\documentclass{beamer}[8pt]
\usepackage{beamerthemesplit}
\usetheme{Warsaw}
%\usepackage{verbatim}
\usepackage{pdfpages}
\usepackage{color}
\usepackage{graphicx}
\usepackage{fontspec}
\setmainfont[Script=Devanagari]{Sanskrit 2003}

\newfontfamily \dv[Script=Devanagari]{Sanskrit 2003}

\title{\textcolor{yellow}{\'{S}ivas{\=u}tra}}
\author{Amba Kulkarni}
\institute{ Fellow IIAS, Shimla}
\date{1st April 2017}

\begin{document}
\setbeamertemplate{footline}[frame number]
%===================================slide 1=========================%
\begin{frame}

\titlepage

\end{frame}
%============================================================%
\begin{frame}
\frametitle{{\it \'{s}ivas\=utr{\=a}\d{n}i}}
Normal Arrangement of Alphabet \\
\vspace{0.5cm}
\begin{tabular}{l l}
\hline
\color{magenta}Vowels \color{black}& {\it a \={a} i \={\i} u \={u} \d{r}  \d{\={r}} \d{l} e ai o au a\d{m} a\d{h} }\\
\hline \hline
\end{tabular}\\
\begin{tabular}{l l l l l l}
\color{magenta}Velar {\dv कण्ठ्य}  \color{black}& {\it k }& {\it kh }& {\it g }& {\it gh }& {\it \.{n} }\\
\color{magenta}Palatal {\dv तालव्य}  \color{black}& {\it c }& {\it ch }& {\it j }& {\it jh }& {\it \~{n} }\\
\color{magenta}Retroflex {\dv मूर्धन्य}  \color{black}& {\it \d{t} }& {\it \d{t}h }& {\it \d{d} }& {\it \d{d}h }& {\it \d{n}}\\
\color{magenta}Dental {\dv दन्त्य}  \color{black}& {\it t }& {\it th }& {\it d }& {\it dh }& {\it n}\\
\color{magenta}Labiel {\dv औष्ट्य}  \color{black}& {\it p }& {\it ph }& {\it b }& {\it bh }& {\it m}\\
\hline \hline 
\end{tabular}\\
\begin{tabular}{l l}
\color{magenta}semi-vowel  {\dv अन्तस्थ} \color{black}& {\it y r l v }\\
\color{magenta}Fricative {\dv उष्म}  \color{black}& {\it \'{s} \d{s} s h }\\
\hline
\end{tabular}
\end{frame}

%============================================================%
\begin{frame}
\frametitle{ \'{s}ivas\=utr{\=a}\d{n}i}
P{\=a}\d{n}ini required several(41-44) subsets of this
alphabet to describe various operations.
Some of these subsets are: \\
\vspace{0.5cm}
	All vowels \\
	All consonants \\
	All vowels + semivowels + nasals \\
	The voiceless stops  \\
	stops and sibilants \\
	Any of {\it ba ga \d{d}a da} \\
        Any of {\it bha gha \d{d}ha dha} \\
        Any of {\it ya va ra la \~na ma \.na \d{n}a na ma jha bha} \\ ....
\end{frame}

