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version 1.2, 2000/01/23 00:41:08

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

Line 134 Tree, Lambda $\in$ CMObject/Basic1. \\

\end{eqnarray*}

/*&C

/*&jp

数式を処理するシステムでは, Tree 構造が一般にもちいられる.

たとえば, $\sin(x+e)$ は,

Line 164 For example,

Line 160 For example,

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

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

as a tree.

We can @@@

Tree may be expressed by putting itself 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.

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 174

Line 180

/*&jp

Lambda は関数を定義するための関数である.

Lisp の Lambda 表現と同じ.

/*&eg

Lambda is used to define functions.

It is the same as the Lambda expression in Lisp.

\noindent

例: $sin(x+e)$ の表現.

//&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_LIST,[size=]1,

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

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

(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

(CMO_STRING, "basic"))

))

)

Line 199 Class.tree [ $plus$ , $Basic$ , [ 123 , 345 ] ]

Line 212 Class.tree [ $plus$ , $Basic$ , [ 123 , 345 ] ]

\bigbreak

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.

\medbreak

\noindent

Line 208 CMObject/Basic1. \\

Line 222 CMObject/Basic1. \\

Monomial, Monomial32, Coefficient, Dpolynomial, DringDefinition,

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

CMObject/DistributedPolynomials. \\

/*&jp

\begin{eqnarray*}

\mbox{Monomial} &:& \mbox{Monomial32}\, |\, \mbox{Zero} \\

\mbox{Monomial32}&:& ({\tt CMO\_MONOMIAL32}, {\sl int32}\, n,

Line 249 $x^e = x_1^{e_1} \cdots x_n^{e_n}$ の各指数 $e_i$

Line 264 $x^e = x_1^{e_1} \cdots x_n^{e_n}$ の各指数 $e_i$

& & \mbox{ --- wvec は order をきめる weight vector,} \\

& & \mbox{ --- outord は出力するときの変数順序.} \\

\end{eqnarray*}

/*&eg

\begin{eqnarray*}

\mbox{Monomial} &:& \mbox{Monomial32}\, |\, \mbox{Zero} \\

\mbox{Monomial32}&:& ({\tt CMO\_MONOMIAL32}, {\sl int32}\, n,

{\sl int32}\, \mbox{e[1]}, \ldots,

{\sl int32}\, \mbox{e[n]}, \\

& & \ \mbox{Coefficient}) \\

& & \mbox{ --- e[i] is the exponent $e_i$ of the monomial

$x^e = x_1^{e_1} \cdots x_n^{e_n}$. } \\

\mbox{Coefficient}&:& \mbox{ZZ} | \mbox{Integer32} \\

\mbox{Dpolynomial}&:& \mbox{Zero} \\

& & |\ ({\tt CMO\_DISTRIBUTED\_POLYNOMIAL},{\sl int32} m, \\

& & \ \ \mbox{DringDefinition}, [\mbox{Monomial32}|\mbox{Zero}], \\

& &\ \

\{\mbox{Monomial32}\}) \\

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

\mbox{DringDefinition}

&:& \mbox{DMS of N variables} \\

& & |\ \mbox{RingByName} \\

& & |\ \mbox{Generic DMS ring} \\

& & \mbox{ --- definition of the ring of distributed polynomials. } \\

\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{DMS of N variables}

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

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

{\sl Integer32}\, \mbox{n}, {\sl Integer32}\, \mbox{p} \\

& & \ \ [,{\sl Cstring}\,\mbox{s}, {\sl List}\, \mbox{vlist},

{\sl List}\, \mbox{wvec}, {\sl List}\, \mbox{outord}]) \\

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

& & \mbox{ --- n is the number of variables, p is the characteristic} \\

& & \mbox{ --- s is the name of the ring, vlist is the list of variables.} \\

& & \mbox{ --- wvec is the weight vector.} \\

& & \mbox{ --- outord is the order of variables to output.} \\

\end{eqnarray*}

/*&jp

RingByName や DMS of N variables はなくても, DMS を定義できる.

したがって, これらを実装してないシステムで DMS を扱うものが

あってもかまわない.

以下, 以上の CMObject にたいする,

xxx = asir, kan の振舞いを記述する.

/*&eg

Note that it is possible to define DMS without RingByName and

DMS of N variables.

In the following we describe how the above CMObjects

are implemented on Asir and Kan.

