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version 1.4, 2000/01/24 02:48:24 version 1.11, 2001/08/27 05:39:15
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 %% $OpenXM: OpenXM/doc/OpenXM-specs/cmo-basic1.tex,v 1.3 2000/01/23 05:28:33 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 }  //&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}  \label{sec:basic1}
Line 27 
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 以下, グループ CMObject/Basic, 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/Basic, 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
Line 46  ZZ, QQ, Zero, Rational, Indeterminate,$\in$ CMObject/B
Line 64  ZZ, QQ, Zero, Rational, Indeterminate,$\in$ CMObject/B
 \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 66  ZZ, QQ, Zero, Rational, Indeterminate,$\in$ CMObject/B
Line 86  ZZ, QQ, Zero, Rational, Indeterminate,$\in$ CMObject/B
 \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 81  ZZ, QQ, Zero, Rational, Indeterminate,$\in$ CMObject/B
Line 103  ZZ, QQ, Zero, Rational, Indeterminate,$\in$ CMObject/B
 */  */
   
 /*&jp  /*&jp
   \subsection{Indeterminate および Tree}
 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,
 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.  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.)  テーブルを作成する必要があるであろう.)
 */  */
 /*&eg  /*&eg
   \subsection{Indetermnate and Tree}
 Indeterminate is a name of a variable.  Indeterminate is a name of a variable.
 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.
Line 106  Group CMObject/Tree requires CMObject/Basic. \\
Line 130  Group CMObject/Tree requires CMObject/Basic. \\
 Tree, Lambda $\in$ CMObject/Basic. \\  Tree, Lambda $\in$ CMObject/Basic. \\
 \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. } \\
Line 123  Group CMObject/Tree requires CMObject/Basic. \\
Line 147  Group CMObject/Tree requires CMObject/Basic. \\
 Tree, Lambda $\in$ CMObject/Basic. \\  Tree, Lambda $\in$ CMObject/Basic. \\
 \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{ --- 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},  \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. } \\
Line 167  stack machine commands for delayed evaluation.
Line 191  stack machine commands for delayed evaluation.
 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 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.  //&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_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) (Basic) [(123).. (345)..]] [(class) (tree)] dc ::
 Class.tree [    $plus$ , $Basic$ , [    123 , 345 ]  ]  Class.tree [    $plus$ , $basic$ , [    123 , 345 ]  ]
 \end{verbatim}  \end{verbatim}
   
   
Line 533  $\in$ CMObject/RecursivePolynomial \\
Line 561  $\in$ CMObject/RecursivePolynomial \\
 \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 563  $\in$ CMObject/RecursivePolynomial \\
Line 591  $\in$ CMObject/RecursivePolynomial \\
 \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 677  $\in$ CMObject/MachineDouble \\
Line 705  $\in$ CMObject/MachineDouble \\
   
 \bigbreak  \bigbreak
 //&jp 次に IEEE 準拠の float および Big float を定義しよう.  //&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}  \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 716  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 conforming to the IEEE standard.
 */  */
   
 \noindent  \noindent
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