Panini's grammar is said to have algebraic rules governing every aspect of the Sanskrit language. If the rules are completely formal, what is the place of this grammar in the Chomsky hierarchy? How come a natural and complex language can have a formal grammar? What makes Sanskrit exhibit this rare (unique?) feature among natural languages?
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1In fact, Sanskrit is not a completely natural language, it was artificially refined and formalized, which is reflected in its name itself: संस्कृत saṃskṛtá “perfected, prepared, constructed, refined” — सम्- (sam-, “together, wholly”) + स्कृ (skṛ, “to do”) + -त (-ta, “-ed”) — which is actually well reflected as con-struct-ed.– Yellow SkyNov 6, 2021 at 1:13
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@YellowSky Thank you for pointing this out. It is still remarkable that a language which was at some point natural, has been refined and formalized. My question still makes sense: either Sanskrit has a place in Chomsky hierarchy (which one?) or it is completely "perpendicular" to this hierarchy.– Gratiela Monica MarcusNov 6, 2021 at 6:23
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Does Panini grammar allow neologisms? If yes, then it does not, mathematically speaking, define a formal language. In that case need first to first define a subset that is formal, and then check if it requires an unrestricted recursion to place it on Chomsky scale– J-msterNov 6, 2021 at 10:58
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@J-mster What is the issue with neologisms? If a word is borrowed from another language it still needs to be written using letters from the original alphabet.– Gratiela Monica MarcusNov 6, 2021 at 13:02
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@GratielaMonicaMarcus Neologisms change language. In a formal language it should be known whether a string is a valid sentence or not, no new words can be borrowed (unless we from the beginning have a pool of strings reserved for new words, with their grammatical properties fixed, but meaning totally undefined)– J-msterNov 6, 2021 at 13:06
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2 Answers
This page explains the concept of "regular expression". Also note this discussion of "regex" features available in e.g. Word's regex search and replace function, which are beyond what are defined for "regular expressions". For example "a string beginning with any of [aeiou] which is terminated by the same letter", which is beyond what constitutes a regular expression. A "regular grammar" uses only regular expressions. For the sake of clarity, a "regular language" would be a language that can be generated using only regular expressions. Another kind of expression is "unrestricted rewrite rule" or "Turing machine", which can do anything that can be called a computer algorithm.
Panini's grammar is available here, with links to the individual chapters. What is provided there is an augmented version of the Astadhyayi, an English translation with commentary. The first rule say "vṛddhirādaich", which is a rule defining [ā au ai] as having the property "vṛddhi". And so forth. Rule 6.1.77 says "ikoyaṇaci", which translates to "of [i u ṛ ḷ ] [y v r l] before vowels". The rules of the grammar must also be interpreted by rules of the grammar to make sense – in this case, the rules omit some things that can be supplied by rule, and a more verbose version would be "in the place of [i u ṛ ḷ ] there is, respectively, [y v r l] before vowels, in connected speech". The Paninian system is very complicated, and exceeds what constitutes a system of regular expressions, possibly going beyond finite state grammar (see this article). While it is possible to model "respectively" in a system of regular expressions, [a,b,c...e] → [A,B,C...E] respectively goes beyond regular expressions in the Chomsky hierarchy sense. You should also note that the Chomsky hierarchy is based on a particular set of mathematical principles that encompass everything that can be called a computation. Wherever the Ashtadhyayi sits on this hierarchy, there will be devices that do more than Panini's grammar does, so there is nothing in the grammar/device taxonomy that exactly describes Panini's grammar.
The popular language-learning notion of "regular" is completely different, and seems to be more based on learning a "base form" such as the infinitive or nominative, then applying simple rules like "replace -ein with -o: to form the 1st singular". Linguistic grammar take a completely different view, relying on abstract underlying forms and sets of rules that don't just say how to convert one surface form to another form. Panini's rules are of the same abstract type as are found in linguistic grammars.
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Thank you for this elaborate answer. I shall take a look at the INRIA paper you cited. I am a computer scientist and not a linguist. At the very last, Panini's grammar sits at the highest level in Chomsky hierarchy, because I cannot imagine the grammar defining a non computable language. I understand it is not regular, maybe not even context-free. Could you explain your affirmation "Wherever the Ashtadhyayi sits on this hierarchy, there will be devices that do more than Panini's grammar does, so there is nothing in the grammar/device taxonomy that exactly describes Panini's grammar" ? Nov 6, 2021 at 18:09
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1What I mean is that a Turing machine, to take the worst case scenario, or context-sensitive (or -free) grammars, can produce conceptually-describable classes of strings that can't be produced by the Ashtadhyayi. One might gratuitously try to define a class of "type 1.5 grammars", the left-context-sensitive rules that have productions of the type αA→αγ (not αAβ→αγβ), which might better match what the Ashtadhyayi does.– user6726Nov 6, 2021 at 18:36
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Even from the beginning, on www.satyavedism.com it is stated that Panini's grammar is a Turing machine, so it is strictly at the fourth (highest) level in Chomsky hierarchy. Thank you @user6726 for the link, where I found the complete answer to my question. Nov 6, 2021 at 18:38
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1The example "a string beginning with any of [aeiou] which is terminated by the same letter" is actually regular, and I can write a finite state automaton accepting this language to prove this. Or, written as a classical regular expression, I can write the language as {a[a-z]*a,e[a-z]*e,i[a-z]*i,o[a-z]*o,u[a-z]*u} (this one does not contain the single letter staring a,e,i,o,u; I can add them easily when I want to have them in, too. Nov 6, 2021 at 20:39
The following paper by Penn and Kiparsky specifically addresses your question:
Penn, Gerald, and Paul Kiparsky. "On Panini and the Generative Capacity of Contextualized Replacement Systems." Proceedings of COLING 2012: Posters. 2012.
A link is provided on Kiparsky's website: https://web.stanford.edu/~kiparsky/Papers/panini-1.pdf
Basically, the form of a standard operational rule (vidhi sutra) in Panini's grammar looks like a context-sensitive rule. However, it does have a constraint on cyclicity which reduces its complexity. This constraint is not that there can be no cyclic applications of rules, which would generate regular languages, but a slightly weaker constraint the authors of the paper call "Nilakanthadikshitar's condition" (after the classical author that gives a statement of it) that prevents cyclic application of rules where a context is reused. The authors develop a more precise understanding of this idea of reusing a context and explore what this entails about the complexity of the resulting system and where it fits in Chomsky's hierarchy.
In brief, the authors argue that (from their abstract) "The formalism behind Paninian grammar, in fact, generates string languages not even contained within any of the multiple-component tree-adjoining languages, MCTAL(k), for any k."
