sl ZZ}\, {\rm b}) \\

\mbox{QQ} &:& ({\tt CMO\_QQ},

{\sl int32}\, {\rm m}, {\sl byte}\, \mbox{a[1]}, \ldots, {\sl byte}\, \mbox{a[$|$m$|$]},

{\sl int32}\, {\rm n}, {\sl byte}\, \mbox{b[1]}, \ldots, {\sl byte}\, \mbox{b[$|$n$|$]})\\

& & \mbox{ --- 有理数 $a/b$ を表す. } \\

\mbox{Rational} &:& ({\tt CMO\_RATIONAL}, {\sl CMObject}\, {\rm a}, {\sl CMObject}\, {\rm b}) \\

& & \mbox{ --- $a/b$ を表す. } \\

Line 60 ZZ, QQ, Zero, Rational, Indeterminate,$\in$ CMObject/B

Line 80 ZZ, QQ, Zero, Rational, Indeterminate,$\in$ CMObject/B

\bigbreak

\noindent

Group CMObject/Basic1 requires CMObject/Basic0. \\

Group CMObject/Basic requires CMObject/Primitive. \\

ZZ, QQ, Zero, Rational, Indeterminate,$\in$ CMObject/Basic1. \\

ZZ, QQ, Zero, Rational, Indeterminate,$\in$ CMObject/Basic. \\

\begin{eqnarray*}

\mbox{Zero} &:& ({\tt CMO\_ZERO}) \\

& & \mbox{ --- Universal zero } \\

\mbox{ZZ} &:& ({\tt CMO\_ZZ},{\sl int32}\, {\rm f}, {\sl byte}\, \mbox{a[1]}, \ldots

{\sl byte}\, \mbox{a[m]} ) \\

{\sl byte}\, \mbox{a[$|$m$|$]} ) \\

&:& \mbox{ --- bignum. The meaning of a[i] will be explained later.}\\

\mbox{QQ} &:& ({\tt CMO\_QQ}, {\sl ZZ}\, {\rm a}, {\sl ZZ}\, {\rm b}) \\

\mbox{QQ} &:& ({\tt CMO\_QQ},

{\sl int32}\, {\rm m}, {\sl byte}\, \mbox{a[1]}, \ldots, {\sl byte}\, \mbox{a[$|$m$|$]},

{\sl int32}\, {\rm n}, {\sl byte}\, \mbox{b[1]}, \ldots, {\sl byte}\, \mbox{b[$|$n$|$]})\\

& & \mbox{ --- Rational number $a/b$. } \\

\mbox{Rational} &:& ({\tt CMO\_RATIONAL}, {\sl CMObject}\, {\rm a}, {\sl CMObject}\, {\rm b}) \\

& & \mbox{ --- Rational expression $a/b$. } \\

Line 81 ZZ, QQ, Zero, Rational, Indeterminate,$\in$ CMObject/B

Line 103 ZZ, QQ, Zero, Rational, Indeterminate,$\in$ CMObject/B

/*&jp

\subsection{Indeterminate および Tree}

Indeterminate は変数名をあらわす.

v はバイト列であればなにを用いてもよいが,

システム毎に変数名として用いられるバイト列は制限がある.

Line 92 escape sequence を用いて実現するのは, 無理があるようで

Line 115 escape sequence を用いて実現するのは, 無理があるようで

テーブルを作成する必要があるであろう.)

/*&eg

\subsection{Indetermnate and Tree}

Indeterminate is a name of a variable.

v may be any sequence of bytes, but each system has its own

restrictions on the names of variables.

Line 102 in one to one correspondence.

Line 126 in one to one correspondence.