\subsection{ Zero}

CMO ではゼロの表現法がなんとうりもあるが,

/*&jp

CMO ではゼロの表現法がなんとおりもあるが,

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.

(たとえば, asir は 0 はただ一つ.)

/*&eg

Though CMO has various representations of zero,

each representation should be translated into zero

in the system.

//&jp \subsection{ 整数 ZZ }

//&eg \subsection{ Integer ZZ }

\subsection{ 整数 ZZ }

\begin{verbatim}

#define CMO_ZZ 20

\end{verbatim}

/*&jp

この節ではOpen xxx 規約における任意の大きさの整数(bignum)の扱いについ

て説明する. Open XM 規約における多重精度整数を表すデータ型 CMO\_ZZ は

GNU MPライブラリなどを参考にして設計されていて, 符号付き絶対値表現を用

いている. (cf. {\tt kan/sm1} の配布ディレクトリのなかの {\tt

plugin/cmo-gmp.c}) CMO\_ZZ は次の形式をとる.

/*&eg

We describe the bignum (multi-precision integer) representation in OpenXM.

In OpenXM {\tt CMO\_ZZ} is used to represent bignum. Its design is similar

to that in GNU MP. (cf. {\tt plugin/cmo-gmp.c} in the {\tt kan/sm1}

distribution). CMO\_ZZ is defined as follows.

この節ではOpen xxx 規約における任意の大きさの整数(bignum)の扱いについて

説明する.

Open XM 規約における多重精度整数を表すデータ型 CMO\_ZZ は GNU MPライブ

ラリなどを参考にして設計されていて, 符号付き絶対値表現を用いている.

(cf. {\tt kan/sm1} の配布ディレクトリのなかの {\tt plugin/cmo-gmp.c})

CMO\_ZZ は次の形式をとる.\\

\begin{tabular}{|c|c|c|c|c|}

\hline

{\tt int32 CMO\_ZZ} & {\tt int32 $f$} & {\tt int32 $b_0$} & $\cdots$ &

{\tt int32 $b_{n}$} \\

\hline

\end{tabular} \\

\end{tabular}

$f$ は32bit整数である.

$b_0, \ldots, b_n$ は unsigned int32 である.

$|f|$ は $n+1$ である.

この CMO の符号は $f$ の符号で定める.

前述したように, 32bit整数の負数は 2 の補数表現で表される.

Open xxx 規約では上の CMO は以下の整数を意味する.

/*&jp

$f$ は32bit整数である. $b_0, \ldots, b_n$ は unsigned int32 である.

$|f|$ は $n+1$ である. この CMO の符号は $f$ の符号で定める. 前述し

たように, 32bit整数の負数は 2 の補数表現で表される.

Open xxx 規約では上の CMO は以下の整数を意味する. ($R = 2^{32}$)

/*&eg

$f$ is a 32bit integer. $b_0, \ldots, b_n$ are unsigned 32bit integers.

$|f|$ is equal to $n+1$.

The sign of $f$ represents that of the above CMO. As stated in Section

\ref{sec:basic0}, a negative 32bit integer is represented by

two's complement.

In OpenXM the above CMO represents the following integer. ($R = 2^{32}$.)

\mbox{sgn}(f)\times (b_0 R^{0}+ b_1 R^{1} + \cdots + b_{n-1}R^{n-1} + b_n R^n).

ここで $R = 2^{32}$ である.

{\tt int32} を network byte order で表現しているとすると,

/*&jp

例えば, 整数 $14$ は CMO\_ZZ で表わすと,

{\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,

/*&eg

If we express {\tt int32} by the network byte order,

a CMO\_ZZ $14$ is expressed by

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

と表わす.

//&jp と表わす. これはバイト列では

これはバイト列では

//&egThe corresponding byte sequence is

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

となる.

//&jp となる.

なお ZZ の 0 ( (ZZ) 0 と書く ) は,

//&jp なお ZZ の 0 ( (ZZ) 0 と書く ) は, {\tt (CMO\_ZZ, 00,00,00,00)} と表現する.

{\tt (CMO\_ZZ, 00,00,00,00)}

//&eg Note that CMO\_ZZ 0 is expressed by {\tt (CMO\_ZZ, 00,00,00,00)}.

と表現する.

\subsection{ 分散表現多項式 Dpolynomial }

//&jp \subsection{ 分散表現多項式 Dpolynomial }

//&eg \subsection{ Distributed polynomial Dpolynomial }

/*&jp

環とそれに属する多項式は次のような考えかたであつかう.

