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1.6     ! noro        1: % $OpenXM: OpenXM/doc/Papers/dag-noro.tex,v 1.5 2001/10/12 05:11:36 noro Exp $
1.3       noro        2: \documentclass{slides}
                      3: \usepackage{color}
                      4: \usepackage{rgb}
                      5: \usepackage{graphicx}
                      6: \usepackage{epsfig}
1.1       noro        7: \newcommand{\qed}{$\Box$}
                      8: \newcommand{\mred}[1]{\smash{\mathop{\hbox{\rightarrowfill}}\limits_{\scriptstyle #1}}}
                      9: \newcommand{\tmred}[1]{\smash{\mathop{\hbox{\rightarrowfill}}\limits_{\scriptstyle #1}\limits^{\scriptstyle *}}}
                     10: \def\gr{Gr\"obner basis }
                     11: \def\st{\, s.t. \,}
                     12: \def\ni{\noindent}
                     13: \def\ve{\vfill\eject}
                     14: \textwidth 9.2in
                     15: \textheight 7.2in
                     16: \columnsep 0.33in
                     17: \topmargin -1in
1.4       noro       18: \def\tc{\color{red}}
                     19: \def\fbc{\bf\color{MediumBlue}}
                     20: \def\itc{\color{brown}}
                     21: \def\urlc{\bf\color{DarkGreen}}
                     22: \def\bc{\color{LightGoldenrod1}}
1.1       noro       23:
1.4       noro       24: \title{\tc A computer algebra system Risa/Asir and OpenXM}
1.1       noro       25:
1.3       noro       26: \author{Masayuki Noro\\ Kobe University, Japan}
1.1       noro       27: \begin{document}
                     28: \setlength{\parskip}{10pt}
                     29: \maketitle
1.3       noro       30:
                     31: %\begin{slide}{}
                     32: %\begin{center}
1.4       noro       33: %\fbox{\fbc \large Part I : OpenXM and Risa/Asir --- overview and history}
1.3       noro       34: %\end{center}
                     35: %\end{slide}
                     36:
                     37: %\begin{slide}{}
1.4       noro       38: %\fbox{\fbc Integration of mathematical software systems}
1.3       noro       39: %
                     40: %\begin{itemize}
                     41: %\item Data integration
                     42: %
                     43: %\begin{itemize}
1.4       noro       44: %\item OpenMath ({\urlc \tt http://www.openmath.org}) , MP [GRAY98]
1.3       noro       45: %\end{itemize}
                     46: %
                     47: %Standards for representing mathematical objects
                     48: %
                     49: %\item Control integration
                     50: %
                     51: %\begin{itemize}
                     52: %\item MCP [WANG99], OMEI [LIAO01]
                     53: %\end{itemize}
                     54: %
                     55: %Protocols for remote subroutine calls or session management
                     56: %
                     57: %\item Combination of two integrations
                     58: %
                     59: %\begin{itemize}
                     60: %\item MathLink, OpenMath+MCP, MP+MCP
                     61: %
1.4       noro       62: %and OpenXM ({\urlc \tt http://www.openxm.org})
1.3       noro       63: %\end{itemize}
                     64: %
                     65: %Both are necessary for practical implementation
                     66: %
                     67: %\end{itemize}
                     68: %\end{slide}
                     69: \begin{slide}{}
1.4       noro       70: \fbox{\fbc A computer algebra system Risa/Asir}
1.3       noro       71:
                     72: \begin{itemize}
1.4       noro       73: \item {\itc Software mainly for polynomial computation}
1.3       noro       74:
1.5       noro       75: Polynomial factorization, Groebner basis computation
1.4       noro       76: \item {\itc User language with C-like syntax}
1.3       noro       77:
                     78: C language without type declaration, with list processing
                     79:
1.4       noro       80: \item {\itc Builtin {\tt gdb}-like debugger for user programs}
1.3       noro       81:
1.5       noro       82: \item {\itc Open source} ({\urlc \tt http://www.math.kobe-u.ac.jp/Asir/asir.html})
1.3       noro       83:
1.5       noro       84: The source and binaries are available via ftp or CVS
1.3       noro       85:
1.5       noro       86: See {\urlc \tt http://www.openxm.org} to get the latest version
1.3       noro       87:
