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version 1.3, 2000/01/23 05:28:33 version 1.12, 2002/01/20 09:26:21
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 %% $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 1.11 2001/08/27 05:39:15 takayama Exp $
 //&jp \section{ 数, 多項式 の  CMO 表現 }  //&jp \section{ 数, 多項式 の  CMO 表現 }
 //&eg \section{ CMOexpressions for numbers and polynomials }  //&eg \section{ CMOexpressions for numbers and polynomials }
   \label{sec:basic1}
 /*&C  /*&C
 @../SSkan/plugin/cmotag.h  @../SSkan/plugin/cmotag.h
 \begin{verbatim}  \begin{verbatim}
Line 24 
Line 24 
 */  */
   
 /*&jp  /*&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  および CMObject/DistributedPolynomial
 に属する CMObject の形式を説明する.  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  /*&eg
 In the sequel, we will explain on the groups  In the sequel, we will explain on the groups
 CMObject/Basic1, CMObject/Tree  CMObject/Basic, CMObject/Tree
 and CMObject/DistributedPolynomial.  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  /*&jp
   
 \bigbreak  \bigbreak
 \noindent  \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*}  \begin{eqnarray*}
 \mbox{Zero} &:& ({\tt CMO\_ZERO}) \\  \mbox{Zero} &:& ({\tt CMO\_ZERO}) \\
 & & \mbox{ --- ユニバーサルな ゼロを表す. } \\  & & \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  \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{ --- 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{ --- 有理数 $a/b$ を表す. } \\
 \mbox{Rational}        &:& ({\tt CMO\_RATIONAL}, {\sl CMObject}\, {\rm a}, {\sl CMObject}\, {\rm b}) \\  \mbox{Rational}        &:& ({\tt CMO\_RATIONAL}, {\sl CMObject}\, {\rm a}, {\sl CMObject}\, {\rm b}) \\
 & & \mbox{ ---  $a/b$ を表す. } \\  & & \mbox{ ---  $a/b$ を表す. } \\
Line 56  ZZ, QQ, Zero, Rational, Indeterminate,$\in$ CMObject/B
Line 76  ZZ, QQ, Zero, Rational, Indeterminate,$\in$ CMObject/B
 & & \mbox{ --- 変数名 $v$ . } \\  & & \mbox{ --- 変数名 $v$ . } \\
 \end{eqnarray*}  \end{eqnarray*}
 */  */
   
   
 /*&eg  /*&eg
   
 \bigbreak  \bigbreak
 \noindent  \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*}  \begin{eqnarray*}
 \mbox{Zero} &:& ({\tt CMO\_ZERO}) \\  \mbox{Zero} &:& ({\tt CMO\_ZERO}) \\
 & & \mbox{ --- Universal zero } \\  & & \mbox{ --- Universal zero } \\
 \mbox{ZZ}         &:& ({\tt CMO\_ZZ},{\sl int32}\, {\rm f}, {\sl byte}\, \mbox{a[1]}, \ldots  \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{ --- 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 number $a/b$. } \\
 \mbox{Rational}        &:& ({\tt CMO\_RATIONAL}, {\sl CMObject}\, {\rm a}, {\sl CMObject}\, {\rm b}) \\  \mbox{Rational}        &:& ({\tt CMO\_RATIONAL}, {\sl CMObject}\, {\rm a}, {\sl CMObject}\, {\rm b}) \\
 & & \mbox{ ---  Rational expression $a/b$. } \\  & & \mbox{ ---  Rational expression $a/b$. } \\
Line 79  ZZ, QQ, Zero, Rational, Indeterminate,$\in$ CMObject/B
Line 103  ZZ, QQ, Zero, Rational, Indeterminate,$\in$ CMObject/B
 /*&C  /*&C
   
 */  */
   /*&C
   
   */
   
 /*&jp  /*&jp
 Indeterminate は変数名をあらわす.  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,  v はバイト列であればなにを用いてもよいが,
Line 92  escape sequence を用いて実現するのは, 無理があるようで
Line 119  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.)  テーブルを作成する必要があるであろう.)
 */  */
 /*&eg  /*&eg
 Indeterminate is a name of a variable.  The name of a variable should be expressed by using Indeterminate.
 v may be any sequence of bytes, but each system has its own  v may be any sequence of bytes, but each system has its own
 restrictions on the names of variables.  restrictions on the names of variables.
 Indeterminates of CMO and internal variable names must be translated  Indeterminates of CMO and internal variable names must be translated
 in one to one correspondence.  in one-to-one correspondence.
 */  */
   
