\section{Introduction}

Morphology systems, lemmatisers and part-of-speech taggers are some of the
basic tools in natural language processing. There are numerous applications,
including syntax parsing, machine translation, automatic indexing and semantic
clustering of words. Unfortunately, for languages other than English, such tools
are rarely available, and different research groups are often forced to
redevelop them over and over again. Considering German, quite a few morphology
systems (Hausser 1996) and taggers (see table \ref{Tagger}) have been developed, which are described in Wothke et al. (1993) (IBM Heidelberg), Steiner (1995) (University of M{\"u}nster), Feldweg (1995) (University of T{\"u}bingen), Schmid (1995) (University of Stuttgart), Armstrong et al. (1995) (ISSCO Geneva), and Lezius (1995) (University of Paderborn).
However, in most cases, the tagger is isolated from the morphology system. It relies on a lexicon of full forms which, of course, may be generated by a morphological
tool. Unfortunately, most German lexicons are not available due to copyright
restrictions and - as far as we know - none of them is public-domain.
Therefore we have decided to make our system Morphy publicly available.
It combines a morphological and tagging module in a single package
and can be downloaded from the World Wide
Web.\footnote{
URL: http://www-psycho.uni-paderborn.de/lezius/morpho.html}

\begin{table}
\caption{\label{Tagger} Comparison of German Taggers}

\begin{footnotesize}

\begin{center}
\begin{tabular}{|l||c|c|c|c|c|c|} \hline
 & IBM & Univer- & Univer- & Univer- & & Univer- \\
 & Heidel- & sity of & sity of & sity of & ISSCO & sity of \\
 & berg & M{\"u}nster & T{\"u}bingen & Stuttgart & Geneva & Paderborn \\
\hline \hline

learning & super- & super- & unsuper- & unsuper- & unsuper- & super-\\
method & vised & vised & vised & vised & vised & vised \\ \hline

context & bi- \& tri- & bi- & bi- & bi- & bi- & tri- \\
method & grams & grams & grams & grams & grams & grams \\ \hline

training & 20.000 & 200.000 & 200.000 & 20.000 & 20.000 & 20.000 \\
corpus & words & words & words & words & words & words \\ \hline

test & 10.000 & 30.000 & 20.000 & 5.000 & 1.850 & 5.000 \\
corpus & words & words & words & words & words & words \\ \hline

size of & 534.514 & 850.000 & 500.000 & 350.000 & 30.000 & 100.000 \\
lexicon & words & words & words & words & words & words \\ \hline

tag sets & 689 & 143 & -  & - & - & 456 \\
large / small & 33 & 54 & 42 & 50 & 56 & 51 \\ \hline

accuracy & 77.7\% & 81.5\% & - & - & - & 84.7\% \\
large / small & 93.4\% & 92.8\% & 96.7\% & 97.0\% & 96.5\% & 95.9\% \\ \hline

\end{tabular}
\end{center}

\end{footnotesize}

\end{table}

Since it has been created not only for linguists, but also for second language
learners, it has been designed for standard PC-platforms and great effort has been put in making it as easy to use as possible.

\vspace{-1ex}

\section{The morphology module of Morphy}

The morphology system is based on the Duden grammar (Drosdowski 1984).
It consists of three parts: {\it Analysis}, {\it generation} and
{\it lexical system}.

The lexical system is more sophisticated than other systems in order to allow a
user-friendly extension of the lexicon.
When entering a new word,
the user is asked the minimal number of questions
which are necessary to infer the new word's grammatical features and
which any native speaker should
be able to answer. In most cases only the root of the word has to be typed in,
questions are answered by pressing the number of the correct alternative
(see figure \ref{Lex} for the dialogue when entering the verb
{\it telefonieren}). Currently,
the lexicon comprises 21.500 words (about 100.000 word forms)
and is extended continuously.

Starting from the root of a word and the inflexion type as stored in the
lexicon, the generation system produces all inflected
forms which are shown on the screen.
Among other morphological features it considers vowel mutation, shifts
between {\it {\ss}} and {\it ss} as well as pre- and infixation
of markers for participles.

The analysis system for each word form determines its root and its part of speech,
and, if appropiate, its gender, case, number, tense and comparative
degree. It also segments compound nouns using a longest-matching rule
which works from right to left. Since the system treats each word form separately,
ambiguities can not be resolved. For ambiguous word forms any possible lemma and
morphological description is given (for some examples
see table \ref{analysis}). If a word form can not be recognised,
its part of speech is predicted by an algorithm which makes use of
statistical data on German suffix frequencies.

\begin{figure}

\begin{center}
\begin{tabular}{|l|} \hline\\
\hspace*{1.2em}1. Geben Sie den Stamm ein: {\bf telefonieren} \\
\hspace*{1.2em}2. Wird das Verb schwach konjugiert?\\
\hspace*{2.7em}{\bf 1}: Ja\\
\hspace*{2.7em}2: Nein\\
\hspace*{1.2em}3. Wie lautet die 2. Person Singular Pr{\"a}sens? \hspace*{1.2em}\\
\hspace*{2.7em}{\bf 1}: du telefonierst\\
\hspace*{2.7em}2: du telefonierest\\
\hspace*{2.7em}3: du telefoniert\\
\hspace*{1.2em}4. Wie lautet das Partizip des Verbs?\\
\hspace*{2.7em}{\bf 1}: telefoniert\\
\hspace*{2.7em}2: getelefoniert\\
\hspace*{1.2em}Verb klassifiziert!\\
\\ \hline
\end{tabular}
\end{center}

\caption{\label{Lex} Dialogue when entering {\it telefonieren}
(user input is printed bold type)}

\end{figure}

Morphy's lookup-mechanism when analyzing texts is not based on a lexicon of
full forms. Instead, there is only a lexicon of roots together with their
inflexion types. When analyzing a word form, Morphy cuts off all
possible suffixes, builds the possible roots, looks up these roots in the lexicon,
and for each root generates all possible inflected forms. Only those roots
which lead to inflected forms identical to the original word
form will be selected (for details see Lezius 1994).

Naturally, this procedure is much slower than the simple lookup-mechanism
in a full form lexicon.\footnote{Morphy's current analysis speed is about 50 word
forms per second on a fast PC, which is sufficient for many purposes. For the
processing of larger corpora we have
used Morphy to generate a full-form lexicon under UNIX. This has led to an
analysis speed of many thousand word forms per second.}
Nevertheless, there are advantages: First, the lexicon
can be kept small,\footnote{Only 750 KB memory is
necessary for the current lexicon.} which is an important consideration
for a PC-based system intended to be widely distributed. Secondly, the
processing of German compound nouns fits in this concept.

The performance of the morphology system has been tested at
the Morpholympics conference 1994
in Erlangen (see Hausser (1996), pp. 13-14, and Lezius (1996)) with a
specially designed test corpus
which had been unknown to the participants. This corpus comprised about 7.800
word forms and consisted of different text types (two political speeches, a fragment of
the Limas-corpus and a list of special word forms). Morphy
recognised 89.2\%, 95.9\%, 86.9\% and 75.8\% of the word forms, respectively.

\begin{table}
\caption{\label{analysis} Some examples of the morphological analysis}

\begin{footnotesize}

\begin{center}
\begin{tabular}{|l|l|l|} \hline

word form & morphological description & root \\ \hline \hline

Fl{\"u}ssen & Substantiv Dativ Plural maskulinum & Flu{\ss} \\ \hline

Bauern- & Substantiv Dativ Plural neutrum & Bauer / Haus \\
h{\"a}usern & & \\ \hline

Schiffahrts- & Substantiv Genitiv Singular maskulinum & Schiff / Fahrt / \\
hafenmeisters &                                       & Hafen / Meister \\
\hline

K{\"u}sse & Substantiv Nominativ Plural maskulinum & Ku{\ss} \\
       & Substantiv Genitiv Plural maskulinum & Ku{\ss} \\
       & Substantiv Akkusativ Plural maskulinum & Ku{\ss} \\
       & Verb 1. Person Singular Pr{\"a}sens & k{\"u}ssen \\
       & Verb 1. Person Singular Konjunktiv 1 & k{\"u}ssen \\
       & Verb 3. Person Singular Konjunktiv 1 & k{\"u}ssen \\
       & Verb Imperativ Singular & k{\"u}ssen \\ \hline
einnahm & Verb 1. Person Singular Pr{\"a}teritum & (ein)nehmen \\
        & Verb 3. Person Singular Pr{\"a}teritum & (ein)nehmen \\ \hline
verspieltest & Verb 2. Person Singular Pr{\"a}teritum & ver-spielen \\
             & Verb 2. Person Singular Konjunktiv 2 & ver-spielen \\ \hline
verspieltes & Adjektiv Nominativ Singular neutrum & verspielt (ver-spielen)\\
            & Adjektiv Akkusativ Singular neutrum & verspielt (ver-spielen)\\ \hline
edlem & Adjektiv Dativ Singular neutrum & edel \\
      & Adjektiv Dativ Singular maskulinum & edel \\ \hline
\end{tabular}
\end{center}

\end{footnotesize}

\end{table}

\section{The tagging module of Morphy}

Since morphological analysis operates on isolated word forms, ambiguities
are not resolved. The task of the tagger is to resolve these ambiguities
by taking into account contextual information. When designing a tagger,
a number of decisions have to be made:

\begin{itemize}
\item Selection of a tag set.
\item Selection of a tagging algorithm.
\item Selection of a training and test corpus.
\end{itemize}

\subsection{Tag Set}

Like the morphology system, the tagger is
based on the classification of the parts of
speech from the Duden grammar. Supplementary additions have been taken from
the system of Bergenholtz and Schaeder (1977). The so-formed tag
set includes grammatical features as gender,
case and number. This results in a very complex system, comprising about
1000 different tags (see Lezius 1995). Since only 456 tags were actually
used in the training corpus, the tag set was reduced to half.
However, most German word forms are highly
ambiguous in this system (about 5 tags per word form on average).