1.4       noro       88: \item {\itc OpenXM interface}
1.3       noro       89:
                     90: \begin{itemize}
1.5       noro       91: \item OpenXM ({\urlc \tt http://www.openxm.org})
1.3       noro       92:
                     93: An infrastructure for exchanging mathematical data
1.5       noro       94: \item Risa/Asir is a main client in OpenXM package
                     95: \item {\tt ox\_asir} is an OpenXM server
                     96: \item {\tt libasir.a} provides OpenXM interface via function call
1.3       noro       97: \end{itemize}
                     98: \end{itemize}
                     99: \end{slide}
                    100:
                    101: \begin{slide}{}
1.4       noro      102: \fbox{\fbc Goal of developing Risa/Asir}
1.3       noro      103:
                    104: \begin{itemize}
1.4       noro      105: \item {\itc Testing new algorithms}
1.3       noro      106:
                    107: \begin{itemize}
                    108: \item Development started in Fujitsu labs
                    109:
                    110: Polynomial factorization, Groebner basis related computation,
                    111: cryptosystems , quantifier elimination , $\ldots$
                    112: \end{itemize}
                    113:
1.4       noro      114: \item {\itc To be a general purpose, open system}
1.3       noro      115:
                    116: Since 1997, we have been developing OpenXM package
                    117: containing various servers and clients
                    118:
                    119: Risa/Asir is a component of OpenXM
                    120:
1.4       noro      121: \item {\itc Environment for parallel and distributed computation}
1.3       noro      122:
                    123: \end{itemize}
                    124: \end{slide}
                    125:
                    126: %\begin{slide}{}
1.4       noro      127: %\fbox{\fbc Capability for polynomial computation}
1.3       noro      128: %
                    129: %\begin{itemize}
                    130: %\item Fundamental polynomial arithmetics
                    131: %
                    132: %recursive representation and distributed representation
                    133: %
                    134: %\item Polynomial factorization
                    135: %
                    136: %\begin{itemize}
                    137: %\item Univariate : over {\bf Q}, algebraic number fields and finite fields
                    138: %
                    139: %\item Multivariate : over {\bf Q}
                    140: %\end{itemize}
                    141: %
                    142: %\item Groebner basis computation
                    143: %
                    144: %\begin{itemize}
                    145: %\item Buchberger and $F_4$ [FAUG99] algorithm
                    146: %
                    147: %\item Change of ordering/RUR [ROUI96] of 0-dimensional ideals
                    148: %
                    149: %\item Primary ideal decomposition
                    150: %
                    151: %\item Computation of $b$-function (in Weyl Algebra)
                    152: %\end{itemize}
                    153: %\end{itemize}
                    154: %\end{slide}
                    155:
                    156: \begin{slide}{}
1.4       noro      157: \fbox{\fbc History of development : Polynomial factorization}
1.3       noro      158:
                    159: \begin{itemize}
1.4       noro      160: \item {\itc 1989}
1.3       noro      161:
                    162: Start of Risa/Asir with Boehm's conservative GC
                    163:
1.4       noro      164: ({\urlc \tt http://www.hpl.hp.com/personal/Hans\_Boehm/gc})
1.3       noro      165:
1.4       noro      166: \item {\itc 1989-1992}
1.3       noro      167:
                    168: Univariate and multivariate factorizers over {\bf Q}
                    169:
1.4       noro      170: \item {\itc 1992-1994}
1.3       noro      171:
                    172: Univariate factorization over algebraic number fields
                    173:
                    174: Intensive use of successive extension, non-squarefree norms
                    175:
1.5       noro      176: Application to splitting field and Galois group computation
                    177:
1.4       noro      178: \item {\itc 1996-1998}
1.3       noro      179:
                    180: Univariate factorization over large finite fields