%============================================================%
\begin{frame}
\frametitle{praty{\=a}h{\=a}ra}
41 sets Panini needed are as follows
\begin{itemize}
\item a\d{N} = \{a i u\}\\
{\dv अ इ उ}
\item aK = \{a i u \d{r} \d{l}\}\\
{\dv अ इ उ ऋ  ऌ }
\item aC = \{a i u \d{r} \d{l} e o ai au\}\\
{\dv अ इ उ ऋ ऌ ए ओ  ऐ औ }
\item a\d{T} = \{a i u \d{r} \d{l} e o ai au ha ya va ra\}\\
{\dv अ इ उ ऋ ऌ ए ओ  ऐ औ  ह य व र}
\item a\d{N} = \{a i u \d{r} \d{l} e o ai au ha ya va ra la\}\\
{\dv अ इ उ ऋ ऌ ए ओ  ऐ औ  ह य व र ल }
\item aM = \{a i u \d{r} \d{l} e o ai au ha ya va ra la \~na ma \.na \d{n}a na\}\\
{\dv अ इ उ ऋ ऌ ए ओ  ऐ औ  ह य व र ल ञ म ङ ण न }
\item a\'{S} = \{a i u \d{r} \d{l} e o ai au ha ya va ra la \~na ma \.na \d{n}a na jha bha gha \d{d}ha dha\}\\
{\dv अ इ उ ऋ ऌ ए ओ  ऐ औ  ह य व र ल ञ म ङ ण न झ भ घ ढ ध}
\end{itemize}
\end{frame}
%============================================================%
\begin{frame}
\begin{itemize}
\item aL = \{a i u \d{r} \d{l} e o ai au ha ya va ra la \~na ma \.na \d{n}a na jha bha gha \d{d}ha dha ja ba ga \d{d}a da kha pha cha \d{t}ha tha ca \d{t}a ta ka pa \'{s}a \d{s}a sa\}\\
{\dv अ इ उ ऋ ऌ ए ओ  ऐ औ  ह य व र ल ञ म ङ ण न झ भ घ ढ ध ज ब ग ड द ख फ छ ठ थ च ट त क प श ष स}
%\item # Last ha is not written, since it is already exists in the set

\item iK = \{i u \d{r} \d{l}\}\\
{\dv इ उ ऋ ऌ }
\item iC = \{i u \d{r} \d{l} e o ai au\}\\
{\dv इ उ ऋ ऌ ए ओ  ऐ औ }
\item i\d{N} = \{i u \d{r} \d{l} e o ai au ha ya va ra la\}\\
{\dv इ उ ऋ ऌ ए ओ  ऐ औ  ह य व र ल }

\item uK = \{u \d{r} \d{l}\}\\
{\dv उ ऋ ऌ}

\end{itemize}
\end{frame}
%============================================================%
\begin{frame}
\begin{itemize}
\item e\.{N} = \{e o\}\\
{\dv ए ओ }
\item eC = \{e o ai au\}\\
{\dv ए ओ  ऐ औ }

\item ha\'{S} = \{ha ya va ra la \~na ma \.na \d{n}a na jha bha gha \d{d}ha dha ja ba ga \d{d}a da\}\\
{\dv ह य व र ल ञ म ङ ण न झ भ घ ढ ध ज ब ग ड द }
\item haL = \{ha ya va ra la \~na ma \.na \d{n}a na jha bha gha \d{d}ha dha ja ba ga \d{d}a da kha pha cha \d{t}ha tha ca \d{t}a ta ka pa \'{s}a \d{s}a sa\}\\
{\dv ह य व र ल ञ म ङ ण न झ भ घ ढ ध ज ब ग ड द ख फ छ ठ थ च ट त क प श ष स }
%\item # Last ha is not written, since it is already exists in the set

\item ya\d{N} = \{ya va ra la\}\\
{\dv य व र ल}
\item yaM = \{ya va ra la \~na ma \.na \d{n}a na\}\\
{\dv य व र ल ञ म ङ ण न}
\item ya\~{N} = \{ya va ra la \~na ma \.na \d{n}a na jha bha\}\\
{\dv य व र ल ञ म ङ ण न झ भ}
\end{itemize}
\end{frame}
%============================================================%
\begin{frame}
\begin{itemize}
\item yaY = \{ya va ra la \~na ma \.na \d{n}a na jha bha gha \d{d}ha dha ja ba ga \d{d}a da kha pha cha \d{t}ha tha ca \d{t}a ta ka pa\}\\
{\dv य व र ल ञ म ङ ण न झ भ घ ढ ध ज ब ग ड द ख फ छ ठ थ च ट त क प}
\item yaR = \{ya va ra la \~na ma \.na \d{n}a na jha bha gha \d{d}ha dha ja ba ga \d{d}a da kha pha cha \d{t}ha tha ca \d{t}a ta ka pa \'{s}a \d{s}a sa\}\\
{\dv य व र ल ञ म ङ ण न झ भ घ ढ ध ज ब ग ड द ख फ छ ठ थ च ट त क प श ष स}