/*&jp

\noindent

Group CMObject/Tree requires CMObject/Basic1. \\

Group CMObject/Tree requires CMObject/Basic. \\

Tree, Lambda $\in$ CMObject/Basic1. \\

Tree, Lambda $\in$ CMObject/Basic. \\

\begin{eqnarray*}

\mbox{Tree} &:& ({\tt CMO\_TREE}, {\sl Cstring}\, {\rm name},

{\sl Cstring}\, {\rm cdname}, {\sl List}\, {\rm leaves}) \\

{\sl List}\, {\rm attributes}, {\sl List}\, {\rm leaves}) \\

& & \mbox{ --- 名前 name の定数または関数. 関数の評価はおこなわない. } \\

& & \mbox{ --- cdname は空文字列でなければ name の意味が説明されている }\\

& & \mbox{ --- attributes は空リストでなければ name の属性を保持している. }\\

& & \mbox{ --- OpenMath CD (content dictionary) の名前. } \\

& & \mbox{ --- 属性リストは, key と値のペアである. }\\

\mbox{Lambda} &:& ({\tt CMO\_LAMBDA}, {\sl List}\, {\rm args},

{\sl Tree} {\rm body}) \\

& & \mbox{ --- body を args を引数とする関数とする. } \\

Line 119 Tree, Lambda $\in$ CMObject/Basic1. \\

Line 143 Tree, Lambda $\in$ CMObject/Basic1. \\

/*&eg

\noindent

Group CMObject/Tree requires CMObject/Basic1. \\

Group CMObject/Tree requires CMObject/Basic. \\

Tree, Lambda $\in$ CMObject/Basic1. \\

Tree, Lambda $\in$ CMObject/Basic. \\

\begin{eqnarray*}

\mbox{Tree} &:& ({\tt CMO\_TREE}, {\sl Cstring}\, {\rm name},

{\sl Cstring}\, {\rm cdname}, {\sl List}\, {\rm leaves}) \\

{\sl List}\, {\rm attributes}, {\sl List}\, {\rm leaves}) \\

& & \mbox{ --- A function or a constant of name. Functions are not evaluated. } \\

& & \mbox{ --- cdname may be a null. If it is not null, it is the name of}\\

& & \mbox{ --- attributes may be a null list. If it is not null, it is a list of}\\

& & \mbox{ --- the OpenMath CD (content dictionary). } \\

& & \mbox{ --- key and value pairs. } \\

\mbox{Lambda} &:& ({\tt CMO\_LAMBDA}, {\sl List}\, {\rm args},

{\sl Tree} {\rm body}) \\

& & \mbox{ --- a function with the arguments body. } \\

Line 160 For example,

Line 184 For example,

$\sin(x+e)$ is expressed as

{\tt (sin, (plus, x, e))}

as a tree.

Tree may be expressed by putting itself between

Tree may be expressed by putting the expression between

{\tt SM\_beginBlock} and {\tt SM\_endBlock}, which are

stack machine commands for delayed evaluation.

(cf. {\tt \{ }, {\tt \} } in PostScript).

However it makes the implementation of stack machines complicated.

It is desirable that CMObject is independent of OX stack machine.

Therefore we introduce an OpenMath like tree representation for CMO

tree object.

Tree object.

This method allows us to implement tree structure easily

on individual OpenXM systems.

Note that CMO Tree corresponds to Symbol and Application in OpenMath.

Line 190 It is the same as the Lambda expression in Lisp.

Line 214 It is the same as the Lambda expression in Lisp.

//&jp 例: $sin(x+e)$ の表現.

//&eg Example: the expression of $sin(x+e)$.

\begin{verbatim}

(CMO_TREE, (CMO_STRING, "sin"), (CMO_STRING, "basic"),

(CMO_TREE, (CMO_STRING, "sin"),

(CMO_LIST,[size=]1,(CMO_LIST,[size=]2,(CMO_STRING, "cdname"),

(CMO_STRING,"basic")))

(CMO_LIST,[size=]1,

(CMO_TREE, (CMO_STRING, "plus"), (CMO_STRING, "basic"),

(CMO_LIST,[size=]2, (CMO_INDETERMINATE,"x"),

//&jp (CMO_TREE,(CMO_STRING, "e"), 自然対数の底

//&eg (CMO_TREE,(CMO_STRING, "e"), Napier's number

//&eg (CMO_TREE,(CMO_STRING, "e"), the base of natural logarithms

(CMO_STRING, "basic"))

))

)

\end{verbatim}

//&jp Leave の成分には, 多項式を含む任意のオブジェクトがきてよい.

//&eg Elements of the leave may be any objects including polynomials.