                    181:
                    182: Motivated by a reseach project in Fujitsu on cryptography
                    183:
1.4       noro      184: \item {\itc 2000-current}
1.3       noro      185:
                    186: Multivariate factorization over small finite fields (in progress)
                    187: \end{itemize}
                    188: \end{slide}
                    189:
                    190: \begin{slide}{}
1.4       noro      191: \fbox{\fbc History of development : Groebner basis}
1.3       noro      192:
                    193: \begin{itemize}
1.4       noro      194: \item {\itc 1992-1994}
1.3       noro      195:
                    196: User language $\Rightarrow$ C version; trace lifting [TRAV88]
                    197:
1.4       noro      198: \item {\itc 1994-1996}
1.3       noro      199:
                    200: Trace lifting with homogenization
                    201:
                    202: Omitting GB check by compatible prime [NOYO99]
                    203:
                    204: Modular change of ordering/RUR[ROUI96] [NOYO99]
                    205:
                    206: Primary ideal decomposition [SHYO96]
                    207:
1.4       noro      208: \item {\itc 1996-1998}
1.3       noro      209:
                    210: Efficient content reduction during NF computation [NORO97]
                    211: Solved {\it McKay} system for the first time
                    212:
1.4       noro      213: \item {\itc 1998-2000}
1.3       noro      214:
                    215: Test implementation of $F_4$ [FAUG99]
                    216:
1.4       noro      217: \item {\itc 2000-current}
1.3       noro      218:
                    219: Buchberger algorithm in Weyl algebra
                    220:
                    221: Efficient $b$-function computation[OAKU97] by a modular method
                    222: \end{itemize}
                    223: \end{slide}
                    224:
                    225: \begin{slide}{}
1.4       noro      226: \fbox{\fbc Timing data --- Factorization}
1.3       noro      227:
1.6     ! noro      228: \underline{\itc Univariate; over {\bf Q}} (on Pentium III, 1GHz; unit : second)
1.3       noro      229:
1.5       noro      230: $N_{i,j}$ : a norm of a polynomial, $\deg(N_i) = i$ with $j$ modular factor
1.3       noro      231: \begin{center}
                    232: \begin{tabular}{|c||c|c|c|c|} \hline
1.5       noro      233:                & $N_{105,23}$ & $N_{120,20}$ & $N_{168,24}$ & $N_{210,54}$ \\ \hline
                    234: {\tc Asir}     & {\tc 0.86}    & {\tc 59} & {\tc 840} & {\tc hard} \\ \hline
1.3       noro      235: Asir NormFactor & 1.6  & 2.2& 6.1& hard \\ \hline
                    236: %Singular& hard?       & hard?& hard? & hard? \\ \hline
                    237: CoCoA 4 & 0.2  & 7.1   & 16 & 0.5 \\ \hline\hline
                    238: NTL-5.2        & 0.16  & 0.9   & 1.4 & 0.4 \\ \hline
                    239: \end{tabular}
                    240: \end{center}
                    241:
1.4       noro      242: \underline{\itc Multivariate; over {\bf Q}}
1.3       noro      243:
                    244: $W_{i,j,k} = Wang[i]\cdot Wang[j]\cdot Wang[k]$ in {\tt asir2000/lib/fctrdata}
                    245: \begin{center}
                    246: \begin{tabular}{|c||c|c|c|c|c|} \hline
                    247:        & $W_{1,2,3}$ & $W_{4,5,6}$ & $W_{7,8,9}$ & $W_{10,11,12}$ & $W_{13,14,15}$ \\ \hline
1.5       noro      248: variables & 3 & 5 & 5 & 5 & 4 \\ \hline
                    249: monomials & 905 & 41369 & 51940 & 30988 & 3344 \\ \hline\hline
                    250: {\tc Asir}     & {\tc 0.2} & {\tc 4.7} & {\tc 14} & {\tc 17} & {\tc 0.4} \\ \hline
1.3       noro      251: %Singular& $>$15min    & ---   & ---& ---& ---\\ \hline
                    252: CoCoA 4 & 5.2 & $>$15min       & $>$15min & $>$15min & 117 \\ \hline\hline
                    253: Mathematica 4& 0.2     & 16    & 23 & 36 & 1.1 \\ \hline
                    254: Maple 7& 0.5   & 18    & 967  & 48 & 1.3 \\ \hline
                    255: \end{tabular}
                    256: \end{center}
                    257:
                    258: %--- : not tested
                    259: \end{slide}
                    260:
                    261: \begin{slide}{}
1.4       noro      262: \fbox{\fbc Timing data --- DRL Groebner basis computation}
1.3       noro      263:
1.4       noro      264: \underline{\itc Over $GF(32003)$}
1.3       noro      265: \begin{center}
                    266: \begin{tabular}{|c||c|c|c|c|c|c|c|} \hline
                    267:                & $C_7$ & $C_8$ & $K_7$ & $K_8$ & $K_9$ & $K_{10}$ & $K_{11}$ \\ \hline
1.5       noro      268: {\tc Asir $Buchberger$}        & {\tc 31} & {\tc 1687}  & {\tc 2.6}  & {\tc 27} & {\tc 294}  & {\tc 4309} & --- \\ \hline
1.3       noro      269: Singular & 8.7 & 278 & 0.6 & 5.6 & 54 & 508 & 5510 \\ \hline
                    270: CoCoA 4 & 241 & $>$ 5h & 3.8 & 35 & 402 &7021  & --- \\ \hline\hline
1.5       noro      271: {\tc Asir $F_4$}       & {\tc 5.3} & {\tc 129} & {\tc 0.5}  & {\tc 4.5} & {\tc 31}  & {\tc 273} & {\tc 2641} \\ \hline
1.3       noro      272: FGb(estimated) & 0.9 & 23 & 0.1 & 0.8 & 6 & 51 & 366 \\ \hline
                    273: \end{tabular}
                    274: \end{center}
                    275:
1.4       noro      276: \underline{\itc Over {\bf Q}}
1.3       noro      277:
                    278: \begin{center}
                    279: \begin{tabular}{|c||c|c|c|c|c|} \hline
                    280:                & $C_7$ & $Homog. C_7$ & $K_7$ & $K_8$ & $McKay$ \\ \hline
1.5       noro      281: {\tc Asir $Buchberger$}        & {\tc 389} & {\tc 594} & {\tc 29} & {\tc 299} & {\tc 34950} \\ \hline
1.3       noro      282: Singular & --- & 15247 & 7.6 & 79 & $>$ 20h \\ \hline
                    283: CoCoA 4 & --- & 13227 & 57 & 709 & --- \\ \hline\hline
1.5       noro      284: {\tc Asir $F_4$}       &  {\tc 989} & {\tc 456} & {\tc 90} & {\tc 991} & {\tc 4939} \\ \hline
1.3       noro      285: FGb(estimated) & 8 &11 & 0.6 & 5 & 10 \\ \hline
                    286: \end{tabular}
                    287: \end{center}
                    288: --- : not tested
                    289: \end{slide}
                    290:
                    291: \begin{slide}{}
1.4       noro      292: \fbox{\fbc Summary of performance}
1.3       noro      293:
                    294: \begin{itemize}
1.4       noro      295: \item {\itc Factorizer}
1.3       noro      296:
                    297: \begin{itemize}
                    298: \item Multivariate : reasonable performance
                    299:
                    300: \item Univariate : obsoleted by M. van Hoeij's new algorithm [HOEI00]
                    301: \end{itemize}
                    302:
1.4       noro      303: \item {\itc Groebner basis computation}
1.3       noro      304:
                    305: \begin{itemize}
                    306: \item Buchberger
                    307:
                    308: Singular shows nice perfomance
                    309:
                    310: Trace lifting is efficient in some cases over {\bf Q}
                    311:
                    312: \item $F_4$
                    313:
                    314: FGb is much faster than Risa/Asir
                    315:
                    316: But we observe that {\it McKay} is computed efficiently by $F_4$
                    317: \end{itemize}
                    318: \end{itemize}
                    319:
                    320: \end{slide}
                    321:
                    322: \begin{slide}{}
1.4       noro      323: \fbox{\fbc What is the merit to use Risa/Asir?}
1.3       noro      324:
                    325: \begin{itemize}
1.4       noro      326: \item {\itc Total performance is not excellent, but not bad}
1.3       noro      327:
1.4       noro      328: \item {\itc A completely open system}
1.3       noro      329:
                    330: The whole source is available
                    331:
1.4       noro      332: \item {\itc It serves as a test bench to try new ideas}
                    333:
                    334: Interactive debugger is very useful
                    335:
                    336: \item {\itc Interface compliant to OpenXM RFC-100}
1.3       noro      337:
                    338: The interface is fully documented
                    339:
                    340: \end{itemize}
                    341:
                    342: \end{slide}
                    343:
                    344:
                    345: %\begin{slide}{}
1.4       noro      346: %\fbox{\fbc CMO = Serialized representation of mathematical object}
1.3       noro      347: %
                    348: %\begin{itemize}
                    349: %\item primitive data
                    350: %\begin{eqnarray*}