Although the amount of information gained by this system is very high,
previous tagging algorithms with such large tag sets led to poor results in the
past (see Wothke et al. 1993; Steiner 1995). This is because different grammatical features
often have the same surface realization (e.g. nominative noun and accusative
noun are difficult to distinguish by the tagger). By grouping together parts of
speech with different grammatical features this kind of error can be
significantly reduced. This is what current small tag sets implicitly do.
However, one has to keep in mind that the gain of information provided by the tagger
is also reduced with a smaller tag set.

Since some applications do not require detailed distinctions, we also constructed
a small tag set comprising 51 tags as shown in table \ref{tagset}.
Both tag sets are constructed in such a way
that the large tag set can be directly mapped onto the small tag set.

\begin{table}

\caption{\label{tagset} The small tag set (51 tags)}

\begin{footnotesize}
\begin{center}
\begin{tabular}{||l|l|l||}\hline\hline

tag name & explanation of the tag & example\\ \hline\hline

\tt SUB & Substantiv & \it (der) Mann\\
\tt EIG & Eigenname & \it Egon, (Herr) Hansen\\ \hline

\tt VER & finite Verbform & \it spielst, l{\"a}uft\\
\tt VER INF & Infinitiv & \it spielen, laufen\\
\tt VER PA2 & Partizip Perfekt & \it gespielt, gelaufen\\
\tt VER EIZ & erweiterter Infinitiv mit \it zu & \it abzuspielen\\
\tt VER IMP & Imperativ & \it lauf', laufe\\
\tt VER AUX & finite Hilfsverbform & \it bin, hast\\
\tt VER AUX INF & Infinitiv & \it haben, sein\\
\tt VER AUX PA2 & Partizip Perfekt & \it gahabt, gewesen\\
\tt VER AUX IMP & Imperativ & \it sei, habe\\
\tt VER MOD & finite Modalverbform & \it kannst, will\\
\tt VER MOD INF & Infinitiv & \it k{\"o}nnen, wollen\\
\tt VER MOD PA2 & Partizip Perfekt & \it gekonnt, gewollt\\
\tt VER MOD IMP & Imperativ & \it k{\"o}nne\\ \hline

\tt ART IND & unbestimmter Artikel & \it ein, eines\\
\tt ART DEF & bestimmter Artikel & \it der, des\\ \hline

\tt ADJ & Adjektivform & \it schnelle, kleinstes\\
\tt ADJ ADV & Adjektiv, adverbiell & \it (Er l{\"a}uft) schnell.\\ \hline

\tt PRO DEM ATT & Demonstrativpronomen, attributiv & \it diese
(Frau)\\
\tt PRO DEM PRO & Demonstrativpronomen, pronominal & \it diese\\
\tt PRO REL ATT & Relativpronomen, attributiv & \it ,dessen
(Frau)\\
\tt PRO REL PRO & Relativpronomen, pronominal & \it ,welcher\\
\tt PRO POS ATT & Possesivpronomen, attributiv & \it mein (Buch)\\
\tt PRO POS PRO & Possesivpronomen, pronominal & \it (Das ist)
meiner.\\
\tt PRO IND ATT & Indefinitpronomen, attributiv & \it alle
(Menschen)\\
\tt PRO IND PRO & Indefinitpronomen, pronominal & \it (Ich mag)
alle.\\
\tt PRO INR ATT & Interrogativpronomen, attributiv & \it Welcher
(Mann)?\\
\tt PRO INR PRO & Interrogativpronomen, pronominal & \it Wer?\\
\tt PRO PER & Personalpronomen & \it er, wir\\
\tt PRO REF & Reflexivpronomen & \it sich, uns\\ \hline

\tt ADV & Adverb & \it schon, manchmal\\
\tt ADV PRO & Pronominaladverb & \it damit, dadurch\\
\tt KON UNT & unterordnende Konjunktion & \it da{\ss}, da\\
\tt KON NEB & nebenordnende Konjunktion & \it und, oder\\
\tt KON INF & Infinitivkonjunktion & \it um (zu spielen)\\
\tt KON VGL & Vergleichskonjunktion & \it als, denn, wie\\
\tt KON PRI & Proportionalkonjunktion & \it desto, um so, je\\
\tt PRP & Pr{\"a}position & \it durch, an\\ \hline

\tt SKZ & Sonderklasse f{\"u}r \it zu & \it (,um) zu (spielen)\\
\tt ZUS & Verbzusatz & \it (spielst) ab\\
\tt INJ & Interjektion & \it Wau, Oh\\
\tt ZAL & Zahlw{\"o}rter & \it eins, tausend\\
\tt ZAN & Zahlen & \it 100, 2\\
\tt ABK & Abk{\"u}rzung & \it Dr., usw.\\ \hline

\tt SZD & Doppelpunkt & :\\
\tt SZE & Satzendezeichen & .!?\\
\tt SZG & Gedankenstrich & -\\
\tt SZK & Komma & ,\\
\tt SZS & Semikolon & ;\\
\tt SZN & sonstige Satzzeichen & ()/ \\ \hline \hline

\end{tabular}
\end{center}
\end{footnotesize}

\end{table}

\subsection{Tagging algorithm}

The tagger uses the Church-trigram-algorithm (Church 1988), which is still
unsurpassed in terms of simplicity, robustness and accuracy. However, since
we assumed that longer n-grams may give more information, and
since we observed that some longer n-grams are rather frequent in corpora
(see figure 2 for some statistics on the Brown-corpus),
we decided to compare
the Church algorithm with a tagging algorithm relying on variable context
widths as described by Rapp (1995).

Starting from an ambiguous word form which is to be tagged, this algorithm
considers the preceding word froms - which have already been
tagged - and the succeeding word forms still to be tagged.
For this ambiguous word form the algorithm constructs all possible tag
sequences composed of the already computed tags on the left, one of the
possible tags of the critical word form and possible tags on the right.

The choice of the tag for the critical word form is a function for the
length of the tag sequences to the left and to the right which can be
found in the training corpus. A detailed description of this algorithm
is given in Rapp (Rapp 1995, pp. 149-154).

Although some authors (Cutting et al. 1992; Schmid 1995; Feldweg 1995) claim
that unsupervised tagging algorithms produce superior results, we chose
supervised learning. These publications pay little attention to
the fact that algorithms for unsupervised tagging require great care
(or even luck) when tuning some initial parameters.
It frequently happens that unsupervised learning with sophisticated tag sets
ends up in local minima, which can lead to poor results without any
indication to the user. Such behavior seemed unacceptable for a standard
tool.

\begin{figure}

$$\beginpicture

\setcoordinatesystem units <0.025em,0.06em>
\setplotarea x from 0 to 1000, y from 0 to 200

\begin{footnotesize}

\axis bottom label {size of the corpus}
      ticks numbered from 0 to 1000 by 200 /

\axis left label {\stack{n,-,g,r,a,m,s}}
      ticks numbered from 0 to 200 by 50 /

\setplotsymbol ({\rm.})
\linethickness0.01em
\setquadratic

\plotsymbolspacing0.05em

\plot 0 0          100 40      200 65
      400 100      600 125     800 145
      1000 155 /

\plot 0 0          100 10      200 15
      600 22.5     1000 23.5 /

\plot 0 0          500 2       1000 4 /

\setlinear
\plotsymbolspacing0.4em

\plot 0 0  200 200 /

\put {n=2} at 1050 7
\put {n=3} at 1050 27
\put {n=4} at 1050 157.5

\end{footnotesize}
\endpicture$$

\vspace{4ex}

\caption{\label{Brown} Statistics on the Brown corpus: number of different
n-grams occuring in the corpus versus size of the corpus (all figures
in thousands)}

\end{figure}

\subsection{Training and test corpus}

For training and testing we took a fragment from the
``Frankfurter-Rundschau''-corpus,\footnote{This corpus was generously donated by
the Druck- und Verlagshaus Frankfurt
am Main and has been included in the CD-ROM of the European
Corpus Initiative. We thank Gisela Zunker for her help with the acquisition and
preparation of the corpus.}
which we have been collecting since 1992. Tables and other non-textual items
were removed manually. A segment of 20.000 word forms was used for training,
another segment of 5.000 word forms for testing. Any word forms not recognised
by the morphology system
were included in the lexicon. Using a special tagging editor which
- on the basis of the morphology module - for each word gives a choice of
possible tags, both corpora were tagged
semiautomatically with the large tag set. A recent version of the editor
additionally predicts the correct tag.