   
 /*&jp  /*&jp
   \subsection{Indeterminate および Tree}
   
 \noindent  \noindent
 Group CMObject/Tree requires CMObject/Basic1. \\  Group CMObject/Tree requires CMObject/Basic. \\
 Tree, Lambda $\in$ CMObject/Basic1. \\  Tree, Lambda $\in$ CMObject/Tree. \\
 \begin{eqnarray*}  \begin{eqnarray*}
 \mbox{Tree}        &:& ({\tt CMO\_TREE}, {\sl Cstring}\, {\rm name},  \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{ --- 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},  \mbox{Lambda}        &:& ({\tt CMO\_LAMBDA}, {\sl List}\, {\rm args},
                           {\sl Tree} {\rm body}) \\                            {\sl Tree} {\rm body}) \\
 & & \mbox{ --- body を args を引数とする関数とする. } \\  & & \mbox{ --- body を args を引数とする関数とする. } \\
 & & \mbox{ --- optional な引数が必要なときは, leaves の後へつづける.} \\  
 \end{eqnarray*}  \end{eqnarray*}
 */  */
 /*&eg  /*&eg
   \subsection{Indeterminate and Tree}
   
 \noindent  \noindent
 Group CMObject/Tree requires CMObject/Basic1. \\  Group CMObject/Tree requires CMObject/Basic. \\
 Tree, Lambda $\in$ CMObject/Basic1. \\  Tree, Lambda $\in$ CMObject/Tree. \\
 \begin{eqnarray*}  \begin{eqnarray*}
 \mbox{Tree}        &:& ({\tt CMO\_TREE}, {\sl Cstring}\, {\rm name},  \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{ --- ``name'' is the name of the node of the tree. } \\
 & & \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},  \mbox{Lambda}        &:& ({\tt CMO\_LAMBDA}, {\sl List}\, {\rm args},
                           {\sl Tree} {\rm body}) \\                            {\sl Tree} {\rm body}) \\
 & & \mbox{ --- a function with the arguments body. } \\  & & \mbox{ --- a function with the arguments body. } \\
 & & \mbox{ --- optional arguments come after leaves.} \\  
 \end{eqnarray*}  \end{eqnarray*}
 */  */
   
Line 160  For example,
Line 188  For example,
 $\sin(x+e)$ is expressed as  $\sin(x+e)$ is expressed as
 {\tt (sin, (plus, x, e))}  {\tt (sin, (plus, x, e))}
 as a tree.  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  {\tt SM\_beginBlock} and {\tt SM\_endBlock}, which are
 stack machine commands for delayed evaluation.  stack machine commands for delayed evaluation.
 (cf. {\tt \{ }, {\tt \} } in PostScript).  (cf. {\tt \{ }, {\tt \} } in PostScript).
 However it makes the implementation of stack machines complicated.  However it makes the implementation of stack machines complicated.
 It is desirable that CMObject is independent of OX stack machine.  It is desirable that CMObject is independent of OX stack machine.
 Therefore we introduce an OpenMath like tree representation for CMO  Therefore we introduce an OpenMath like tree representation for CMO
 tree object.  Tree object.
 This method allows us to implement tree structure easily  This method allows us to implement tree structure easily
 on individual OpenXM systems.  on individual OpenXM systems.
 Note that CMO Tree corresponds to Symbol and Application in OpenMath.  Note that CMO Tree corresponds to Symbol and Application in OpenMath.
Line 183  Lisp の Lambda 表現と同じ.
Line 211  Lisp の Lambda 表現と同じ.
 */  */
 /*&eg  /*&eg
 Lambda is used to define functions.  Lambda is used to define functions.
 It is the same as the Lambda expression in Lisp.  The notion ``lambda'' is borrowed from the language Lisp.
 */  */
   