                    351: %\mbox{Integer32} &:& ({\tt CMO\_INT32}, {\sl int32}\ \mbox{n}) \\
                    352: %\mbox{Cstring}&:& ({\tt CMO\_STRING},{\sl int32}\,  \mbox{ n}, {\sl string}\, \mbox{s}) \\
                    353: %\mbox{List} &:& ({\tt CMO\_LIST}, {\sl int32}\, len, ob[0], \ldots,ob[m-1])
                    354: %\end{eqnarray*}
                    355: %
                    356: %\item numbers and polynomials
                    357: %\begin{eqnarray*}
                    358: %\mbox{ZZ}         &:& ({\tt CMO\_ZZ},{\sl int32}\, {\rm f}, {\sl byte}\, \mbox{a[1]}, \ldots
                    359: %{\sl byte}\, \mbox{a[$|$f$|$]} ) \\
                    360: %\mbox{Monomial32}&:& ({\tt CMO\_MONOMIAL32}, n, \mbox{e[1]}, \ldots, \mbox{e[n]}, \mbox{Coef}) \\
                    361: %\mbox{Coef}&:& \mbox{ZZ} | \mbox{Integer32} \\
                    362: %\mbox{Dpolynomial}&:& ({\tt CMO\_DISTRIBUTED\_POLYNOMIAL},\\
                    363: %                  & & m, \mbox{DringDefinition}, \mbox{Monomial32}, \ldots)\\
                    364: %\mbox{DringDefinition}
                    365: %                  &:& \mbox{DMS of N variables} \\
                    366: %                  & & ({\tt CMO\_RING\_BY\_NAME}, name) \\
                    367: %                  & & ({\tt CMO\_DMS\_GENERIC}) \\
                    368: %\end{eqnarray*}
                    369: %\end{itemize}
                    370: %\end{slide}
                    371: %
                    372: %\begin{slide}{}
1.4       noro      373: %\fbox{\fbc Stack based communication}
1.3       noro      374: %
                    375: %\begin{itemize}
                    376: %\item Data arrived a client
                    377: %
                    378: %Pushed to the stack
                    379: %
                    380: %\item Result
                    381: %
                    382: %Pushd to the stack
                    383: %
                    384: %Written to the stream when requested by a command
                    385: %
                    386: %\item The reason why we use the stack
                    387: %
                    388: %\begin{itemize}
                    389: %\item Stack = I/O buffer for (possibly large) objects
                    390: %
                    391: %Multiple requests can be sent before their execution
                    392: %
                    393: %A server does not get stuck in sending results
                    394: %\end{itemize}
                    395: %\end{itemize}
                    396: %\end{slide}
                    397:
                    398: \begin{slide}{}
1.4       noro      399: \fbox{\fbc OpenXM (Open message eXchange protocol for Mathematics) }
1.3       noro      400:
                    401: \begin{itemize}
1.4       noro      402: \item {\itc An environment for parallel distributed computation}
1.3       noro      403:
                    404: Both for interactive, non-interactive environment
                    405:
1.4       noro      406: \item {\itc OpenXM RFC-100 = Client-server architecture}
1.3       noro      407:
                    408: Client $\Leftarrow$ OX (OpenXM) message $\Rightarrow$ Server
                    409:
                    410: OX (OpenXM) message : command and data
                    411:
1.4       noro      412: \item {\itc Data}
1.3       noro      413:
                    414: Encoding : CMO (Common Mathematical Object format)
                    415:
                    416: Serialized representation of mathematical object
                    417:
                    418: --- Main idea was borrowed from OpenMath
                    419:
1.4       noro      420: ({\urlc \tt http://www.openmath.org})
1.3       noro      421:
1.4       noro      422: \item {\itc Command}
1.3       noro      423:
                    424: stack machine command --- server is a stackmachine
                    425:
                    426: + server's own command sequences --- hybrid server
                    427: \end{itemize}
                    428: \end{slide}
                    429:
                    430: \begin{slide}{}
1.4       noro      431: \fbox{\fbc Example of distributed computation --- $F_4$ vs. $Buchberger$ }
1.3       noro      432:
                    433: \begin{verbatim}
                    434: /* competitive Gbase computation over GF(M) */