\section{Results}

Using the probabilities from the manually annotated training corpus,
the test corpus was tagged automatically. The results were compared with the
previous manual annotation of the test corpus. This was done for both tagging
algorithms and tag sets. For the small tag set, the Church algorithm achieved
an accuracy of 95.9\%, whereas with the variable-context algorithm an accuracy
of 95.0\% was obtained. For the large tag set the respective figures are 84.7\%
and 81.8\%.

In comparison with other research groups, the results are
similar for the small tagset and slightly better for the large tagset (see
table \ref{Tagger}). Surprisingly, inspite of considering less context, the Church
algorithm performs better than the variable-context algorithm in both cases.

This is the reason why the current implementation
of Morphy only includes the Church algorithm.\footnote{The speed of the tagger
(including morphological analysis) is about 20 word forms per second for
the large and 100 word forms per second for the small tag set on a fast PC.}
As an example, figure \ref{Example}
gives the annotation results of a few test sentences for both tag sets.

\begin{figure}

$$\beginpicture

\setcoordinatesystem units <0.001275em,0.2em>
\setplotarea x from 0 to 20000, y from 0 to 100

\begin{footnotesize}

\axis bottom label {size of the training corpus}
      ticks numbered from 0 to 20000 by 5000 /

\axis left label {\stack{a,c,c,u,r,a,c,y}}
      ticks numbered from 0 to 100 by 10 /

\setplotsymbol ({\rm.})
\setquadratic

\plotsymbolspacing0.05em

\plot 0 0          500 70      1000 93.8
      2500 94.3    5000 94.6   7500 94.7
      10000 94.8   12500 94.9  15000 95.0
      17500 94.8   20000 95.0 /

\plotsymbolspacing0.05em

\plot 0 0          500 70      1000 93.9
      2500 94.7    5000 94.9   7500 95.2
      10000 95.4   12500 95.5  15000 95.5
      17500 95.7   20000 95.9 /

\put{Church} at 17500 99
\put{var.context} at 17500 92

\put{small} at 21250 96
\put{tag set} at 21250 93

\setplotsymbol ({\rm.})
\setquadratic

\plotsymbolspacing0.05em

\plot 0 0          500 51      1000 69.2
      2500 74.0    5000 75.9   7500 77.2
      10000 78.6   12500 79.4  15000 80.1
      17500 80.8   20000 81.8 /

\plotsymbolspacing0.05em

\plot 0 0          500 51      1000 71.4
      2500 75.6    5000 78.9   7500 81.8
      10000 82.7   12500 83.3  15000 83.6
      17500 84.2   20000 84.7 /

\put{Church} at 17500 87
\put{var.context} at 17500 77

\put{large} at 21250 84
\put{tag set} at 21250 81

\end{footnotesize}
\endpicture$$

\vspace{4ex}

\caption{\label{Results} Accuracy versus size of the training corpus for
Church's trigram algorithm and the variable-context algorithm and both tag
sets.}

\end{figure}

However, there are also some advantages on the side of the
variable-context algorithm. First, its potential when using larger training
corpora seems to be slightly higher (see figure \ref{Results}). Secondly, when
the algorithm
is modified in such a way that sentence boundaries are not assumed to be known
beforehand, the performance degrades only minimally. This means that this
algorithm can actually contribute to finding sentence boundaries. And third,
if there are sequences of unknown word forms in the text, the algorithm takes
better guesses than the Church algorithm (examples are given in
Rapp 1995, p. 155). When about 2\% of the word forms in the test corpus were
randomly replaced by unknown word forms, the quality of the results for the
Church algorithm decreased by 0.7\% for the small and by 2.0\% for the large
tag set. The respective figures for the variable-context algorithm are 0.9\%
and 1.3\%, which is better overall.

In a further experiment, the contribution of the lexical probabilities
to the quality of the results was examined. Without the lexical
probabilities, the results decreased by 0.3\% (small) and
0.6\% (large tag set) for the Church algorithm, the respective
figures for the variable-context algorithm were 0.9\% and 0.0\%.

\begin{figure}[t]

\begin{footnotesize}
\begin{center}

\begin{tabular}{|c|c|c|c|c|c|} \hline
Die & Frau & bringt & das & Essen & . \\
\hline
ART DEF & SUB & VER & ART DEF & SUB & SZE \\ \hline
\end{tabular}

\vspace{1ex}

\begin{tabular}{|c|c|c|c|c|} \hline
Ich & meine & meine & Frau & . \\ \hline
PER PRO & VER & POS ATT & SUB &
SZE \\ \hline
\end{tabular}

\vspace{1ex}

\begin{tabular}{|c|c|c|c|c|c|c|c|c|} \hline
Winde & das & im & Winde & flatternde & Segel & um & die & Winde \\ \hline
\bf SUB & ART DEF & PRP & SUB & ADJ & SUB & PRP & ART & SUB \\ \hline
\end{tabular}

\vspace{6ex}

\begin{tabular}{|c|c|c|} \hline
Die & Frau & bringt \\ \hline
ART DEF NOM SIN FEM & SUB NOM FEM SIN & VER 3PE SIN \\ \hline
\end{tabular}

\vspace{1ex}

\begin{tabular}{|c|c|c|c|} \hline
das & Essen & . & Ich \\ \hline
ART DEF AKK SIN NEU & SUB AKK NEU SIN & SZE & PER NOM SIN 1PE \\
\hline
\end{tabular}

\vspace{1ex}

\begin{tabular}{|c|c|c|c|} \hline
meine & meine & Frau & .\\ \hline
VER 1PE SIN & POS AKK SIN FEM ATT & SUB AKK FEM SIN & SZE \\
\hline
\end{tabular}

\vspace{1ex}

\begin{tabular}{|c|c|c|} \hline
Winde & das & im \\ \hline
VER \begin{bf}3PE \end{bf} SIN & \begin{bf} DEM NOM \end{bf} SIN NEU \begin{bf}
PRO \end{bf} & PRP DAT \\ \hline
\end{tabular}

\vspace{1ex}

\begin{tabular}{|c|c|c|} \hline
Winde & flatternde & Segel \\ \hline
SUB DAT MAS SIN  & PA1 \begin{bf} SOL \end{bf} NEU AKK \begin{bf} PLU \end{bf} &
SUB AKK NEU \begin{bf} PLU \end{bf} \\ \hline
\end{tabular}

\vspace{1ex}

\begin{tabular}{|c|c|c|c|} \hline
um & die & Winde & . \\ \hline
PRP AKK & ART DEF AKK SIN FEM & SUB AKK FEM SIN & SZE \\ \hline
\end{tabular}

\end{center}

\caption{\label{Example} Tagging example for both tag sets - the
ambiguity rates amount to 2.4 tags per word for the small and 8.8 tags per word
for the large tag set (errors are printed bold type).}
\end{footnotesize}
\end{figure}

\section{Conclusions}

We have compared two different tagging algorithms and two different tag sets.
The first tagging algorithm is the Church algorithm
which uses trigrams to compute contextual probabilities. The second algorithm,
the so-called variable-context algorithm, has been described in paragraph 3.
The smaller of the two tag sets contains 51 parts-of-speech, the larger tag set
includes additional grammatical features such as case, number and gender.
The small tag set is a subset of the large tag set.

In comparison with the Church algorithm, the variable-context algorithm produces
similar results for the small tag set, but significantly inferior results for the
large tag set. On the other hand, the performance of the variable-context
algorithm for the large tag set improves faster with increasing size of the
training corpus than the performance of the Church algorithm.
Thus, with tagging more training
texts manually, similar results are to be expected for the two algorithms.

Considering the two tag sets, the results for the small tag set are
significantly better. Nevertheless, with increasing size of the training corpus
an approximation of the results seems to be possible.

One of our aims for the near future is to use the output of the tagger for
lemma\-tization. In this way a sentence like {\it Ich meine meine Frau.} could be
unambiguously reduced to {\it ich / meinen / mein / Frau}.

\vspace{-2ex}

\section*{Bibliography}

\vspace{-2ex}

S. Armstrong, G. Russell, D. Petitpierre and G. Robert (1995). An open architecture for multilingual text processing. In: {\it Proceedings of the ACL SIGDAT Workshop. From
Texts to Tags: Issues in Multilingual Language Analysis}, Dublin.

\vspace{-0.75ex}

H. Bergenholtz and B. Schaeder (1977). {\it Die
Wortarten des Deutschen}. Klett, Stuttgart.

\vspace{-0.75ex}

K. Church (1988). A stochastic parts program and noun phrase parser
for unrestricted text. In:
{\it Second Conference on Applied Natural Language Processing}, pp. 136-143.
Austin, Texas.