\item va\'{S} = \{va ra la \~na ma \.na \d{n}a na jha bha gha \d{d}ha dha ja ba ga \d{d}a da\}\\
{\dv व र ल ञ म ङ ण न झ भ घ ढ ध ज ब ग ड द}

\item vaL = \{va ra la \~na ma \.na \d{n}a na jha bha gha \d{d}ha dha ja ba ga \d{d}a da kha pha cha \d{t}ha tha ca \d{t}a ta ka pa \'{s}a \d{s}a sa ha\}\\
{\dv य व र ल ञ म ङ ण न झ भ घ ढ ध ज ब ग ड द ख फ छ ठ थ च ट त क प श ष स ह}

\end{itemize}
\end{frame}
%============================================================%
\begin{frame}
\begin{itemize}
\item raL = \{ra la \~na ma \.na \d{n}a na jha bha gha \d{d}ha dha ja ba ga \d{d}a da kha pha cha \d{t}ha tha ca \d{t}a ta ka pa \'{s}a \d{s}a sa ha\}\\
{\dv र ल ञ म ङ ण न झ भ घ ढ ध ज ब ग ड द ख फ छ ठ थ च ट त क प श ष स ह}

\item maY = \{ma \.na \d{n}a na jha bha gha \d{d}ha dha ja ba ga \d{d}a da kha pha cha \d{t}ha tha ca \d{t}a ta ka pa\}\\
{\dv म ङ ण न झ भ घ ढ ध ज ब ग ड द ख फ छ ठ थ च ट त क प}

\item \.naM = \{\.na \d{n}a na\}\\
{\dv ङ ण न}

\item jhaR = \{jha bha gha \d{d}ha dha\}\\
{\dv झ भ घ ढ ध}
\item jhaS = \{jha bha gha \d{d}ha dha ja ba ga \d{d}a da\}\\
{\dv झ भ घ ढ ध ज ब ग ड द}
\end{itemize}
\end{frame}
%============================================================%
\begin{frame}
\begin{itemize}
\item jhaY = \{jha bha gha \d{d}ha dha ja ba ga \d{d}a da kha pha cha \d{t}ha tha ca \d{t}a ta ka pa\}\\
{\dv झ भ घ ढ ध ज ब ग ड द ख फ छ ठ थ च ट त क प}
\item jhaR = \{jha bha gha \d{d}ha dha ja ba ga \d{d}a da kha pha cha \d{t}ha tha ca \d{t}a ta ka pa \'{s}a \d{s}a sa\}\\
{\dv झ भ घ ढ ध ज ब ग ड द ख फ छ ठ थ च ट त क प श ष स}
\item jhaL = \{jha bha gha \d{dh}a dha ja ba ga \d{d}a da kha pha cha \d{t}ha tha ca \d{t}a ta ka pa \'{s}a \d{s}a sa ha\}\\
{\dv झ भ घ ढ ध ज ब ग ड द ख फ छ ठ थ च ट त क प श ष स}

\item bhaR = \{bha gha \d{d}ha dha\}\\
{\dv भ घ ढ ध }

\end{itemize}
\end{frame}
%============================================================%
\begin{frame}
\begin{itemize}
\item jaS = {ja ba ga \d{d}a da}\\
{\dv ज ब ग ड द}

\item baS = {ba ga \d{d}a da}\\
{\dv ब ग ड द}

\item CaV = {cha \d{t}ha tha ca \d{t}a ta}\\
{\dv छ ठ थ च ट }

\item KaY = {kha pha cha \d{t}ha tha ca \d{t}a ta ka pa}\\
{\dv छ ठ थ च ट त क प}
\item KaR = {kha pha cha \d{t}ha tha ca \d{t}a ta ka pa \'{s}a \d{s}a sa}\\
{\dv ख फ छ ठ थ च ट त क प श ष स}

