Pâninian Linguistics

https://web.stanford.edu/class/linguist289/encyclopaedia001.pdf

Is Pânini's formal system (which appears to be in the form of a term-rewriting system and is the true source of generative grammar) Turing complete ?  That is to say, could the same formal mechanism Pânini used to specify the structure of Sanskrit be used to generate the expressions of any recursive set of strings ?

Here is a paper addressing this question:

https://aclanthology.org/C12-2092.pdf

This would make Pânini of supreme logical and philosophical interest. There might be a connection to Vedic rituals: the development of a formal language (a programing language) to codify rituals.

https://ijirt.org/publishedpaper/IJIRT187010_PAPER.pdf

We note that the syntactic transformation of our natural term logic can be expressed more easily as a term-rewriting system than standard quantifier logics.

In fact Pânini is of immense interest to our positions expounded in our paper on Analyticity and the A Priori (and also our paper on Ancient Quantifier Logic).   One view is that true logic (let us say in the form of second-order logic) comprises only universal quantification and implication and their natural deduction rules (cf. Prawitz, Natural Deduction, p.67, Dover edition). This is the logic which can be transcendentally examined to be the pre-condition to understand, check and apply rules, formal term-rewriting systems (or similar systems), such as that of Pânini. We must also examine Pânini as a logic of relations.

But derivations in formal languages are like proofs and are instantiations of rules (a derivation is called a prakriyâ in Pânini's system).  So even the introduction of the universal quantifier is itself the application of the rule of the elimination of the universal quantifier. Pânini's system is a sophisticated system of formal rule-based proofs which is close to the sequent calculus in that memory of past rule applications carries conditions the application of rules later on in the derivation (as in control structures or focusing).