\vspace{-0.75ex}

D. Cutting, J. Kupiec, J. Pedersen and P. Sibun (1992). A
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\end{document}


\section{Introduction}
Hartshorne conjectured that every smooth, codimension $c$ subvariety of $\bP^n$, with $c<\frac{1}{3}n$, is a complete intersection~\cite[p.~1017]{hartshorne-bulletin}.
We prove this in the special case when $n\gg \deg X$.
\begin{theorem}\label{thm:more general}
There is a function $N(c,e)$ such that if $X\subseteq \bP^n_{\bk}$ is closed, equidimensional of
codimension $c$, and has degree~$e$, and if $X$ is nonsingular in codimension $\geq N(c,e)$, then $X$ is a complete intersection.  In particular, $N(c,e)$ does not depend on $n$ or on the field $\bk$.
\end{theorem}

\noindent In characteristic zero, Hartshorne showed this in~\cite[Theorem 3.3]{hartshorne-bulletin}.  In parallel, and also in characteristic zero, Barth and Van de Ven proved an effective version of this result, showing that $N=\frac{5}{2}e(e-7)+c$ works~\cite{barth-icm}.
 Later improvements include: work of Ran~\cite[Theorem]{ran}, sharpening the bound and extending it to arbitrary characteristic, but only when $c=2$; and Bertram-Ein-Lazarsfeld's \cite[Corollary~3]{bertram-ein-lazarsfeld}, which sharpens the bound in arbitrary codimension, but only holds in characteristic zero.  See also~\cites{ballico-chiantini, holme-schneider}.
 Our result is new in the case of positive characteristic and $c>2$.

All of these previous results are proved by geometric means. For instance, the main ingredient in ~\cite{barth-icm} is an analysis of the variety of lines in $X$ through a point, and many of the proofs make use of Kodaira Vanishing and topological results like Lefschetz-type restriction theorems.

Our proof, on the other hand, is algebraic and employs quite different methods: we derive Theorem~\ref{thm:more general} as an elementary consequence of our result in ~\cite{ess-stillman}  that the graded ultraproduct of polynomial rings is isomorphic to a polynomial ring (see Theorem~\ref{thm:ultra poly} below).  Although Theorem~\ref{thm:more general} is ineffective, it holds in arbitrary characteristic and is independent of the field.

Our approach also explicitly connects Hartshorne's Conjecture with the circle of ideas initiated by Ananyan and Hochster in~\cite{ananyan-hochster} in their proof of Stillman's Conjecture.
Though we do not rely on the results of~\cite{ananyan-hochster}, those ideas motivated our approach.

Our proof also has some overlap with the Babylonian tower theorems, like ~\cite[Theorems~I and IV]{BVdV} and those in~\cite{coandua, flenner, sato} among others. From an algebraic perspective, the natural setting for such statements is an inverse limit of polynomial rings, and~\cite{ess-stillman} shows such an inverse limit shares many properties with the ultraproduct ring.

\section{Setup and Background} \label{sec:background}
Each closed subscheme $X\subseteq \bP^n_\bk$ determines a homogeneous ideal $I_X\subseteq \bk[x_0,\dots,x_n]$.  The scheme $X$, or the ideal $I_X$, is {\bf equidimensional of codimension $c$} if all associated primes of $I_X$ have codimension $c$, and $X$ is a {\bf complete intersection} if $I_X$ is defined by a regular sequence.  Since the minimal free resolution of $I_X$ is stable under extending the ground field $\bk$, the property of being a complete intersection is also stable under field extension.

From here on, $\bk$ and $\bk_i$ will denote fields. If $R$ is a graded ring with $R_0=\bk$, then as in~\cite{ananyan-hochster}, we define the {\bf strength} of a homogeneous element $f\in R$ to be the minimal integer $k \ge -1$ for which there is a decomposition $f=\sum_{i=1}^{k+1} g_i h_i$ with $g_i$ and $h_i$ homogeneous elements of $R$ of positive degree, or $\infty$ if no such decomposition exists. The {\bf collective strength} of a set of homogeneous elements $f_1,\dots,f_r\in R$ is the minimal strength of a non-trivial homogeneous $\bk$-linear combination of the $f_i$.

 \begin{lemma}\label{lem:decreasing strength}
Let $R$ be a graded ring with $R_0=\bk$.  If $I\subseteq R$ is homogeneous and finitely generated, then $I$ has a generating set $g_1,\dots,g_r$ where the strength of $g_k$ equals the collective strength of $g_1,\dots,g_k$ for each $1\leq k \leq r$.
\end{lemma}

 \begin{proof}
 Choose any homogeneous generators $f_1,\dots,f_r$ of $I$.  We prove the statement by induction on $r$.  For $r=1$ the statement is tautological.  Now let $r>1$.  By definition of collective strength, we have a $\bk$-linear combination $g_r=\sum_{i=1}^r a_if_i$ such that the strength of $g_r$ equals the collective strength of $f_1,\dots,f_r$.  After relabeling, we can assume that $a_r\ne 0$ and it follows that $f_1,\dots,f_{r-1},g_r$ generate $I$.  Applying the induction hypothesis to the ideal $(f_1,\dots,f_{r-1})$ yields the desired result.
 \end{proof}

Let $Q=(f_1,\dots,f_r) \subseteq \bk[x_1,x_2,\dots]$.  The ideal of $c\times c$ minors of the Jacobian matrix of $(\frac{\partial f_i}{\partial x_j})$ does not depend on the choice of generators of $Q$.  We denote this ideal by $J_c(Q)$.

\begin{lemma}\label{lem:strength and J}
Let $Q=(f_1,\dots,f_r)$ be a homogeneous ideal in $\bk[x_1,x_2,\dots]$.  If the strength of $f_i$ is at most $s$ for $c\leq i \leq r$,  then $\codim J_c(Q)\leq (r-c+1)(2s+2)$.
\end{lemma}

\begin{proof}
For each $c \leq i \leq r$, we write $f_i=\sum_{j=0}^{s} a_{i,j}g_{i,j}$ where  $a_{i,j}$ and $g_{i,j}$ have positive degree for all $i,j$.  Write $L_i$ for the ideal $(a_{i,j}, g_{i,j} \mid 0\leq j \leq s)$ and let $L =L_c+L_{c+1}+\dots+L_r$.  The $i$th row of the Jacobian matrix has entries $\frac{\partial f_i}{\partial x_k}$; thus by the product rule, every entry in this row is in $L_i$.  Since every $c\times c$ minor of the Jacobian matrix will involve row $i$ for some $c\leq i \leq r$, it follows that $J_c(Q) \subseteq L$.  Thus $\codim J_c(Q)\leq \codim L$, which by the Principal Ideal Theorem is at most $(r-c+1)(2s+2)$, as this is the number of generators of $L$.
\end{proof}

We briefly recall the definition of the ultraproduct ring, referring to~\cite[\S4.1]{ess-stillman} for a more detailed discussion.  Let $\cI$ be an infinite set and let $\cF$ be a non-principal ultrafilter on $\cI$.  We refer to subsets of $\cF$ as {\bf neighborhoods of $\ast$}, where $\ast$ is an imaginary point of $\cI$. For each $i\in \cI$, let $\bk_i$ be an infinite perfect field.  Let $\bS$ denote the graded ultraproduct of $\{\bk_i[x_1,x_2,\dots]\}$, where each polynomial ring is given the standard grading.  An element $g\in \bS$ of degree $d$ corresponds to a collection $(g_i)_{i\in \cI}$ of degree $d$ elements $g_i\in \bk_i[x_1,x_2,\dots]$, modulo the relation that $g=0$ if and only if $g_i=0$ for all $i$ in some neighborhood of $\ast$.  For a homogeneous $g\in \bS$ we write $g_i$ for the corresponding element in $\bk_i[x_1,x_2,\dots]$, keeping in mind that this is only well-defined for $i$ in some neighborhood of $\ast$.  The following comes from~\cite[Theorems~1.2 and 4.6]{ess-stillman}:
\begin{theorem}\label{thm:ultra poly}
Let $K$ be the ultraproduct of perfect fields $\{\bk_i\}$ and fix $f_1,\dots, f_r\in \bS$ of infinite collective strength.  There is a set $\cU$, containing the $f_i$, such that
$\bS$ is isomorphic to the polynomial ring $K[\cU]$.
\end{theorem}

The following result was first proven in~\cite[Theorem~4.2]{cmpv} (though that result is more general than this lemma). We provide an alternate proof, to illustrate how it also follows quickly from Theorem~\ref{thm:ultra poly}.

\begin{lemma}\label{lem:numgens and degree}
Fix $c$ and $e$.  There exist positive integers $d_1,\dots,d_r$, depending only on $c$ and $e$, such that any homogeneous, equidimensional, and radical ideal $Q\subseteq \bk[x_1,\dots,x_n]$ (with $\bk$ perfect) of codimension $c$ and degree $e$ can be generated (not necessarily minimally) by homogeneous polynomials $f_1,\dots,f_r$ and where $\deg(f_i) \le d_i$. Neither $r$ nor the $d_i$ depend on $n$ or $\bk$.
\end{lemma}

\begin{proof}
For a homogeneous ideal $I$ we write $\nu(I)$ for the sum of the degrees of the minimal generators of $I$.  If the statement were false, then for each $j\in \bN$ we could find a homogeneous, equidimensional, and radical ideal $Q'_j\subset \bk_j[x_1,x_2,\dots]$ of codimension $c$ and degree $e$, such that  $\nu(Q'_j)\to \infty$ as $j\to \infty$.  Choose some function $m \colon \cI \to \bN$ where $m(i)$ is unbounded in any neighborhood of $\ast$.  For each $i\in \cI$, choose some $j$ such that $\nu(Q'_j)\ge m(i)$ and set $Q_i=Q_j'$.  By construction, the function $i\mapsto \nu(Q_i)$ is unbounded in every neighborhood of $\ast$.