Generic DMS ring に属する元は,

Line 343 Ring by Name を用いた場合, 現在の名前空間で変数 yyy に

Line 437 Ring by Name を用いた場合, 現在の名前空間で変数 yyy に

{\tt kan/sm1} の場合, 環の定義は ring object として格納されており,

この ring object を変数 yyy で参照することにより CMO としてうけとった

多項式をこの ring の元として格納できる.

/*&eg

We treat polynomial rings and their elements as follows.

An element of a generic DMS ring is an element of

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.

When a server has received an element in a 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} 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

the term order are all ignored.

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.

This is done in {\tt kan/sm1}.

\medbreak \noindent

{\bf Example}:

//&jp {\bf Example}: (すべての数の表記は 16 進表記)

(すべての数の表記は 16 進表記)

//&eg {\bf Example}: (all numbers are represented in hexadecimal notation)

{\footnotesize \begin{verbatim}

Z/11Z [6 variables]

(kxx/cmotest.sm1) run

Line 369 ff omc ::

Line 488 ff omc ::

(CMO_DISTRIBUTED_POLYNOMIAL[1f],[size=]1,(CMO_DMS_GENERIC[18],),

(CMO_MONOMIAL32[13],3*x^2*y),),

\end{verbatim} }

length は, monomial の数$+2$ である.

/*&jp

$ 3 x^2 y$ は 6 変数の多項式環の元としてみなされている.

%%Prog: (3x^2 y). cmosave ===> debug/cmodata1.cmo

%%\\ 反省: 分散多項式の定義で,

/*&eg

%%{\tt CMO\_LIST} でなく, {\tt CMO\_DMS} がはじめにくるべきだったのでは?

$3 x^2 y$ is regarded as an element of a six-variate polynomial ring.

%%あたらしい分散多項式の定義は次のようにすべき:

%% 修正済み. 1999, 9/13

//&jp \subsection{再帰表現多項式の定義}

//&eg \subsection{Recursive polynomials}

\subsection{再帰表現多項式の定義}

\begin{verbatim}

#define CMO_RECURSIVE_POLYNOMIAL 27

#define CMO_POLYNOMIAL_IN_ONE_VARIABLE 33

Line 391 Polynomial in 1 variable, Coefficient, Name of the mai

Line 509 Polynomial in 1 variable, Coefficient, Name of the mai

Recursive Polynomial, Ring definition for recursive polynomials

$\in$ CMObject/RecursivePolynomial \\

/*&jp

\begin{eqnarray*}

\mbox{Polynomial in 1 variable} &:&

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

Line 409 $\in$ CMObject/RecursivePolynomial \\

Line 528 $\in$ CMObject/RecursivePolynomial \\

& & \mbox{ --- v は変数番号 (0 からはじまる) を表す. } \\

\mbox{Recursive Polynomial} &:&

\mbox{ ( {\tt CMO\_RECURSIVE\_POLYNOMIAL}, } \\

& & \quad \mbox{ Ring definition for

& & \quad \mbox{ RringDefinition, } \\

recursive polynomials, } \\

& & \quad

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

\mbox{Ring definition for recursive polynomials }

\mbox{RringDefinition}

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

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

& & \quad \mbox{ --- 順序の高い順. } \\

\end{eqnarray*}

/*&eg

\begin{eqnarray*}

\mbox{Polynomial in 1 variable} &:&

\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{ --- A pair of e and Coefficieint 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 } \\

& & \mbox{ \quad the main variable. } \\

\mbox{Coefficient} &:& \mbox{ ZZ} \,|\, \mbox{ QQ } \,|\,

\mbox{ integer32 } \,|\,

\mbox{ Polynomial in 1 variable } \\

& & \quad \,|\, \mbox{Tree} \,|\, \mbox{Zero} \,|\,\mbox{Dpolynomial}\\

\mbox{Name of the main variable } &:&

\mbox{ {\sl int32} v } \\

& & \mbox{ --- v denotes a variable number. } \\

\mbox{Recursive Polynomial} &:&

\mbox{ ( {\tt CMO\_RECURSIVE\_POLYNOMIAL}, } \\

& & \quad \mbox{ RringDefinition, } \\

& & \quad

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

\end{eqnarray*}

\bigbreak

\noindent

実例:

Example:

\begin{verbatim}

(CMO_RECURSIEVE_POLYNOMIAL, ("x","y"),

(CMO_POLYNOMIAL_IN_ONE_VARIABLE, 2, 0, <--- "x"

Line 432 recursive polynomials, } \\

Line 580 recursive polynomials, } \\

10, 1,

5, 31)))

\end{verbatim}

これは,

//&jp これは,

$$ x^3 (1234 y^5 + 17 ) + x^1 (y^10 + 31 y^5) $$

//&eg This represents

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

/*&jp

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

すること. つまり, 主変数かける係数の順番.