\item caR = {ca \d{t}a ta ka pa \'{s}a \d{s}a sa}\\
{\dv च ट त क प श ष स}

\item SaR = {\'{s}a \d{s}a sa}\\
{\dv श ष स}
\item Sal = {\'{s}a \d{s}a sa ha}\\
{\dv श ष स  ह }
\end{itemize}
\end{frame}
%============================================================%
\begin{frame}
Panini organised these partially overlapping sets in a linear order, and devised a naming scheme to remember the members of each set.
\end{frame}
%============================================================%
\begin{frame}
\frametitle{Simple Example}
Let me take a small example with 6 members instead of 41, and explain what P{\=a}\d{n}ini did. \\
\vspace{0.2cm}
In a certain class in a school there are 6 students who participate in various games. Let us name them from `a' to `f'.\\
The games they participate are as under.\\
\begin{itemize}
\item Cricket: b, c
\item Tennis: d, e, f
\item Football: a, d, f
\item Chess: b, e, f
\end{itemize}
\vspace{0.2cm}
%As is the case with all students, these students also like to gossip and
%chat whenever they get an opportunity. Hence 
Students playing the same games want to sit next to each other.
Now if these students sit in an alphabetic order then student `b' can
not talk with `e' on chess, since he would be intervened by two other
students `c' and `d'. Similarly `a' will not be in a position to talk to
his other friends `d' and `f' on football. \\
\vspace{0.2cm}
In what order should these students sit so that there are no cross-talks.
\end{frame}
%============================================================%
\begin{frame}
\frametitle{Simple Example ..}
\begin{itemize}
\item Cricket: b, c
\item Tennis: d, e, f
\item Football: a, d, f
\item Chess: b, e, f
\end{itemize}
\vspace{0.2cm}
If they are seated in the following order\\
\begin{center}
c b e f d a
\end{center}
then everybody is happy. All the players having common interests are
adjacent to each other.

\vspace{0.2cm}
Can we have some acronyms so that we can also remember which students play which games?
\end{frame}
%============================================================%
\begin{frame}
\frametitle{Simple Example ..}
\begin{itemize}
\item Cricket: b, c
\item Tennis: d, e, f
\item Football: a, d, f
\item Chess: b, e, f
\end{itemize}
\vspace{0.2cm}
We introduce markers $M_1$, $M_2$, $M_3$ and $M_4$ indicating the boundary for Cricket, Chess, Tennis and Football players respectively.
\begin{center}
c b $M_1$ e f $M_2$ d $M_3$ a $M_4$
\end{center}
The slice starting with `c' and ending with $M_1$ gives us the names of
players playing cricket. Let us name this slice by `c$M_1$'. The slice `b$M_2$'
gives the names of Chess players, the slice `e$M_3$' gives the names of
Tennis players, and finally `f$M_4$' gives the names of football players.