By~\cite[Proposition~3.5]{eisenbud-huneke-vasconcelos} (see also~\cite[Theorem~1]{mumford}), each $Q_i$ can be generated up to radical by a regular sequence $f_{1,i},f_{2,i},\dots,f_{c,i}\in \bk_i[x_1,x_2,\dots]$ with $\deg(f_{j,i})=e$ for all $i,j$.  Let $f_j=(f_{j,i})\in \bS$ and let $J=(f_1,f_2,\dots,f_c)\subseteq \bS$.  Since $P=\sqrt{J}$ is finitely generated by Theorem~\ref{thm:ultra poly}, we can write $P=(g_1,\dots,g_r)\subseteq \bS$.  We let $P_i=(g_{1,i},\dots,g_{r,i})$ be the corresponding ideal in $\bk_i[x_1,x_2,\dots]$.

For any $x=(x_i)\in \bS$, we have:
\[
\begin{matrix}\text{$x_i\in P_i$}\\\text{for $i$ near $\ast$}\end{matrix} \iff  x\in P \iff
\begin{matrix}x^n\in J\\ \text{ for some $n$}\end{matrix}
 \iff \begin{matrix}\text{$x_i^n \in J_i$}\\\text{ for $i$ near $\ast$} \end{matrix}
 \iff \begin{matrix}x_i\in Q_i\\ \text{ for $i$ near $\ast$}\end{matrix}.
\]
Thus $P_i=Q_i$ for $i$ near $\ast$.  It follows that in a neighborhood of $\ast$, $\nu(Q_i)$ is bounded by $\sum_{k=1}^r \deg(g_k)$, providing our contradiction.
\end{proof}

 \section{Proof of the main result}

\begin{theorem}\label{thm:Nreg}
There is a function $N(c,e)$ such that if $Q\subseteq \bk[x_1,\dots,x_n]$ is a homogeneous, equidimensional ideal of codimension $c$ and degree $e$ and if $V(Q)$ is nonsingular in codimension $\geq N(c,e)$, then $Q$ is defined by a regular sequence of length $c$.  In particular, $N(c,e)$ does not depend on $n$ or on the field $\bk$.
\end{theorem}
\begin{remark}
Since an equidimensional ideal of codimension $c$ that is nonsingular in codimension $2c+1$ must be prime, it would be equivalent to rephrase Theorem~\ref{thm:Nreg} in terms of prime ideals.  We stick with equidimensional and radical ideals because some of the auxiliary results in this paper might be of interest with this added generality.
\end{remark}

\begin{proof}[Proof of Theorem~\ref{thm:Nreg}]
We first reduce to the case where $\bk$ is perfect.  Extending the field will change neither the minimal number of generators of $Q$, nor the codimension of the singular locus.  By taking $N(c,e)\geq 1$, we can also assume that $Q$ is radical, even after extending the field.  Finally, since a field extension will not change the codimension of any minimal prime of $Q$~\cite[00P4]{stacks}, we can assume that $\bk$ is perfect and that $Q$ is radical and equidimensional of codimension $c$.

Suppose that the theorem were false.  Then for some fixed $c,e$ and for each $j\in \bN$ we can find an equidimensional, radical ideal $Q_j'\subseteq \bk_j[x_1,x_2,\dots]$ (with $\bk_j$ perfect) of codimension $c$ and degree $e$ that is not a complete intersection, but where the codimension of the singular locus of $V(Q_j')$ tends to $\infty$ as $j\to \infty$.  Since the singular locus of $V(Q_j')$ is defined by $Q_j'+J_c(Q_j')$, this implies that $\codim J_c(Q_j') \to \infty$ as $j\to \infty$.  We choose a function $m \colon \cI \to \bN$ where $m(i)$ is unbounded in each neighborhood of $\ast$.  For each $i \in \cI$, define $Q_i$ to be any of the $Q_j'$ satisfying $\codim J_c(Q_j')\geq m(i)$.  By construction, $\codim J_c(Q_i)$ is unbounded in every neighborhood of $\ast$.

By Lemma~\ref{lem:numgens and degree}, there are positive integers $d_1,\dots,d_r$ satisfying: for each $i\in \cI$, there are homogeneous $f_{1,i}, \dots, f_{r,i}$ of degrees $d_1,\dots,d_r$ which generate $Q_i$.  Let $f_1=(f_{1,i}), \dots, f_r=(f_{r,i})$ be the corresponding elements in $\bS$ and let $Q=(f_1,\dots,f_r)$.  By Lemma~\ref{lem:decreasing strength}, we can assume that the strength of $f_k$ is the collective strength of $f_1,\dots, f_k$ for each $1\leq k\leq r$.

If $f_c$ had strength at most $s$, then the same holds for $f_{c,i}$ in a neighborhood of $\ast$; for if $f_c=\sum_{j=0}^s a_jh_j$ then $f_{c,i}=\sum_{j=0}^s (a_j)_i(h_j)_i$ for $i$ near $\ast$.  But by Lemma~\ref{lem:strength and J}, this would imply that $\codim J_c(Q_i)$ is bounded in a neighborhood of $\ast$.  Since this cannot happen, $f_c$ must have infinite strength.  Thus the collection $f_1,\dots,f_c$ has infinite collective strength and so by Theorem~\ref{thm:ultra poly}, $f_1,\dots,f_c$ are independent variables in $\bS$.  In particular, $(f_1,\dots,f_c)$ defines a prime ideal of codimension $c$ and we must have $f_{c+1}=\dots=f_{r}=0$.  However, this implies that in a neighborhood of $\ast$, each $Q_i$ is a complete intersection, contradicting our original assumption.
\end{proof}

\begin{proof}[Proof of Theorem~\ref{thm:more general}]
As in the beginning of the proof of Theorem~\ref{thm:Nreg}, we can quickly reduce to the case where $\bk$ is perfect.  For a fixed $c$ and $e$, we let $N$ equal the bound from Theorem~\ref{thm:Nreg}. Fix some $X\subseteq \bP^n$ satisfying the hypotheses of Theorem~\ref{thm:more general}, and let $Q\subseteq \bk[x_1,\dots,x_{n+1}]$ be the defining ideal of $X$.  By Theorem~\ref{thm:Nreg}, $Q$ is defined by a regular sequence, and thus $X$ is a complete intersection.
\end{proof}

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\section{Introduction\label{sec:Introduction}}

For a Fermi gas at zero temperature,  a Fermi surface(FS) naturally arises as the boundary separating
the occupied  and empty states in $(\omega,\mathbf{k})$-space.
As is known, when  weak interactions/perturbations
and disorders are introduced, some FSs still survive though the occupation
of states may be shifted dramatically, while some others are easily
gapped. Such kind of FS stability originates from its
topological property of the Green's function or the Feynman propagator
for fermionic particles, $G(\omega,\mathbf{k})=\left(i\omega-\mathcal{H}\right)^{-1}$,
which was first pointed out by Volovik in Ref.\cite{Volovik-Book}.
Generally speaking, an FS is robust against weak interactions/perturbations and disorders, if it has a nontrivial topological charge that
provides the protection; otherwise it is vulnerable and easy to be gapped.
The most general case is that a Hamiltonian is not subject to any
symmetry, where the topological charge is formulated by the homotopy
group $\pi_{p}(GL(N,\mathbf{C}))$. This general case was addressed
in Refs.\cite{Volovik-Book,Volovik Vacuum} and analyzed
 in the framework of the K theory \cite{Horava}. Notably, real physical systems
have normally certain symmetries, making it necessary and significant to develop
a corresponding theory for symmetry preserving cases.

It is known that the symmetry
of a quantum system can always have either a unitary representation
or an anti-unitary one in the corresponding Hilbert space. The unitary symmetries, such as rotation, translation, and parity symmetries, are easy to be broken by weak interactions/perturbations and disorders, while the anti-unitary symmetries, such as the time-reversal symmetry(TRS) and charge
conjugate or particle-hole symmetry(PHS)  can usually be preserved. Thus we are motivated to
develop a unified theory to classify the FSs of systems with the two so-called reality symmetries, namely TRS and PHS, which seems to be fundamentally important.
In this Letter, taking into account the two reality symmetries and introducing six types of topological charges,  a new kind of complete classification of all FSs is obtained, being explicitly illustrated in Tab.{[}\ref{tab:periodicity table}{]}. Moreover, an intrinsic relationship between the symmetry class index and the codimension number is established, which provides us a unique way for realizing any type of FS with a high codimension.