\noindent

Example:

\begin{verbatim}

sm1> [(plus) (Basic) [(123).. (345)..]] [(class) (tree)] dc ::

Class.tree [ $plus$ , $Basic$ , [ 123 , 345 ] ]

Class.tree [ $plus$ , $basic$ , [ 123 , 345 ] ]

\end{verbatim}

\bigbreak

//&jp 次に, 分散表現多項式に関係するグループを定義しよう.

//&eg Let us define a group for distributed polynomials.

/*&eg

Let us define a group for distributed polynomials. In the following

DMS stands for Distributed Monomial System.

\medbreak

\noindent

Group CMObject/DistributedPolynomials requires CMObject/Basic0,

Group CMObject/DistributedPolynomials requires CMObject/Primitive,

CMObject/Basic1. \\

CMObject/Basic. \\

Monomial, Monomial32, Coefficient, Dpolynomial, DringDefinition,

Generic DMS ring, RingByName, DMS of N variables $\in$

CMObject/DistributedPolynomials. \\

Line 289 $x^e = x_1^{e_1} \cdots x_n^{e_n}$ の各指数 $e_i$

Line 320 $x^e = x_1^{e_1} \cdots x_n^{e_n}$ の各指数 $e_i$

\mbox{Generic DMS ring}

&:& ({\tt CMO\_DMS\_GENERIC}) \\

\mbox{RingByName}&:& ({\tt CMO\_RING\_BY\_NAME}, {\sl Cstring} s) \\

& & \mbox{ --- The ring definition refered by the name ``s''.} \\

& & \mbox{ --- The ring definition referred by the name ``s''.} \\

\mbox{DMS of N variables}

&:& ({\tt CMO\_DMS\_OF\_N\_VARIABLES}, \\

& & \ ({\tt CMO\_LIST}, {\sl int32}\, \mbox{m},

Line 394 a CMO\_ZZ $14$ is expressed by

Line 425 a CMO\_ZZ $14$ is expressed by

\mbox{(CMO\_ZZ, 1, 0, 0, 0, e)},

//&jp と表わす. これはバイト列では

//&egThe corresponding byte sequence is

//&eg The corresponding byte sequence is

\mbox{\tt 00 00 00 14 00 00 00 01 00 00 00 0e}

Line 442 Ring by Name を用いた場合, 現在の名前空間で変数 yyy に

Line 473 Ring by Name を用いた場合, 現在の名前空間で変数 yyy に

/*&eg

We treat polynomial rings and their elements as follows.

An element of a generic DMS ring is an element of

Generic DMS ring is an $n$-variate polynomial ring $K[x_1, \ldots, x_n]$,

an $n$-variate polynomial ring $K[x_1, \ldots, x_n]$,

where $K$ is some coefficient set. $K$ is unknown in advance

and it is determined when coefficients are received.

and it is determined when coefficients of an element are received.

When a server has received an element in a generic DMS ring,

When a server has received an element in Generic DMS ring,

the server has to translate it into the corresponding local object

on the server. Each server has its own translation scheme.

In Asir such an element are translated into a distributed polynomial.

In {\tt kan/sm1} things are complicated.

{\tt kan/sm1} does not have any class corresponding to a generic DMS ring.

{\tt kan/sm1} does not have any class corresponding to Generic DMS ring.

{\tt kan/sm1} translates a DMS of N variables into an element of

the CurrentRing.

If the CurrentRing is $n'$-variate and $n' < n$, then

a polynomial ring is newly created. Optional informations such as

an $n$-variate polynomial ring is newly created. Optional informations such as

the term order are all ignored.

If RingbyName ({\tt CMO\_RING\_BY\_NAME}, yyy)

If RingByName ({\tt CMO\_RING\_BY\_NAME}, yyy)

is specified as the second field of DMS,

it requests a sever to use a ring object whose name is yyy

as the destination ring for the translation.