\vspace{0.2cm}
Such a naming scheme, which is a sort of abbreviation
is called a \textbf{praty{\=a}h{\=a}ra}.
\vspace{0.2cm}
In order to know the members of a set or to know the name of a set
given the elements, one needs to memorise this linear list. That's
all! \\
\end{frame}
%============================================================%
\begin{frame}
\frametitle{Simple Example ..}
This is how P{\=a}\d{n}ini solved his problem.
He arranged 41 Sanskrit sounds in a linear order
and inserted some markers (called {\sl anubandha}s) or {\sl it} markers, in between that
serve as the right boundary. P{\=a}\d{n}ini needed 14 markers in order to describe all 41 subsets.
\end{frame}
%============================================================%
\begin{frame}
How did Panini arrived at this \'{s}ivas{\=u}tra?\\
\vspace{0.2cm}
Tradition: Legend\\
\vspace{0.2cm}
In recent years there are several efforts to justify this arrangement\\
\vspace{0.2cm}
Historically (Cardona)\\
Logically (Kiparsky, Staal)\\
Mathematically (Petersen).\\
\end{frame}
%============================================================%
\begin{frame}
\frametitle{ Logical Approach}
Logical approach to build the \'{s}ivas{\=u}tra\\
\vspace{0.5cm}
From the given sets, we try to arrange the sets in a linear order with partial overlaps.\\
Let us try to decide the order from the 41 sets Panini needed.\\
Consider the following sets.\\
\vspace{0.2cm}
A = \{a i u \d{r} \d{l}\}\\
B = \{i u \d{r} \d{l}\}\\
C = \{u \d{r} \d{l}\}\\
D = \{a i u\}\\
\vspace{0.2cm}
This fixes the order as\\
a i u $M_1$ \d{r} \d{l} $M_2$\\
\end{frame}
%============================================================%
\begin{frame}
\frametitle{ Logical Approach ...}
Suppose, in addition to \\
A = \{a i u \d{r} \d{l}\}\\
B = \{i u \d{r} \d{l}\}\\
C = \{u \d{r} \d{l}\}\\
D = \{a i u\}\\
\vspace{0.1cm}
we also have\\
\vspace{0.1cm}
E = \{a i u \d{r} \d{l} e o ai au\}\\
F = \{  i u \d{r} \d{l} e o ai au\}\\
G = \{          e o ai au\}\\
H = \{              ai au\}\\
I = \{          e o\}\\
\vspace{0.1cm}
This fixes the order:\\
a i u $M_1$ \d{r} \d{l} $M_2$ \{e o\} $M_3$ \{ai au\} $M_4$, or as\\
a i u $M_1$ \d{r} \d{l} $M_2$ \{ai au\} $M_4$ \{e o\} $M_3$\\
\vspace{0.1cm}
or in short as\\
a i u $M_1$ \d{r} \d{l} $M_2$ \{(\{e o\} $M_3$) (\{ai au\} $M_4$)\}\\
where order of elements within{} is not fixed.
\end{frame}
%============================================================%
\begin{frame}
\frametitle{ Logical Approach ...}
Now suppose, we consider the following sets\\
J = \{y v r l\}\\
K = \{a i u \d{r} \d{l} e o ai au h y v r l\}\\
L = \{  i u \d{r} \d{l} e o ai au h y v r l\}\\
M = \{a i u \d{r} \d{l} e o ai au h y v r\}\\

\vspace{0.2cm}
This fixes the further order:\\
        h \{y v r\} l
order within {} is not fixed.\\
\vspace{0.2cm}
Thus so far we have the following order.\\
a i u $M_1$ \d{r} \d{l} $M_2$  \{( \{e o \} $M_3$) ( \{ai au \} $M_4$) \} h  \{ y v r  \} l $M_5$\\

\end{frame}
%============================================================%
\begin{frame}
\frametitle{ Logical Approach ...}
Now consider following sets\\
N =  \{h y v r l ....  \}\\
O =  \{ y v r l ....  \}\\
P =  \{   v r l .... \}\\
Q =  \{     r l .... \}\\