Let us first introduce how to classify all sets of
Hamiltonian involving the above-mentioned two reality symmetries, as done in the random matrix theory\cite{Ramdom Matrix 0,Ramdom Matrix I}. It turns out that the chiral symmetry(CS) has to be introduced for a
complete set of classification.
If a unitary symmetry represented by $\mathcal{K}$ can anti-commute
with the Hamiltonian $\mathcal{H}$, $i.e.$, $\left\{ \mathcal{K},\mathcal{H}\right\} =0$,
it corresponds the CS. In fact, the combined symmetry of TRS and PHS
is a kind of chiral symmetry. On the other hand, two chiral symmetries can be combined to commute
with $\mathcal{H}$, which makes $\mathcal{H}$ diagonal. Thus it
is sufficient to consider only one chiral symmetry. Based on these considerations\cite{Ramdom Matrix II,TI Classification III,TI Classification I},
a kind of complete classification of all sets of Hamiltonian can be obtained with
respect to the three symmetries. As is known, the TRS and PHS are both anti-unitary,
 and can  be expressed in a unified form
\begin{equation}
\mathcal{H}(\mathbf{k})=\epsilon_{c}C\mathcal{H}^{T}(-\mathbf{k})C^{-1},\quad CC^{\dagger}=1,\quad C^{T}=\eta_{c}C,\label{eq:reality transformation}
\end{equation}
where $\epsilon_{c}=\pm1$ denotes TRS/PHS, and $\eta_{c}=\pm1$. As a result, each reality
symmetry may take three possible types (even, odd, and absent), and thus there
are nine classes. In addition,  considering that the chiral symmetry may be preserved or not
when both reality symmetries are
 absent,  we can have
 ten classes, as summarized in Tab.{[}\ref{tab:classification}{]},
which is the famous Cartan classification of Hamiltonians in the random matrix theory\cite{Ramdom Matrix 0,Ramdom Matrix I}, while may be easily understood in the present framework \cite{TI Classification I,TI Classification III,Ramdom Matrix II,Tl Classification II}.

\begin{table}
\begin{tabular}{|>{\centering}p{0.5cm}|>{\centering}p{0.6cm}>{\centering}p{0.6cm}>{\centering}p{0.6cm}>{\centering}p{0.6cm}>{\centering}p{0.6cm}|>{\centering}p{0.7cm}>{\centering}p{0.7cm}>{\centering}p{0.7cm}>{\centering}p{0.6cm}>{\centering}p{0.6cm}|}
\hline
 & \multicolumn{5}{c|}{Non-chiral case} & \multicolumn{5}{c|}{Chiral case}\tabularnewline
\hline
\hline
 & A & AI & D & AII & C & AIII & BDI & DIII & CII & CI\tabularnewline
\hline
T & 0 & +1 & 0 & -1 & 0 & 0 & +1 & -1 & -1 & +1\tabularnewline
\hline
C & 0 & 0 & +1 & 0 & -1 & 0 & +1 & +1 & -1 & -1\tabularnewline
\hline
S & 0 & 0 & 0 & 0 & 0 & 1 & 1 & 1 & 1 & 1\tabularnewline
\hline
\end{tabular}
\caption{Classification of Hamiltonian. T and C denote respectively TRS and PHS, with $0$ indicating
the absence of TRS(PHS) and $\pm1$ denoting the sign of TRS(PHS). S denotes CS, with $0$ and $1$ representing the absence and presence of CS. \label{tab:classification}}
\end{table}

For a system with spatial dimension $d$, the FS is a compact submanifold
with dimension $d-p$ if the $(\omega,\mathbf{k})$-space is compact. We may define the codimension of the FS
as $p$.  Topological charges can be defined on a chosen $p$-dimensional submanifold in the $(\omega,\mathbf{k})$-space enclosing the FS in its transverse dimension. The two reality symmetries are essentially different from the CS, which lies in that each of the reality symmetries relates a $\mathbf{k}$ point
to the $-\mathbf{k}$ point in $\mathbf{k}$-space,
while the CS acts on every $\mathbf{k}$. This difference leads
to the requirement that a chosen submanifold in $(\omega,\mathbf{k})$-space
has to be centrosymmetric around the origin in order to preserve either of the
reality symmetries.

Generally, the topological property of an FS depends on its codimension and symmetry~\cite{note1}. After the detailed analysis, we are able to classify all classes in an
appropriate form illustrated in Tab.{[}\ref{tab:periodicity table}{]}, which is one of main results of this work. In each case, as seen in
Tab.{[}\ref{tab:periodicity table}{]}, a kind of topological charge is designated to the FS. Topological charges $0$, $\mathbf{Z}$,  $\mathbf{Z}^{(1,2)}_{2}$, and $2\mathbf{Z}$
 correspond respectively to $0$, an integer,  an integer of $\mod2$, and  an even integer. As illustrated in the table, the ten classes can be divided into real and complex cases according to whether or not they have either of the reality symmetries. On the other hand, they can also be put into the chiral and non-chiral cases depending on whether or not they have the CS. Actually, from this kind of classification, we can find that there are six types of topological charges, which form two groups in terms of the CS, with each group consisting of an original one ($\mathbf{Z}$) and the two descendants ($\mathbf{Z}^{(1)}_2$ and $\mathbf{Z}^{(2)}_2$) \cite{note}.  In the chiral or non-chiral real case, all of the four types of FSs can be realized for any given number of codimension.  It is also clear that the complex and real cases have different periodicities: one is of a two-fold periodicity, while the other is of an eight-fold one. Obviously, for the complex case, we can introduce a matrix element $C(p,i)$ to denote the topological charges listed in Tab.{[}\ref{tab:periodicity table}{]}, where $i=$ odd(even) number corresponds to the class A (AIII), and $p$ is the number of codimension. In this way,  $C(p,i)$ satisfies the relation $C(p,i)=C(p+n,i+n)$ with $n$ an integer and $i\mod 2$. For the real case, we can also introduce another matrix element $R(p,i)$, where $i=1, ...,8$ denote the classes AI, BDI, D, DIII, AII, CII, C and CI respectively. Intriguingly, we can also find
 \begin{equation}
 R(p,i)=R(p+n,i+n)
 \end{equation}
with $i\mod 8$, which establishes an intrinsic relationship between the symmetry class index and the codimension number. More significantly,  based on this relationship, we are able to realize any type of FS with a high codimension by reducing its codimension to an experimentally accessible one with the same CS, $i.e.$, $p=1, 2 , 3$. Actually, the above relation and the two-fold periodicity as well as the eight-fold one are originated from the Bott periodicity of $GL(N,\mathbf{C})$. In particular, the present eight-fold periodicity  is induced by the two reality symmetries enforcing on the two-fold Bott periodicity, and thus is different from the eight-fold Bott periodicity for $O(N)$ and $Sp(N)$\cite{Nakahara}.

\begin{table}
\begin{tabular}{|>{\centering}p{0.6cm}|>{\centering}p{0.6cm}>{\centering}p{0.6cm}>{\centering}p{0.6cm}>{\centering}p{0.6cm}|>{\centering}p{0.7cm}>{\centering}p{0.7cm}>{\centering}p{0.7cm}>{\centering}p{0.7cm}|}
\hline
 & \multicolumn{4}{c|}{Non-chiral case} & \multicolumn{4}{c|}{Chiral case}\tabularnewline
\hline
\hline
 & \multicolumn{8}{c|}{Complex case}\tabularnewline
\hline
 & \multicolumn{4}{c|}{A} & \multicolumn{4}{c|}{AIII}\tabularnewline
\hline
$p\diagdown i$ & \multicolumn{4}{c|}{$\mathbf{1}$} & \multicolumn{4}{c|}{$\mathbf{2}$}\tabularnewline
\hline
1 & \multicolumn{4}{c|}{$\mathbf{Z}$} & \multicolumn{4}{c|}{$0$}\tabularnewline
\hline
2 & \multicolumn{4}{c|}{$0$} & \multicolumn{4}{c|}{$\mathbf{Z}$}\tabularnewline
\hline
$\vdots$ & \multicolumn{4}{c|}{$\vdots$} & \multicolumn{4}{c|}{$\vdots$}\tabularnewline
\hline
\hline
 & \multicolumn{8}{c|}{Real case}\tabularnewline
\hline
 & AI & D & AII & C & BDI & DIII & CII & CI\tabularnewline
\hline
$p\diagdown i$ & $\mathit{1}$ & $\mathit{3}$ & $\mathit{5}$ & $\mathit{7}$ & $\mathit{2}$ & $\mathit{4}$ & $\mathit{6}$ & $\mathit{8}$\tabularnewline
\hline
1 & $0$ & $\mathbf{Z}$ & $\mathbf{Z}^{(2)}_{2}$ & $2\mathbf{Z}$ & $0$ & $\mathbf{Z}^{(1)}_{2}$ & $0$ & $0$\tabularnewline
\hline
2 & $0$ & $0$ & $\mathbf{Z}^{(1)}_{2}$ & $0$ & $0$ & $\mathbf{Z}$ & $\mathbf{Z}^{(2)}_{2}$ & $2\mathbf{Z}$\tabularnewline
\hline
3 & $2\mathbf{Z}$ & $0$ & $\mathbf{Z}$ & $\mathbf{Z}^{(2)}_{2}$ & $0$ & $0$ & $\mathbf{Z}^{(1)}_{2}$ & $0$\tabularnewline
\hline
4 & $0$ & $0$ & $0$ & $\mathbf{Z}^{(1)}_{2}$ & $2\mathbf{Z}$ & $0$ & $\mathbf{Z}$ & $\mathbf{Z}^{(2)}_{2}$\tabularnewline
\hline
5 & $\mathbf{Z}^{(2)}_{2}$ & $2\mathbf{Z}$ & $0$ & $\mathbf{Z}$ & $0$ & $0$ & $0$ & $\mathbf{Z}^{(1)}_{2}$\tabularnewline
\hline
6 & $\mathbf{Z}^{(1)}_2$ & $0$ & $0$ & $0$ & $\mathbf{Z}^{(2)}_{2}$ & $2\mathbf{Z}$ & $0$ & $\mathbf{Z}$\tabularnewline
\hline
7 & $\mathbf{Z}$ & $\mathbf{Z}^{(2)}_{2}$ & $2\mathbf{Z}$ & $0$ & $\mathbf{Z}^{(1)}_{2}$ & $0$ & $0$ & $0$\tabularnewline
\hline
8 & $0$ & $\mathbf{Z}^{(1)}_{2}$ & $0$ & $0$ & $\mathbf{Z}$ & $\mathbf{Z}^{(2)}_{2}$ & $2\mathbf{Z}$ & $0$\tabularnewline
\hline
$\vdots$ & $\vdots$ & $\vdots$ & $\vdots$ & $\vdots$ & $\vdots$ & $\vdots$ & $\vdots$ & $\vdots$\tabularnewline
\hline
\end{tabular}
\caption{Classification of Fermi surfaces. $p$ is the codimension of an FS and $i$ is the index of symmetry classes.
\label{tab:periodicity table}}
\end{table}

