A new approch to the Ashtâdhyâyî

The goal of this approach is to be able to study rigorously the computational and logical power of Pâninian formal grammar. 

The basis $B$ (representing phonetic atoms) is generated from certain finite sets $I_i$ by taking finite cartesian products and finite ordered disjoint unions.  The sets can be seen as including dualities and oppositions as well as other kinds of 'scales'. Thus $I_1$ might contain two elements representing the opposition between voiced and non-voiced, $I_2$, representing aspirated vs. non-aspirated, $I_3$ represents the place of articulation so that part of the consonants would be represented by $I_1 \times I_2 \times I_3$ (this is obviously similar to some aspects of the mathematical formulation of field theory or particle physics). To include the nasal consonants we take the ordered disjoint union with $I_3$, perhaps like $(I_1\times I_2 \times I_3, 0) \cup (I_3,1)$. We treat the sibilants and the voiced h similarly.

We identify $W = B^\star$ with general expressions (which include padas in particular, actually grammatically correct words). Dhatus will be a subset $D \subset W$. 

Pânini defines a recursive subset of $P \subset W$ which are not actual words or suffixes but technical expressions (pratyayas) used to define and carry out the rules of his generative grammar. Due to the particularities of Sanskrit and the combinatory capacities of $B$ this is possible in an elegant way (for instance he makes use of nasalized vowels).

We will work with  one-sided sequents which are elements of $W^*$. Later on we will work with two-sided and more complex sequents which take into account the history of a derivation.

Some rules (vidhis) can be formulated for instance as:

$\Gamma, wa, bw', \Gamma' \longrightarrow \Gamma, wc, dw', \Gamma' $

$\Gamma, w, w', \Gamma' \longrightarrow \Gamma, v, w', \Gamma'$ 

and in the phonetic part of the proviso we use instead of pratyahara,  a conjunction of conditions involving projections (for instance $\pi_2 a = x$).  There will also be more provisos depending on the class of wa, bw' and even following expressions.

The rule above contains in particular the case in which $b = d$ or some of the $a,b,c,d$ are empty ($\epsilon$, zero, adarshanam). 

Another type of rule would be:

$\Gamma, w, \Gamma' \longrightarrow \Gamma, w, s, \Gamma'$ for $w \in D$ and $s$ some concrete suffix (like lat or tip),

$\Gamma, w, w', \Gamma' \longrightarrow \Gamma, w, s, w' ,\Gamma'$. Also: 

 $\Gamma, w, \Gamma' \longrightarrow \Gamma, w', \Gamma'$ for $w$ belonging to a particular set in $W$ and $w'$ resulting from $w$ by deletion of certain well-defined elements in $B$ (the tasya lopah rule).