/*&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.

\noindent

\begin{verbatim}

Line 446 sm1>ff ::

Line 602 sm1>ff ::

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

\end{verbatim}

//&jp \subsection{CPU依存の double }

//&eg \subsection{CPU dependent double}

int32 と Integer32 の違い.

次にくるデータがかならず int32 とわかっておれば,

int32 を用いる.

次のデータ型がわからないとき Integer32 を用いる.

\subsection{CPU依存の double }

\begin{verbatim}

#define CMO_64BIT_MACHINE_DOUBLE 40

#define CMO_ARRAY_OF_64BIT_MACHINE_DOUBLE 41

Line 469 Group CMObject/MachineDouble requires CMObject/Basic0.

Line 618 Group CMObject/MachineDouble requires CMObject/Basic0.

128bit machine double, Array of 128bit machine double

$\in$ CMObject/MachineDouble \\

/*&jp

\begin{eqnarray*}

\mbox{64bit machine double} &:&

\mbox{({\tt CMO\_64BIT\_MACHINE\_DOUBLE}, } \\

Line 495 $\in$ CMObject/MachineDouble \\

Line 645 $\in$ CMObject/MachineDouble \\

& & \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.}

\end{eqnarray*}

/*&eg

\begin{eqnarray*}

\mbox{64bit machine double} &:&

\mbox{({\tt CMO\_64BIT\_MACHINE\_DOUBLE}, } \\

& & \quad \mbox{ {\sl byte} s1 , \ldots , {\sl byte}} s8)\\

& & \mbox{ --- s1, $\ldots$, s8 は {\tt double} (64bit). } \\

& & \mbox{ --- This depends on CPU.}\\

&& \mbox{\quad\quad Add informations on CPU to the mathcap.} \\

\mbox{Array of 64bit machine double} &:&

\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])\\

& & \mbox{ --- s*[1], $\ldots$ s*[m] are 64bit double's. } \\

& & \mbox{ --- This depends on CPU.}\\

& & \mbox{\quad\quad Add informations on CPU to the mathcap.} \\

\mbox{128bit machine double} &:&

\mbox{({\tt CMO\_128BIT\_MACHINE\_DOUBLE}, } \\

& & \quad \mbox{ {\sl byte} s1 , \ldots , {\sl byte}} s16)\\

& & \mbox{ --- s1, $\ldots$, s16 は {\tt long double} (128bit). } \\

& & \mbox{ --- This depends on CPU.}\\

& & \mbox{\quad\quad Add informations on CPU to the mathcap.} \\

\mbox{Array of 128bit machine double} &:&

\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])\\

& & \mbox{ --- s*[1], $\ldots$ s*[m] are 128bit long double's. } \\

& & \mbox{ --- This depends on CPU.}\\

& & \mbox{\quad\quad Add informations on CPU to the mathcap.} \\

\end{eqnarray*}

\bigbreak

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.

\begin{verbatim}

#define CMO_BIGFLOAT 50

#define CMO_IEEE_DOUBLE_FLOAT 51

\end{verbatim}

IEEE 準拠の float については,

/*&jp

IEEE 754 double precision floating-point format

IEEE 準拠の float については, IEEE 754 double precision floating-point

(64 bit) の定義を見よ.

format (64 bit) の定義を見よ.

/*&eg

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

conformant float.

\noindent

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

Line 516 $\in$ CMObject/Bigfloat \\

Line 701 $\in$ CMObject/Bigfloat \\

\mbox{Bigfloat} &:&

\mbox{({\tt CMO\_BIGFLOAT}, } \\

& & \quad \mbox{ {\sl ZZ} a , {\sl ZZ} e})\\

& & \mbox{ --- $a \times 2^e$ をあらわす. } \\

& & \mbox{ --- $a \times 2^e$. } \\

\end{eqnarray*}

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