\vspace{0.2cm}
This fixes the order of y,v,r also, as\\
        h y v r l\\

\vspace{0.2cm}
a i u $M_1$ \d{r} \d{l} $M_2$ \{(\{e o\} $M_3$) (\{ai au\} $M_4$)\} h  y v r l $M_5$\\
\end{frame}
%============================================================%
\begin{frame}
\frametitle{ \'{s}ivas\=utr{\=a}\d{n}i}
Following this process, we finally get the following order.\\
\begin{columns}
\begin{column}{0.1\textwidth}
1\\
2\\
3\\
4\\
5\\
6\\
7\\
8\\
9\\
10\\
11\\
12\\
13\\
14\\
\end{column}
\begin{column}{0.9\textwidth}
\begin{center}
\color{blue}{\it a i u \color{red}(\d{N})}\\
\color{blue}{\it \d{r} \d{l}\color{red}(K)}\\
\color{blue}{\it e o \color{red}(\.N) }\\
\color{blue}{\it ai au \color{red}(C) }\\
\color{blue}{\it ha ya va ra\color{red} (\d{T}) }\\
\color{blue}{\it la \color{red}(\d{N})}\\
\color{blue}{\it \~na ma \.na \d{n}a na \color{red}(M) }\\
\color{blue}{\it jha bha\color{red}(\~N) }\\
\color{blue}{\it gha \d{d}ha dha \color{red}(\d{S})}\\
\color{blue}{\it ja ba ga \d{d}a da \color{red}(\'{S}) }\\
\color{blue}{\it kha pha cha \d{t}ha tha ca \d{t}a ta \color{red}(V)}\\
\color{blue}{\it ka pa \color{red}(Y)}\\
\color{blue}{\it \'{sa} \d{sa} s \color{red}(R)}\\
\color{blue}{\it ha \color{red}(L)}\\
\end{center}
\end{column}
\end{columns}
There are 14 s{\=u}tras, which are also called praty{\=a}h{\=a}ra s{\=u}tras
\end{frame}
%============================================================%
\begin{frame}
\frametitle{ \'{s}ivas\=utr{\=a}\d{n}i}
It is not advisable to give 41 names to these sets.  \\
It will be difficult to memorize the association. \\
\vspace{1cm}
Panini devised a special scheme for naming these sets.\\
{\=a}di\d{h} antyena sahit{\=a} (1.1.71)\\
{\dv आदिरन्त्येन सहेता}\\
An initial letter with an `it' letter as a final, is the name of itself and of the intervening letters.\\
According to `hal antyam', final `hal' is an `it'.\\
Thus in the above arrangement, all the letters marked in red color are `it's.\\
\end{frame}
%============================================================%
\begin{frame}
The naming of the sets now is as follows.\\
The name consists of two letters: {\=a}di (the first one) and the antya (the last one) which is an `it'.\\
\begin{columns}
\begin{column}{0.1\textwidth}
1\\
2\\
3\\
4\\
5\\
\end{column}
\begin{column}{0.9\textwidth}
\begin{center}
{\it \color{yellow}a \color{blue}i u \color{red}(\d{N})}\\
\color{blue}{\it \d{r} \d{l}\color{red}(K)}\\
\color{blue}{\it e o \color{red}(\.N) }\\
\color{blue}{\it ai au \color{yellow}(C) }\\
\color{blue}{\it ha ya va ra\color{red} (\d{T}) }\\
\end{center}
\end{column}
\end{columns}
For example \textbf{aC} represents all the letters that occur between (and including) `a' and the `it' marker `C', i.e. the set \{a i u \d{r} \d{l} e o ai au\}\\

With this arrangement now it is trivial to associate the set names with its members.\\
\end{frame}
%============================================================%
\begin{frame}
\frametitle{Praty{\=a}h{\=a}ra}s
A Praty{\=a}h{\=a}ra consists of a phoneme-letter (var\d{n}a) and an anubandha (or a marker) and signify all of the intervening phonemes (more than one)  from the \'{s}ivas{\=u}tra, excluding the anubandha.\\
\end{frame}
%============================================================%
\begin{frame}
With 14 markers, how many Praty{\=a}h{\=a}ras are possible with the 14 \'{s}ivas{\=u}tras?\\
There are 3 var\d{n}as in the first s{\=u}tra viz. a,i,u. And there are 14 markers after these. So there are 14*3 possible praty{\=a}h{\=a}ras. Similarly in the second s{\=u}tra, there are 2 var\d{n}as, and 13 markers following it. So there are total 2*13 possible praty{\=a}h{\=a}ras. Continuing in this way, we have\\
14*3 + 13*2 + 12*2 + 11*2 + 10*4 + 9*1 + 8*5 + 7*2 + 6*3 * 5*5 + 4*8 + 3*2 + 2*3 +1*1\\
possibilities. But of these, all praty{\=a}h{\=a}ras with single phoneme are to be deleted. Further, due to the duplication of `h', the sets aR =aL, iR = iL, and so on. These are 10 in number with a,i,u,\d{r},\d{l},e,o,ai,au and h combining with either `R' or `L'. Thus there are 305-14-10 = 281 praty{\=a}h{\=a}ras possible. But P{\=a}\d{n}ini needed only 41-44 of them.
\end{frame}
%============================================================%
\end{document}