 \noindent  \noindent
 //&jp 例: $sin(x+e)$ の表現.  //&jp 例: $sin(x+e)$ の表現.
 //&eg Example: the expression of $sin(x+e)$.  //&eg Example: the expression of $sin(x+e)$.
 \begin{verbatim}  \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_LIST,[size=]1,
         (CMO_TREE, (CMO_STRING, "plus"), (CMO_STRING, "basic"),          (CMO_TREE, (CMO_STRING, "plus"), (CMO_STRING, "basic"),
             (CMO_LIST,[size=]2, (CMO_INDETERMINATE,"x"),              (CMO_LIST,[size=]2, (CMO_INDETERMINATE,"x"),
 //&jp                                (CMO_TREE,(CMO_STRING, "e"),  自然対数の底  //&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"))      (CMO_LIST,[size=]1,(CMO_LIST,[size=]2,(CMO_STRING, "cdname"),
                                             (CMO_STRING,"basic")))
         ))          ))
     )      )
 )  )
 \end{verbatim}  \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  \noindent
 Example:  Example:
 \begin{verbatim}  \begin{verbatim}
 sm1> [(plus) (Basic) [(123).. (345)..]] [(class) (tree)] dc ::  sm1> [(plus) [[(cdname) (basic)]] [(123).. (345)..]] [(class) (tree)] dc ::
 Class.tree [    $plus$ , $Basic$ , [    123 , 345 ]  ]  Class.tree [$plus$ , [[$cdname$ , $basic$ ]], [ 123 , 345 ]  ]
 \end{verbatim}  \end{verbatim}
   
   \noindent
   Example:
   \begin{verbatim}
   asir
   [753] taka_cmo100_xml_form(quote(sin(x+1)));
   <cmo_tree>  <cmo_string>"sin"</cmo_string>
    <cmo_list><cmo_int32 for="length">1</cmo_int32>
      <cmo_list><cmo_int32 for="length">2</cmo_int32>
        <cmo_string>"cdname"</cmo_string>
        <cmo_string>"basic"</cmo_string>
      </cmo_list> </cmo_list>
   <cmo_tree>    <cmo_string>"plus"</cmo_string>
     <cmo_list><cmo_int32 for="length">1</cmo_int32>
       <cmo_list><cmo_int32 for="length">2</cmo_int32>
         <cmo_string>"cdname"</cmo_string>
         <cmo_string>"basic"</cmo_string>
       </cmo_list> </cmo_list>
    <cmo_indeterminate> <cmo_string>"x"</cmo_string>  </cmo_indeterminate>
    <cmo_zz>1</cmo_zz>
   </cmo_tree></cmo_tree>
   \end{verbatim}
   
   
 \bigbreak  \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.  //&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  \medbreak
 \noindent  \noindent
 Group CMObject/DistributedPolynomials requires CMObject/Basic0,  Group CMObject/DistributedPolynomials requires CMObject/Primitive,
 CMObject/Basic1. \\  CMObject/Basic. \\
 Monomial, Monomial32, Coefficient, Dpolynomial, DringDefinition,  Monomial, Monomial32, Coefficient, Dpolynomial, DringDefinition,
 Generic DMS ring, RingByName, DMS of N variables $\in$  Generic DMS ring, RingByName, DMS of N variables $\in$
 CMObject/DistributedPolynomials. \\  CMObject/DistributedPolynomials. \\
Line 234  $x^e = x_1^{e_1} \cdots x_n^{e_n}$ の各指数 $e_i$
Line 291  $x^e = x_1^{e_1} \cdots x_n^{e_n}$ の各指数 $e_i$
 をあらわす.} \\  をあらわす.} \\
 \mbox{Coefficient}&:& \mbox{ZZ} | \mbox{Integer32} \\  \mbox{Coefficient}&:& \mbox{ZZ} | \mbox{Integer32} \\
 \mbox{Dpolynomial}&:& \mbox{Zero} \\  \mbox{Dpolynomial}&:& \mbox{Zero} \\
 & & |\ ({\tt CMO\_DISTRIBUTED\_POLYNOMIAL},{\sl int32} m, \\  & & |\ ({\tt CMO\_DISTRIBUTED\_POLYNOMIAL},{\sl int32}\, m, \\
 & & \ \ \mbox{DringDefinition},  & & \ \ \mbox{DringDefinition},
 [\mbox{Monomial32}|\mbox{Zero}], \\  [\mbox{Monomial32}|\mbox{Zero}], \\
 & &\ \  & &\ \
Line 276  $x^e = x_1^{e_1} \cdots x_n^{e_n}$ の各指数 $e_i$
Line 333  $x^e = x_1^{e_1} \cdots x_n^{e_n}$ の各指数 $e_i$
                       $x^e = x_1^{e_1} \cdots x_n^{e_n}$. } \\                        $x^e = x_1^{e_1} \cdots x_n^{e_n}$. } \\
 \mbox{Coefficient}&:& \mbox{ZZ} | \mbox{Integer32} \\  \mbox{Coefficient}&:& \mbox{ZZ} | \mbox{Integer32} \\
 \mbox{Dpolynomial}&:& \mbox{Zero} \\  \mbox{Dpolynomial}&:& \mbox{Zero} \\
                  & & |\ ({\tt CMO\_DISTRIBUTED\_POLYNOMIAL},{\sl int32} m, \\                   & & |\ ({\tt CMO\_DISTRIBUTED\_POLYNOMIAL},{\sl int32}\, m, \\
                  & & \ \ \mbox{DringDefinition}, [\mbox{Monomial32}|\mbox{Zero}], \\                   & & \ \ \mbox{DringDefinition}, [\mbox{Monomial32}|\mbox{Zero}], \\
                  & &\ \                   & &\ \
                     \{\mbox{Monomial32}\})  \\                      \{\mbox{Monomial32}\})  \\
Line 289  $x^e = x_1^{e_1} \cdots x_n^{e_n}$ の各指数 $e_i$
Line 346  $x^e = x_1^{e_1} \cdots x_n^{e_n}$ の各指数 $e_i$
 \mbox{Generic DMS ring}  \mbox{Generic DMS ring}
                  &:& ({\tt CMO\_DMS\_GENERIC}) \\                   &:& ({\tt CMO\_DMS\_GENERIC}) \\
 \mbox{RingByName}&:& ({\tt CMO\_RING\_BY\_NAME}, {\sl Cstring} s) \\  \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}  \mbox{DMS of N variables}
                  &:& ({\tt CMO\_DMS\_OF\_N\_VARIABLES}, \\                   &:& ({\tt CMO\_DMS\_OF\_N\_VARIABLES}, \\
                  & & \ ({\tt CMO\_LIST}, {\sl int32}\, \mbox{m},                   & & \ ({\tt CMO\_LIST}, {\sl int32}\, \mbox{m},
Line 322  are implemented on Asir and Kan.
Line 379  are implemented on Asir and Kan.
   