Let us look at the most general case, $i.e.$, the class A, in the absence of any
symmetry. The basic idea to classify FSs in terms of the topology
may be learnt from this case, which was already discussed in Ref.\cite{Horava}. We here follow it for a pedagogical
purpose. The Green's function can be written as
\begin{equation}
G(\omega,\mathbf{k})=\frac{1}{i\omega-\mathcal{H}(\mathbf{k})},\label{eq:Green's function}
\end{equation}
which is regarded as an N-dimensional matrix. FSs are defined
to be connected to the zero energy. Formulating topological charges in terms of the Green's function has an advantage when handling interacting systems, which is addressed in \cite{Order-parameter of TI,Horava,Interacting}.
Generally the FS consists of branches
of compact manifolds in $\mathbf{k}$-space. For a specific branch
with dimension $d-p$,  a $p$-dimensional sphere
$S^{p}$ can always be picked up from $(\omega,\mathbf{k})$-space to enclose this branch in its transverse dimension. $G^{-1}(\omega,\mathbf{k})$ is nonsingular for each $(\omega,\mathbf{k})$
restricted on the $S^{p}$; in other words, it is a member of $GL(N,\mathbf{C})$.
Then $G^{-1}(\omega,\mathbf{k})$ restricted on the $S^{p}$ can be
regarded as a mapping from $S^{p}$ to $GL(N,\mathbf{C})$. As all
these mappings can be classified by the homotopy group $\pi_{p}(GL(N,\mathbf{C}))$,
$G^{-1}(\omega,\mathbf{k})$ on the $S^{p}$ is in a certain homotopy
class. In the so-called stable regime, $N>\frac{p}{2}$, $\pi_{p}(GL(N,\mathbf{C}))$
satisfies the Bott periodicity\cite{Nakahara},
\[
\pi_{p}\left(GL(N,\mathbf{C})\right)\cong\begin{cases}
\mathbf{Z} & p\in odd\\
0 & p\in even
\end{cases}.
\]
Since physical systems are always in the stable regime, only the mappings
on $S^{p}$ for FSs with the odd codimension $p=2n+1$ can
be in a nontrivial homotopy class, while ones for the even codimension
$p$ are always trivial. In other words, FSs with the even
codimension $p$ are always trivial and vulnerable against weak perturbations,
while ones with odd codimension can be classified by $\pi_{p}(GL(N,\mathbf{C}))\cong\mathbf{Z}$
and its stability is topologically protected
against any weak perturbations for the nontrivial one. We emphasize that since the class
A is not subject to any symmetry, the perturbation can be quite
arbitrary. In addition, the concrete shape of the $S^{p}$ is irrelevant
in principle, and the only requirement is that it is an orientable
compact submanifold in $(\omega,\mathbf{k})$-space with dimension
$p$, which is distinctly different from the cases of reality symmetries. The
homotopy number of a mapping for an FS is named as the
topological charge of the FS, which can be calculated from
the following formula,
\begin{equation}
N_{p}=C_{p}\int_{S^{p}}\mathbf{tr}\left(G\mathbf{d}G^{-1}\right)^{p}\label{eq:homotopic number}
\end{equation}
with
\[
C_{p}=-\frac{n!}{(2n+1)!(2\pi i)^{n+1}}.
\]

The $\mathbf{Z}$-type of FSs in the class A is
thus obtained. Once an FS has nontrivially this topological
charge, it can survive under any weak perturbations
in the absence of any symmetry\cite{Hubbard}. But the topologically trivial FSs
can be gapped due to the perturbations. Both cases can be seen in
$^{3}He$\cite{Volovik-Book}, where two Fermi points in $^{3}He$-$A$ phase are topologically protected with charge $\pm2$ individually, while the Fermi line in the
planar phase cannot be topologically charged because its codimension is even. The FS of another simple model Hamiltonian $\mathcal{H}=\mathbf{g}(\mathbf{k})\cdot\mathbf{\sigma}$ belongs also to this topological class\cite{Supp}, where $\sigma_i$ are Pauli matrices.

We now turn to consider the case when the system has a chiral
symmetry denoted by $\mathcal{K}$, $i.e.$,
\[
\left\{ \mathcal{H},\mathcal{K}\right\} =0.
\]
Systems with this symmetry have a crucial property: for any eigenstate
$|\alpha\rangle$ of $H$ with the energy $E_{\alpha}$, $K|\alpha\rangle$
is also an eigenstate of $H$ but with the energy $-E_{\alpha}$. The
Hamiltonian associated with this symmetry can alway be diagonalized in $\mathbf{k}$-space
on the basis that diagonalizes $\mathcal{K}$ and thus can be written as
\[
h(\mathbf{k})=\begin{pmatrix} & u(\mathbf{k})\\
u^{\dagger}(\mathbf{k})
\end{pmatrix}.
\]
On the $S^{p}$ that is chosen to enclose an FS of dimension $d-p$,
$h(\mathbf{k})$ also takes this form, which makes the topological charge
defined in Eq(\ref{eq:homotopic number}) always vanishing. However
$u(\mathbf{k})$ can be regarded as a mapping from $S^{p-1}$(set
$\omega=0$) to $GL(N/2,\mathbf{C})$, which may be in a nontrivial
homotopy class of $\pi_{p-1}(GL(N/2),\mathbf{C})$. In this sense, there exist
nontrivial FSs for the even number of codimension, $i.e.$, $p=2n$. The
homotopy number may be calculated as
\[
\nu_{p}=C_{p-1}\int_{S^{p-1}}\mathbf{tr}\left(u(\mathbf{k})\mathbf{d}u^{-1}(\mathbf{k})\right)^{p-1}.
\]
As the homotopic number is real, it can also be calculated by $u^{\dagger}(\mathbf{k})$.
Thus the homotopic number can also be expressed by Green's function:
\begin{equation}
\nu_{p}=\frac{C_{p-1}}{2}\int_{S^{p-1}}\mathbf{tr}\left(\mathcal{K}\left(G\mathbf{d}G^{-1}\right)^{p-1}|_{\omega=0}\right).\label{eq:general chiral charge}
\end{equation}
The homotopic number is referred to as the chiral topological charge of
the FS.

For this $\mathbf{Z}$-type topological charge in the class AIII, if
it is nontrivial, the FS is stable
against any perturbations that do not break  the CS. However the
topological protection is not as stable as that induced by Eq.(\ref{eq:homotopic number}),
because  the FS is gapped if ever the CS is broken. For instance,   the honeycomb lattice model  has two Dirac cones with the low-energy effective Hamiltonian $\mathcal{H}_\alpha=k^{\pm}_x\sigma_1\pm k_y^{\pm}\sigma_2$, respectively, with  $\sigma_3$ representing the chiral symmetry that is the sublattice symmetry here \cite{Haldane model}. The two Dirac cones have $\nu_2=\pm1$ respectively\cite{Supp}, and thus they are topologically stable as far as  this chiral symmetry is preserved.  More remarkably, according to our present theory, it is noted that the stability of FSs in this model was recently  verified in an ultra-cold atom experiment\cite{Dirac Points}.  In addition, it was also seen clearly from this experiment that if the sublattice symmetry is broken, a gap may be opened in the whole Brillouin zone. It is noted that this type of topological charge was also seen in a superconducting system\cite{SConductor}. Another interesting model Hamiltonian $\mathcal{H}=(k^2_x-k^2_y)\sigma_1\pm2k_xk_y\sigma_2$ also gives out this type of topological charge $\nu_2=\pm2$ \cite{Supp}.