                    435: /* Cf. A.28 in SINGULAR Manual */
1.4       noro      436: /* Process list is specified as buch_vs_f4_mod(...|proc=P) */
                    437: def buch_vs_f4_mod(G,V,M,O)
1.3       noro      438: {
                    439:   P = getopt(proc);
                    440:   if ( type(P) == -1 ) return dp_f4_mod_main(G,V,M,O);
                    441:   P0 = P[0]; P1 = P[1]; P = [P0,P1];
1.4       noro      442:   map(ox_reset,P); /* resets the both servers */
                    443:   ox_cmo_rpc(P0,"dp_f4_mod_main",G,V,M,O);  /* for F4 */
                    444:   ox_cmo_rpc(P1,"dp_gr_mod_main",G,V,0,M,O); /* for Buchberger */
1.3       noro      445:   map(ox_push_cmd,P,262); /* 262 = OX_popCMO */
1.4       noro      446:   F = ox_select(P); /* waits a server to return the result */
                    447:   R = ox_get(F[0]); /* gets the result from the winner */
1.3       noro      448:   if ( F[0] == P0 ) { Win = "F4"; Lose = P1;}
                    449:   else { Win = "Buchberger"; Lose = P0; }
                    450:   ox_reset(Lose); /* simply resets the loser */
                    451:   return [Win,R];
                    452: }
                    453: \end{verbatim}
                    454: \end{slide}
                    455:
                    456: \begin{slide}{}
1.4       noro      457: \fbox{\fbc Real speedup by parallelism --- polynomial multiplication}
                    458:
                    459: {\itc Product of dense univariate polynomials with 3000bit coefficients}
                    460:
                    461: {\itc Algorithm : FFT+Chinese remainder (by Shoup)}
                    462:
                    463: \epsfxsize=20cm
                    464: \epsffile{3k.ps}
                    465:
                    466: {\itc Communication cost}
                    467:
                    468: \begin{itemize}
                    469: \item $O(n{\color{red}\log L})$ with server-server communication (OX-RFC102)
                    470: \item $O(n{\color{red}L})$ without server-server communication (OX-RFC100)
                    471: \end{itemize}
                    472: ($L$: number of processes, $n$: degree)
                    473:
                    474: \end{slide}
                    475:
                    476: \begin{slide}{}
                    477: \fbox{\fbc References}
1.3       noro      478:
                    479: [BERN97] L. Bernardin, On square-free factorization of
                    480: multivariate polynomials over a finite field, Theoretical
                    481: Computer Science 187 (1997), 105-116.
                    482:
                    483: [FAUG99] J.C. Faug\`ere,
                    484: A new efficient algorithm for computing Groebner bases  ($F_4$),
                    485: Journal of Pure and Applied Algebra (139) 1-3 (1999), 61-88.
                    486:
                    487: [GRAY98] S. Gray et al,
                    488: Design and Implementation of MP, A Protocol for Efficient Exchange of
                    489: Mathematical Expression,
                    490: J. Symb. Comp. {\bf 25} (1998), 213-238.
                    491:
                    492: [HOEI00] M. van Hoeij, Factoring polynomials and the knapsack problem,
                    493: to appear in Journal of Number Theory (2000).
                    494:
                    495: [LIAO01] W. Liao et al,
                    496: OMEI: An Open Mathematical Engine Interface,
                    497: Proc. ASCM2001 (2001), 82-91.
                    498: [NORO97] M. Noro, J. McKay,
                    499: Computation of replicable functions on Risa/Asir.
                    500: Proc. PASCO'97, ACM Press (1997), 130-138.
                    501: \end{slide}
                    502:
                    503: \begin{slide}{}
                    504:
                    505: [NOYO99] M. Noro, K. Yokoyama,
                    506: A Modular Method to Compute the Rational Univariate
                    507: Representation of Zero-Dimensional Ideals.
                    508: J. Symb. Comp. {\bf 28}/1 (1999), 243-263.
                    509:
                    510: [OAKU97] T. Oaku, Algorithms for $b$-functions, restrictions and algebraic
                    511: local cohomology groups of $D$-modules.
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1.1       noro      526: \end{document}

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