 \subsection{ Zero}  \subsection{ Zero}
 /*&jp  /*&jp
 CMO では ゼロの表現法がなんとおりもあるが,  CMO では ゼロの表現法がなんとおりもあることに注意.
 どのようなゼロをうけとっても,  %% どのようなゼロをうけとっても,
 システムのゼロに変換できるべきである.  %% システムのゼロに変換できるべきである.
 */  */
 /*&eg  /*&eg
 Though CMO has various representations of zero,  Note that CMO has various representations of zero.
 each representation should be translated into zero  
 in the system.  
 */  */
   
   
Line 348  GNU MPライブラリなどを参考にして設計されていて, 符号付
Line 403  GNU MPライブラリなどを参考にして設計されていて, 符号付
 plugin/cmo-gmp.c}) CMO\_ZZ は次の形式をとる.  plugin/cmo-gmp.c}) CMO\_ZZ は次の形式をとる.
 */  */
 /*&eg  /*&eg
 We describe the bignum (multi-precision integer) representation in OpenXM.  We describe the bignum (multi-precision integer) representation
 In OpenXM {\tt CMO\_ZZ} is used to represent bignum. Its design is similar  {\tt CMO\_ZZ} in OpenXM.
   The format is similar
 to that in GNU MP. (cf. {\tt plugin/cmo-gmp.c} in the {\tt kan/sm1}  to that in GNU MP. (cf. {\tt plugin/cmo-gmp.c} in the {\tt kan/sm1}
 distribution). CMO\_ZZ is defined as follows.  distribution). CMO\_ZZ is defined as follows.
 */  */
Line 371  Open xxx 規約では上の CMO は以下の整数を意味する. ($R 
Line 427  Open xxx 規約では上の CMO は以下の整数を意味する. ($R 
 /*&eg  /*&eg
 $f$ is a 32bit integer. $b_0, \ldots, b_n$ are unsigned 32bit integers.  $f$ is a 32bit integer. $b_0, \ldots, b_n$ are unsigned 32bit integers.
 $|f|$ is equal to $n+1$.  $|f|$ is equal to $n+1$.
 The sign of $f$ represents that of the above CMO. As stated in Section  The sign of $f$ represents that of the above integer to be expressed.
   As stated in Section
 \ref{sec:basic0}, a negative 32bit integer is represented by  \ref{sec:basic0}, a negative 32bit integer is represented by
 two's complement.  two's complement.
   