When TRS and PHS are considered, many $\mathbf{Z}$-type topological charges vanish. As for the first descendant ($i.e.\,\,\mathbf{Z}^{(1)}_2$) of a $\mathbf{Z}$-type  in a non-chiral real case ,  $e.g.$
 the case of codimension 2 that is labeled one row above that of codimension 3 ($\mathbf{Z}$-type) in the class AII in Tab.{[}\ref{tab:periodicity table}{]},
the Green's function restricted on the chosen $S^{p}$ can be classified by a $\mathbf{Z}_2$-type topological charge.
The key idea is that $G(\omega,\mathbf{k})|_{S^p}$ can be continuously extended
to $\tilde{G}(\omega,\mathbf{k};u)$ on the $(p+1)$-dimensional disk  by introducing
an auxiliary parameter $u$ (ranging from $0$ to $1$) with the two
requirements\label{sub: requirement I}: $i)$ $\tilde{G}(\omega,\mathbf{k};0)|_{S^{p}}=G(\omega,\mathbf{k})|_{S^{p}}$,
and $ii)$ $\tilde{G}(\omega,\mathbf{k};1)|_{S^{p}}$ is a diagonal
matrix, such that $\tilde{G}_{\alpha\alpha}=\left(i\omega-\Delta\right)^{-1}$for
empty bands and $\tilde{G}_{\beta\beta}=(i\omega+\Delta)^{-1}$ for
occupied bands, where $\Delta$ is a positive constant\cite{Order-parameter of TI,Xiao-Liang PRB}.
The validity of the extension is based on the fact that $G(\omega,\mathbf{k})$
restricted on $S^{p}$ is always trivial in the homotopic sense in
this case. This topological charge in terms of Green's function is
formulated as
\begin{equation}
N_{p}^{(1)}=C'_{p}\int_{S^{p}}\int_{0}^{1}du\;\mathbf{tr}\left(\left(G\mathbf{d}G^{-1}\right)^{p}G\partial_{u}G^{-1}\right)\mod\:2\label{eq:Z_2 son for non-chiral case}
\end{equation}
with
\[
C_{p}^{'}=-\frac{2(p/2)!}{p!(2\pi i)^{(p/2)+1}},
\]
where ``$\sim$'' has been dropped for brevity. The topological
charge is defined in a similar way to the WZW-term in quantum field theory\cite{WZW-term}
and the $\mathbf{Z}_{2}$ character comes from the fact that two different
extensions differ to an even integer, as demonstrated in Ref.\cite{Order-parameter of TI}. As a concrete illustration, we here exemplify this $\mathbf{Z}_2$-type topological charge by a model Hamiltonian defined in a two spatial dimension: $\mathcal{H}(\mathbf{k})=k_x\sigma_1+k_y\sigma_2+(k_x+k_y)\sigma_3$, which  has only a TRS with $C=i\sigma_2$ and $\eta=-1$ according to Eq.(\ref{eq:reality transformation}),  and thus belongs to the class AII in Tab.{[}\ref{tab:classification}{]} and corresponds to the case of $R(2,5)$ in Tab.{[}\ref{tab:periodicity table}{]}. The corresponding FS at the origin in the $(\omega,\mathbf{k})$-space
is found to have a nontrivial topological charge $N_{2}^{(1)}=1 $ from Eq.(\ref{eq:Z_2 son for non-chiral case}) \cite{Supp} .

For the second descendant with the codimension $p$ ($i.e.\,\,\mathbf{Z}^{(2)}_2$), $e.g.$ the case
of codimension 3 with two rows above that of codimension 5 in the class C in Tab.{[}\ref{tab:periodicity table}{]},
 a $\mathbf{Z}_{2}$-type topological charge can also be defined on the
chosen $S^{p}$. To define the $\mathbf{Z}_{2}$-type topological charge,
the $G(\omega,\mathbf{k})|_{S^{p}}$ is smoothly extended to a two-dimensional
torus $T^{2}$ parameterized by the two auxiliary parameters $u$
and $v$ (both ranging from $-1$ to $1$ ) with the three requirements:
$i)$ $\tilde{G}(\omega,\mathbf{k};0,0)|_{S^{p}}=G(\omega,\mathbf{k})|_{S^{p}}$; and
$ii)$ $\tilde{G}(\omega,\mathbf{k};u,v)|_{S^{p}}=\epsilon_{c}C\tilde{G}^{T}(\omega,-\mathbf{k};-u,-v)|_{S^{p}}C^{-1}$,
referring to Eq.(\ref{eq:reality transformation}); $iii)$$\tilde{G}(\omega,\mathbf{k};1,1)|_{S^{p}}$
corresponds to a trivial system, such as $\tilde{G}_{\alpha\alpha}=\left(i\omega-\Delta\right)^{-1}$for
empty bands and $\tilde{G}_{\beta\beta}=(i\omega+\Delta)^{-1}$ for
occupied bands. The corresponding topological
charge is defined as\cite{Xiao-Liang PRB,Order-parameter of TI}
\begin{equation}
N_{p}^{(2)}=C{}_{p+2}\int_{S^{p}\times T^{2}}\:\mathbf{tr}\left(G\mathbf{d}G^{-1}\right)^{p+2}\mod2,\label{eq:Z_2 grandson fro non-chiral case}
\end{equation}
where ``$\sim$'' has been dropped and
the $C_{p+2}$ is defined in Eq.(\ref{eq:homotopic number}). As a topological charge of  FSs, its physical meaning is analogous to that
of Eq.(\ref{eq:Z_2 son for non-chiral case}).

Similar to the non-chiral case, a $\mathbf{Z}$ chiral topological
charge is associated with two descendants: son and grandson. The $\mathbf{Z_{2}}$
topological charge of the son is originated from the chiral topological
charge defined in Eq.(\ref{eq:general chiral charge}) in the same
spirit of that Eq.(\ref{eq:Z_2 son for non-chiral case}) is originated from
Eq.(\ref{eq:homotopic number}), which is given by
\begin{equation}
\begin{split}
\nu_{p}^{(1)}=&\frac{C'_{p-1}}{2}\int_{S^{p-1}}\int_{0}^{1}du\;\\
&\mathbf{tr}\left(\mathcal{K}\left(G\mathbf{d}G^{-1}\right)^{p-1}G\partial_{u}G^{-1}|_{\omega=0}\right)\mod\:2,\label{eq:Chiral son charge}
\end{split}
\end{equation}
where $\mathcal{K}$ is the matrix to represent the chiral symmetry,
$S^{p-1}$ is the chosen $S^{p}$ restricted on $\omega=0$, and $G$
is extended by an auxiliary parameter ranging from $0$ to $1$ with
the same requirements for introducing Eq.(\ref{eq:Z_2 son for non-chiral case}). The $\mathbf{Z}_{2}$ topological charge for the grandson can also
be defined in the same spirit of writing out Eq.(\ref{eq:Z_2 grandson fro non-chiral case}).
We extend the $G(\omega,\mathbf{k})|_{S^{p}}$ to a two-dimensional
torus $T^{2}$ parameterized by two auxiliary parameters $u$ and
$v$ (both ranging from $-1$ to $1$) with the four requirements:
the first three requirements are the same as those of the non-chiral
counterpart in Eq.(\ref{eq:Z_2 grandson fro non-chiral case}); while the
fourth one is that the chiral symmetry is preserved on $T^{2}$, which
also permits that either of reality symmetries is applicable in the second
requirement. The $\mathbf{Z}_{2}$ topological charge is written as
\begin{equation}
\nu_{p}^{(2)}=\frac{C_{p+1}}{2}\int_{S^{p-1}\times T^{2}}\:\mathbf{tr}\left(\mathcal{K}\left(G\mathbf{d}G^{-1}\right)^{p+1}|_{\omega=0}\right)\mod2,\label{eq:chiral grandson charge}
\end{equation}
where $C_{p+1}$ is defined in Eq.(\ref{eq:homotopic number}).

The physical meanings of topological charges for the eight classes with TRS and/or PHS are elaborated as
follows. The FS(s) with a certain codimension is always distributed centrosymmetrically, since either TRS or PHS relates  $\mathbf{k}$ to $-\mathbf{k}$. Thus there exist two possibilities:  $i)$ the FS with codimension $p$ resides at the origin in $(\omega,\mathbf{k})$-space; $ii)$the FS(s) is centrosymmetric outside of the origin.  For the first case, we can choose an $S^p$ in $(\omega,\mathbf{k})$-space to enclose the FS, and use the corresponding formula to calculate the topological charge. If the topological charge is nontrivial, the FS is stable against perturbations provided the corresponding symmetries are preserved.
For the second case, the FS(s) is usually spherically distributed, so two $S^{p}$s can be chosen
to sandwich the FS(s) in its transverse dimension. The difference of the topological charges calculated from the two $S^p$s is the topological charge of the FS(s). If it is nontrivial, the FS(s) is topologically stable when the corresponding symmetries are preserved.

Before concluding this paper, we wish to emphasize that the topological charges of FSs addressed here are closely connected to topological insulators/superconductors\cite{Zhao1}, and therefore this work may provide a new and deep insight for studying them.

To conclude, FSs
in all of the ten symmetry
classes have been classified appropriately in terms of topological charges.
It has been revealed that when an FS is associated with a nontrivial topological charge,  this FS is topologically protected
by the corresponding symmetry.

\begin{acknowledgments}
We thank G. E. Volovik for helpful discussions. This work was supported
by the GRF (HKU7058/11P),  the CRF (HKU8/11G) of Hong Kong, the URC fund of HKU, and
the SKPBR of China (Grant No. 2011CB922104).
\end{acknowledgments}