Line 504 $3 x^2 y$ is regarded as an element of a six-variate p

Line 532 $3 x^2 y$ is regarded as an element of a six-variate p

#define CMO_POLYNOMIAL_IN_ONE_VARIABLE 33

\end{verbatim}

Group CMObject/RecursivePolynomial requires CMObject/Basic0, CMObject/Basic1.\\

Group CMObject/RecursivePolynomial requires CMObject/Primitive, CMObject/Basic.\\

Polynomial in 1 variable, Coefficient, Name of the main variable,

Recursive Polynomial, Ring definition for recursive polynomials

$\in$ CMObject/RecursivePolynomial \\

Line 533 $\in$ CMObject/RecursivePolynomial \\

Line 561 $\in$ CMObject/RecursivePolynomial \\

\mbox{ Polynomial in 1 variable}\, | \, \mbox{Coefficient} \\

\mbox{RringDefinition}

& : & \mbox{ {\sl List} v } \\

& & \quad \mbox{ --- v は, 変数名(indeterminate) のリスト. } \\

& & \quad \mbox{ --- v は, 変数名(indeterminate) または Tree のリスト. } \\

& & \quad \mbox{ --- 順序の高い順. } \\

\end{eqnarray*}

Line 543 $\in$ CMObject/RecursivePolynomial \\

Line 571 $\in$ CMObject/RecursivePolynomial \\

\mbox{({\tt CMO\_POLYNOMIAL\_IN\_ONE\_VARIABLE},\, {\sl int32}\, m, } \\

& & \quad \mbox{ Name of the main variable }, \\

& & \quad \mbox{ \{ {\sl int32} e, Coefficient \}} \\

& & \mbox{ --- m is the number of monimials. } \\

& & \mbox{ --- m is the number of monomials. } \\

& & \mbox{ --- A pair of e and Coefficieint represents a monomial. } \\

& & \mbox{ --- A pair of e and Coefficient represents a monomial. } \\

& & \mbox{ --- The pairs of e and Coefficient are sorted in the } \\

& & \mbox{ \quad decreasing order, usually with respect to e.} \\

& & \mbox{ --- e denotes an exponent of a monomial with respect to } \\

Line 563 $\in$ CMObject/RecursivePolynomial \\

Line 591 $\in$ CMObject/RecursivePolynomial \\

\mbox{ Polynomial in 1 variable}\, | \, \mbox{Coefficient} \\

\mbox{RringDefinition}

& : & \mbox{ {\sl List} v } \\

& & \quad \mbox{ --- v is a list of names of indeterminates. } \\

& & \quad \mbox{ --- v is a list of names of indeterminates or trees. } \\

& & \quad \mbox{ --- It is sorted in the decreasing order. } \\

\end{eqnarray*}

Line 591 $$ x^3 (1234 y^5 + 17 ) + x^1 (y^{10} + 31 y^5) $$

Line 619 $$ x^3 (1234 y^5 + 17 ) + x^1 (y^{10} + 31 y^5) $$

/*&eg

We intend to represent non-commutative polynomials with the

same form. In such a case, the order of products are defined

as above, that is a power of the mail variable $\times$ a coeffcient.

as above, that is a power of the main variable $\times$ a coeffcient.

\noindent

Line 613 Class.recursivePolynomial h * ((-1)) + (x^2 * (1))

Line 641 Class.recursivePolynomial h * ((-1)) + (x^2 * (1))

\end{verbatim}

\noindent

Group CMObject/MachineDouble requires CMObject/Basic0.\\

Group CMObject/MachineDouble requires CMObject/Primitive.\\

64bit machine double, Array of 64bit machine double

128bit machine double, Array of 128bit machine double

$\in$ CMObject/MachineDouble \\

Line 677 $\in$ CMObject/MachineDouble \\

Line 705 $\in$ CMObject/MachineDouble \\

\bigbreak

//&jp 次に IEEE 準拠の float および Big float を定義しよう.

//&eg We define IEEE conformant float and big float.

//&eg We define float and big float conforming to the IEEE standard.

\begin{verbatim}

#define CMO_BIGFLOAT 50

#define CMO_IEEE_DOUBLE_FLOAT 51

Line 688 IEEE 準拠の float については, IEEE 754 double precisio

Line 716 IEEE 準拠の float については, IEEE 754 double precisio

format (64 bit) の定義を見よ.

/*&eg

See IEEE 754 double precision floating-point (64 bit) for the details of IEEE

See IEEE 754 double precision floating-point (64 bit) for the details of

conformant float.

float conforming to the IEEE standard.

\noindent

Group CMObject/Bigfloat requires CMObject/Basic0, CMObject/Basic1.\\

Group CMObject/Bigfloat requires CMObject/Primitive, CMObject/Basic.\\

Bigfloat

$\in$ CMObject/Bigfloat \\

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