Line 383  In OpenXM the above CMO represents the following integ
Line 440  In OpenXM the above CMO represents the following integ
 \]  \]
   
 /*&jp  /*&jp
   \noindent  例:
 {\tt int32} を network byte order で表現  {\tt int32} を network byte order で表現
 しているとすると,例えば, 整数 $14$ は CMO\_ZZ で表わすと,  しているとすると,例えば, 整数 $14$ は CMO\_ZZ で表わすと,
 */  */
 /*&eg  /*&eg
   \noindent Example:
 If we express {\tt int32} by the network byte order,  If we express {\tt int32} by the network byte order,
 a CMO\_ZZ $14$ is expressed by  a CMO\_ZZ $14$ is expressed by
 */  */
Line 394  a CMO\_ZZ $14$ is expressed by
Line 453  a CMO\_ZZ $14$ is expressed by
 \mbox{(CMO\_ZZ, 1, 0, 0, 0, e)},  \mbox{(CMO\_ZZ, 1, 0, 0, 0, e)},
 \]  \]
 //&jp と表わす. これはバイト列では  //&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
 \[  \[
 \mbox{\tt 00 00 00 14 00 00 00 01 00 00 00 0e}  \mbox{\tt 00 00 00 14 00 00 00 01 00 00 00 0e}
 \]  \]
Line 442  Ring by Name を用いた場合, 現在の名前空間で変数 yyy に
Line 501  Ring by Name を用いた場合, 現在の名前空間で変数 yyy に
 /*&eg  /*&eg
 We treat polynomial rings and their elements as follows.  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 a coefficient set. $K$ is unknown in advance
 where $K$ is some coefficient set. $K$ is unknown in advance  and it is determined when coefficients of an element are received.
 and it is determined when coefficients are received.  When a server has received an element in Generic DMS ring,
 When a server has received an element in a generic DMS ring,  
 the server has to translate it into the corresponding local object  the server has to translate it into the corresponding local object
 on the server. Each server has its own translation scheme.  on the server. Each server has its own translation scheme.
   
 In Asir such an element are translated into a distributed polynomial.  In Asir such an element are translated into a distributed polynomial.
   
 In {\tt kan/sm1} things are complicated.  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  {\tt kan/sm1} translates a DMS of N variables into an element of
 the CurrentRing.  the CurrentRing.
 If the CurrentRing is $n'$-variate and $n' < n$, then  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.
 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,  is specified as the second field of DMS,
 it requests a sever to use a ring object whose name is yyy  it requests a sever to use a ring object whose name is yyy
 as the destination ring for the translation.  as the destination ring for the translation.
 This is done in {\tt kan/sm1}.  
 */  */
   
 \medbreak \noindent  \medbreak \noindent
Line 504  $3 x^2 y$ is regarded as an element of a six-variate p
Line 559  $3 x^2 y$ is regarded as an element of a six-variate p
 #define CMO_POLYNOMIAL_IN_ONE_VARIABLE  33  #define CMO_POLYNOMIAL_IN_ONE_VARIABLE  33
 \end{verbatim}  \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,  Polynomial in 1 variable, Coefficient, Name of the main variable,
 Recursive Polynomial, Ring definition for recursive polynomials  Recursive Polynomial, Ring definition for recursive polynomials
 $\in$ CMObject/RecursivePolynomial \\  $\in$ CMObject/RecursivePolynomial \\
Line 514  $\in$ CMObject/RecursivePolynomial \\
Line 569  $\in$ CMObject/RecursivePolynomial \\
 \mbox{Polynomial in 1 variable} &:&  \mbox{Polynomial in 1 variable} &:&
 \mbox{({\tt CMO\_POLYNOMIAL\_IN\_ONE\_VARIABLE},\, {\sl int32}\, m, } \\  \mbox{({\tt CMO\_POLYNOMIAL\_IN\_ONE\_VARIABLE},\, {\sl int32}\, m, } \\
 & & \quad \mbox{ Name of the main variable }, \\  & & \quad \mbox{ Name of the main variable }, \\
 & & \quad \mbox{ \{ {\sl int32} e, Coefficient \}} \\  & & \quad \mbox{ \{ {\sl int32} e, Coefficient \}} ) \\
 & & \mbox{ --- m はモノミアルの個数. } \\  & & \mbox{ --- m はモノミアルの個数. } \\
 & & \mbox{ --- e, Coefficieint はモノミアルを表現している. } \\  & & \mbox{ --- e, Coefficieint はモノミアルを表現している. } \\
 & & \mbox{ --- 順序の高い順にならべる. 普通は巾の高い順.} \\  & & \mbox{ --- 順序の高い順にならべる. 普通は巾の高い順.} \\
Line 530  $\in$ CMObject/RecursivePolynomial \\
Line 585  $\in$ CMObject/RecursivePolynomial \\
 \mbox{ ( {\tt CMO\_RECURSIVE\_POLYNOMIAL}, } \\  \mbox{ ( {\tt CMO\_RECURSIVE\_POLYNOMIAL}, } \\
 & & \quad \mbox{ RringDefinition, } \\  & & \quad \mbox{ RringDefinition, } \\
 & & \quad  & & \quad
 \mbox{ Polynomial in 1 variable}\, | \, \mbox{Coefficient}   \\  \mbox{ Polynomial in 1 variable}\, | \, \mbox{Coefficient} )  \\
 \mbox{RringDefinition}  \mbox{RringDefinition}
 & : &  \mbox{ {\sl List} v } \\  & : &  \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. } \\  & & \quad \mbox{ --- 順序の高い順. } \\
 \end{eqnarray*}  \end{eqnarray*}
 */  */
Line 542  $\in$ CMObject/RecursivePolynomial \\
Line 597  $\in$ CMObject/RecursivePolynomial \\
 \mbox{Polynomial in 1 variable} &:&  \mbox{Polynomial in 1 variable} &:&
 \mbox{({\tt CMO\_POLYNOMIAL\_IN\_ONE\_VARIABLE},\, {\sl int32}\, m, } \\  \mbox{({\tt CMO\_POLYNOMIAL\_IN\_ONE\_VARIABLE},\, {\sl int32}\, m, } \\
 & & \quad \mbox{ Name of the main variable }, \\  & & \quad \mbox{ Name of the main variable }, \\
 & & \quad \mbox{ \{ {\sl int32} e, Coefficient \}} \\  & & \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{ --- The pairs of e and Coefficient are sorted in the } \\
 & & \mbox{ \quad decreasing order, usually with respect to e.} \\  & & \mbox{ \quad decreasing order, usually with respect to e.} \\
 & & \mbox{ ---  e denotes an exponent of a monomial with respect to } \\  & & \mbox{ ---  e denotes an exponent of a monomial with respect to } \\
Line 560  $\in$ CMObject/RecursivePolynomial \\
Line 615  $\in$ CMObject/RecursivePolynomial \\
 \mbox{ ( {\tt CMO\_RECURSIVE\_POLYNOMIAL}, } \\  \mbox{ ( {\tt CMO\_RECURSIVE\_POLYNOMIAL}, } \\
 & & \quad \mbox{ RringDefinition, } \\  & & \quad \mbox{ RringDefinition, } \\
 & & \quad  & & \quad
 \mbox{ Polynomial in 1 variable}\, | \, \mbox{Coefficient}   \\  \mbox{ Polynomial in 1 variable}\, | \, \mbox{Coefficient} )  \\
 \mbox{RringDefinition}  \mbox{RringDefinition}
 & : &  \mbox{ {\sl List} v } \\  & : &  \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. } \\  & & \quad \mbox{ --- It is sorted in the decreasing order. } \\
 \end{eqnarray*}  \end{eqnarray*}
 */  */
Line 585  Example:
Line 640  Example:
 $$   x^3 (1234 y^5 + 17 ) +  x^1 (y^{10} + 31 y^5)  $$  $$   x^3 (1234 y^5 + 17 ) +  x^1 (y^{10} + 31 y^5)  $$
 /*&jp  /*&jp
 をあらわす.  をあらわす.
 非可換多項式もこの形式であらわしたいので, 積の順序を上のように  %%非可換多項式もこの形式であらわしたいので, 積の順序を上のように
 すること. つまり, 主変数かける係数の順番.  %%すること. つまり, 主変数かける係数の順番.
 */  */
 /*&eg  /*&eg
 We intend to represent non-commutative polynomials with the  %%We intend to represent non-commutative polynomials with the
 same form. In such a case, the order of products are defined  %%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  \noindent
Line 613  Class.recursivePolynomial h * ((-1)) + (x^2  * (1))
Line 669  Class.recursivePolynomial h * ((-1)) + (x^2  * (1))
 \end{verbatim}  \end{verbatim}
   
 \noindent  \noindent
 Group CMObject/MachineDouble requires CMObject/Basic0.\\  Group CMObject/MachineDouble requires CMObject/Primitive.\\
 64bit machine double, Array of 64bit machine double  64bit machine double, Array of 64bit machine double
 128bit machine double, Array of 128bit machine double  128bit machine double, Array of 128bit machine double
 $\in$ CMObject/MachineDouble \\  $\in$ CMObject/MachineDouble \\
Line 622  $\in$ CMObject/MachineDouble \\
Line 678  $\in$ CMObject/MachineDouble \\
 \begin{eqnarray*}  \begin{eqnarray*}
 \mbox{64bit machine double} &:&  \mbox{64bit machine double} &:&
 \mbox{({\tt CMO\_64BIT\_MACHINE\_DOUBLE}, } \\  \mbox{({\tt CMO\_64BIT\_MACHINE\_DOUBLE}, } \\
 & & \quad \mbox{ {\sl byte} s1 , \ldots , {\sl byte}} s8)\\  & & \quad \mbox{ {\sl byte} s1 , \ldots , {\sl byte} s8})\\
 & & \mbox{ --- s1, $\ldots$, s8 は {\tt double} (64bit). } \\  & & \mbox{ --- s1, $\ldots$, s8 は {\tt double} (64bit). } \\
 & & \mbox{ --- この表現はCPU依存である.}\\  & & \mbox{ --- この表現はCPU依存である.}\\
 &&  \mbox{\quad\quad mathcap に CPU 情報を付加しておく.} \\  &&  \mbox{\quad\quad mathcap に CPU 情報を付加しておく.} \\
Line 634  $\in$ CMObject/MachineDouble \\
Line 690  $\in$ CMObject/MachineDouble \\
 & & \mbox{ \quad\quad mathcap に CPU 情報を付加しておく.} \\  & & \mbox{ \quad\quad mathcap に CPU 情報を付加しておく.} \\
 \mbox{128bit machine double} &:&  \mbox{128bit machine double} &:&
 \mbox{({\tt CMO\_128BIT\_MACHINE\_DOUBLE}, } \\  \mbox{({\tt CMO\_128BIT\_MACHINE\_DOUBLE}, } \\
 & & \quad \mbox{ {\sl byte} s1 , \ldots , {\sl byte}} s16)\\  & & \quad \mbox{ {\sl byte} s1 , \ldots , {\sl byte} s16})\\
 & & \mbox{ --- s1, $\ldots$, s16 は {\tt long double} (128bit). } \\  & & \mbox{ --- s1, $\ldots$, s16 は {\tt long double} (128bit). } \\
 & & \mbox{ --- この表現はCPU依存である.}\\  & & \mbox{ --- この表現はCPU依存である.}\\
 &&  \mbox{\quad\quad mathcap に CPU 情報を付加しておく.} \\  &&  \mbox{\quad\quad mathcap に CPU 情報を付加しておく.} \\
 \mbox{Array of 128bit machine double} &:&  \mbox{Array of 128bit machine double} &:&
 \mbox{({\tt CMO\_ARRAY\_OF\_128BIT\_MACHINE\_DOUBLE}, {\sl int32} m, } \\  \mbox{({\tt CMO\_ARRAY\_OF\_128BIT\_MACHINE\_DOUBLE}, {\sl int32} m, } \\
 & & \quad \mbox{ {\sl byte} s1[1] , \ldots , {\sl byte}} s16[1], \ldots , {\sl byte} s16[m])\\  & & \quad \mbox{ {\sl byte} s1[1] , \ldots , {\sl byte} s16[1], \ldots , {\sl byte} s16[m]})\\
 & & \mbox{ --- s*[1], $\ldots$ s*[m] は m 個の long double (128bit) である. } \\  & & \mbox{ --- s*[1], $\ldots$ s*[m] は m 個の long double (128bit) である. } \\
 & & \mbox{ --- この表現はCPU依存である.}\\  & & \mbox{ --- この表現はCPU依存である.}\\
 & & \mbox{ \quad\quad mathcap に CPU 情報を付加しておく.}  & & \mbox{ \quad\quad mathcap に CPU 情報を付加しておく.}
Line 650  $\in$ CMObject/MachineDouble \\
Line 706  $\in$ CMObject/MachineDouble \\
 \begin{eqnarray*}  \begin{eqnarray*}
 \mbox{64bit machine double} &:&  \mbox{64bit machine double} &:&
 \mbox{({\tt CMO\_64BIT\_MACHINE\_DOUBLE}, } \\  \mbox{({\tt CMO\_64BIT\_MACHINE\_DOUBLE}, } \\
 & & \quad \mbox{ {\sl byte} s1 , \ldots , {\sl byte}} s8)\\  & & \quad \mbox{ {\sl byte} s1 , \ldots , {\sl byte} s8})\\
 & & \mbox{ --- s1, $\ldots$, s8 は {\tt double} (64bit). } \\  & & \mbox{ --- s1, $\ldots$, s8 は {\tt double} (64bit). } \\
 & & \mbox{ --- This depends on CPU.}\\  & & \mbox{ --- Encoding depends on CPU.}\\
 &&  \mbox{\quad\quad Add informations on CPU to the mathcap.} \\  &&  \mbox{\quad\quad Add informations on CPU to the mathcap.} \\
 \mbox{Array of 64bit machine double} &:&  \mbox{Array of 64bit machine double} &:&
 \mbox{({\tt CMO\_ARRAY\_OF\_64BIT\_MACHINE\_DOUBLE}, {\sl int32} m, } \\  \mbox{({\tt CMO\_ARRAY\_OF\_64BIT\_MACHINE\_DOUBLE}, {\sl int32} m, } \\
 & & \quad \mbox{ {\sl byte} s1[1] , \ldots , {\sl byte}}\, s8[1], \ldots , {\sl byte}\, s8[m])\\  & & \quad \mbox{ {\sl byte} s1[1] , \ldots , {\sl byte}}\, s8[1], \ldots , {\sl byte}\, s8[m])\\
 & & \mbox{ --- s*[1], $\ldots$ s*[m] are 64bit double's. } \\  & & \mbox{ --- s*[1], $\ldots$ s*[m] are 64bit double's. } \\
 & & \mbox{ --- This depends on CPU.}\\  & & \mbox{ --- Encoding depends on CPU.}\\
 & & \mbox{\quad\quad Add informations on CPU to the mathcap.} \\  & & \mbox{\quad\quad Add informations on CPU to the mathcap.} \\
 \mbox{128bit machine double} &:&  \mbox{128bit machine double} &:&
 \mbox{({\tt CMO\_128BIT\_MACHINE\_DOUBLE}, } \\  \mbox{({\tt CMO\_128BIT\_MACHINE\_DOUBLE}, } \\
 & & \quad \mbox{ {\sl byte} s1 , \ldots , {\sl byte}} s16)\\  & & \quad \mbox{ {\sl byte} s1 , \ldots , {\sl byte} s16})\\
 & & \mbox{ --- s1, $\ldots$, s16 は {\tt long double} (128bit). } \\  & & \mbox{ --- s1, $\ldots$, s16 は {\tt long double} (128bit). } \\
 & & \mbox{ --- This depends on CPU.}\\  & & \mbox{ --- Encoding depends on CPU.}\\
 & & \mbox{\quad\quad Add informations on CPU to the mathcap.} \\  & & \mbox{\quad\quad Add informations on CPU to the mathcap.} \\
 \mbox{Array of 128bit machine double} &:&  \mbox{Array of 128bit machine double} &:&
 \mbox{({\tt CMO\_ARRAY\_OF\_128BIT\_MACHINE\_DOUBLE}, {\sl int32} m, } \\  \mbox{({\tt CMO\_ARRAY\_OF\_128BIT\_MACHINE\_DOUBLE}, {\sl int32} m, } \\
 & & \quad \mbox{ {\sl byte} s1[1] , \ldots , {\sl byte}} s16[1], \ldots , {\sl byte} s16[m])\\  & & \quad \mbox{ {\sl byte} s1[1] , \ldots , {\sl byte} s16[1], \ldots , {\sl byte} s16[m]})\\
 & & \mbox{ --- s*[1], $\ldots$ s*[m] are 128bit long double's. } \\  & & \mbox{ --- s*[1], $\ldots$ s*[m] are 128bit long double's. } \\
 & & \mbox{ --- This depends on CPU.}\\  & & \mbox{ --- Encoding depends on CPU.}\\
 & & \mbox{\quad\quad Add informations on CPU to the mathcap.} \\  & & \mbox{\quad\quad Add informations on CPU to the mathcap.} \\
 \end{eqnarray*}  \end{eqnarray*}
 */  */
   
 \bigbreak  \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.  
 \begin{verbatim}  \begin{verbatim}
 #define CMO_BIGFLOAT   50  #define CMO_BIGFLOAT   50
 #define CMO_IEEE_DOUBLE_FLOAT 51  #define CMO_IEEE_DOUBLE_FLOAT 51
Line 688  IEEE 準拠の float については, IEEE 754 double precisio
Line 743  IEEE 準拠の float については, IEEE 754 double precisio
 format (64 bit) の定義を見よ.  format (64 bit) の定義を見よ.
 */  */
 /*&eg  /*&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 compliant to the IEEE standard.
 */  */
   
 \noindent  \noindent
 Group CMObject/Bigfloat requires CMObject/Basic0, CMObject/Basic1.\\  Group CMObject/Bigfloat requires CMObject/Primitive, CMObject/Basic.\\
 Bigfloat  Bigfloat
 $\in$ CMObject/Bigfloat \\  $\in$ CMObject/Bigfloat \\
   
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