Treatment of optional forms in Mathematical modelling of P āṇini (2020)

Abstract: Pāṇini in his Aṣṭādhyāyī has written the grammar of Sanskrit in an extremely concise manner in the form of about 4000 sūtras. We have attempted to mathematically remodel the data produced by these sūtras. The mathematical modelling is a way to show that the Pāṇinian approach is a minimal method of capturing the grammatical data for Sanskrit which is a natural language. The sūtras written by Pāṇini can be written as functions, that is for a single input the function produces a single output of the form y=f(x), where x and y is the input and output respectively. However, we observe that for some input dhātus, we get multiple outputs. For such cases, we have written multivalued functions that is the functions which give two or more outputs for a single input. In other words, multivalued function is a way to represent optional output forms which are expressed in Pāṇinian grammar with the help of 3 terms i.e. vā, vibhaṣā, and anyatarasyam. Comparison between the techniques employed by Pāṇini and our notation of functions helps us understand how Pāṇinian techniques ensure brevity and terseness, hence illustrating that Pāṇinian grammar is minimal.

## introduction pāṇini's aṣṭādhyāyī is 'almost an exhaustive grammar for any human language with meticulous details yet small enough to memorize it' @xcite . such an exhaustive grammar is ideal to be used for artificial language processing. briggs @xcite even demonstrated in his article the salient feature of sanskrit language that can make it serve as an artificial language. although, this is not a new concept, various efforts in mathematical modelling of indian languages have been done before. joseph kallrath in his book 'modeling languages in mathematical optimization' says that 'a modeling language serves the need to pass data and a mathematical model description to a solver in the same way that people especially mathematicians describe those problems to each other' @xcite . mathematical modelling of languages also impacts our understanding of the language and its grammar. as scholars are delving into the question of formalizing various natural languages, it is also having an impact on how we understand the language itself. recent work in theoretical and computational linguistics has influenced the interpretation of grammar @xcite . we have followed a similar approach, wherein we have modelled the pratyayas in sanskrit in the form of functions with the help of pāṇinian sūtras. similar to mathematical functions which can be expressed as f(x)=y where x is the input and y is the output of function f; 'the sūtras too look for their preconditions in an input environment. the effects produced by sūtras become part of an ever-evolving environment which may trigger other' @xcite . for the grammar to fit mathematical functions, we 'need a strong and unambiguous grammar which is provided by maharishi pāṇini in the form of aṣṭādhyāyī' @xcite . statistical analysis of a language is a vital part of natural language processing @xcite . according to how components of the target linguistic phenomenon are realized ## methodology we are here attempting to mathematically model the data produced by the sūtras for which we started with compiling the list of dhātus and their respective derived dhātus with different pratyayas like from the kridantkosh of pushpa dikhshita vol.1 @xcite , sanskritworld.in (dhaval patel, n.d.), siddhananta kaumudi of bhattoji dikshita (s.c. @xcite , the madhaviya dhātuvritti @xcite and the roots, verb-forms and primary derivatives of the sanskrit language by w.d.whitney @xcite . the list of dhātus without the application of any pratyaya are considered as x, after the application of the concept of anubandhas. anubandhas have a very prominent role to play in the pāṇinian system of sanskrit grammar. it literally means 'what is attached to'. it has been used by all ancient authorities on sanskrit grammar who have come after pāṇini, right from kātyāyana to nageśa. however, pāṇini has used the term 'it' to describe the anubandhas. m. williams dictionary @xcite defines anubandhas as an indicatory letter or syllable attached to roots etc., marking some peculiarity in their inflection e.g. an 'i' attached to roots denotes the insertion of a nasal before their final consonant. according to nyāyakosa, anubandha is a letter that is attached to the stem (prakṛti), termination (pratyaya), augment (āgama) or a substitute (ādesha) to indicate the occurrence of some special modifications such as vikaraṇa, āgama, guṇa or vṛddhi, accent etc. but it is dropped from the finished word i.e. pada. the use of anubandha is one of the crucial steps pāṇini has taken to ensure the brevity and terseness of his work. we can say that anubandhas do form part of the pratyayas etc. to which they are found appended @xcite . but before we directly start writing our functions, we need to define the input set which comprises of dhātus from the dhātupatha as well as the derived dhātus without anubandhas. let a be a set of all the dhātus after the anubandhas have been removed. these primary dhātus are 1943 in total. however, the input dhātus are not limited to these dhātus in set a. we can also derive a new dhātu set b by adding a san pratyaya to the dhātus of set a. in this paper we will focus on the multivalued functions that give two or more outputs for the same input dhātu of the form f(x)= if there are two optional forms; f(x)= if there are three optional forms and so on. ## notation let x be the input dhātu. for the purpose of writing these functions, we start enumerating the syllables from left to right or from right to left depending upon that particular function. we can denote x as, @xmath0.,x(2),x(1))= (x'(1),x'(2),….). for example: if @xmath1, then conversion are denoted by a right arrow with a number on the top. the number denotes the location of the conversion. for example, x[a ā] denotes that in the dhātu x, a which is at the 2nd place from the right is getting replaced with ā. we also define a '+ operator' to explain the change of syllables when two syllables combine. in sanskrit language when two syllables come closer, for the ease of pronunciation (in most cases) it gets replaced by another syllable or a combination of syllables. for example: ū+@xmath2, e+@xmath3, o+@xmath4, d+@xmath5, ch+@xmath6, j+@xmath7, dh+@xmath8, bh+@xmath9, h+@xmath10. note that although the '+ operator' may look similar to the concept of sandhi in sanskrit, it is totally based on our need to fit our dataset and does not encompass the broad concept of sandhi. ## a function p(x) this function is not a pratyaya function, but it is required to write the pratyaya function. thus, it would be helpful to define it here. the dhātus which have two or more vowels are called udātta, and when a suffix is added to them an additional 'i' comes. such dhātus are called seṭ (literally meaning 'with iṭ'). for dhātus which have one vowel, we need to see the instructions given in the dhātupaṭha. they can either be seṭ or aniṭ depending upon the given instructions given. example of one such instruction is 'bhu sattayām| 'udāttḥ parasmaibhāṣḥ'| it says that 'i' will come as the prayogsamavāyī svara is udāttḥ. the function p(x) is defined by, ## multivalued functions the words used for optionality by pāṇini are vā, vibhaṣā, anyatarasyām. vā appears 136 times, vibhaṣā appears 258 times and, anyatarasyām appears 161 times respectively in aṣṭādhyāyī; including the ones that occur in anuvritti 1 . pāṇini and all the commentators have given us no indication that they are supposed to be anything but synonyms. but the modern scholar paul kiparsky has wondered how could this be so, because pāṇini has vowed to eliminate every needless extraneous syllable and there must be a deeper reason to suggest the use of three different terms. hence, he has propounded the hypothesis in his well-argued study pāṇini as a 'variationist' that the three terms vā, vibhaṣā, anyatarasyām refer respectively to three different kinds of options: those that are preferable (vā), those that are marginal (vibhaṣā) and those that are simple options(anyatarasyām) @xcite . one such case which results in such optional forms is represented in the table below where the addition and absence of 'i' results in two forms and the change of 'h' syllable to two different syllables further results in two forms. thus, we end up with three forms of the same word. let us look at an example for this case for @xmath11: sheeba with the help of rsvp shabdabodha (2006-08). program to generate the text from markings: pawan goyal, ph.d. student, iit kanpur version dated: 18th august, 2008 6 cases for multivalued functions some cases for multivalued functions are displayed below 2 . some multivalued functions for tumun pratyaya case i: if x є {svṛ sū dhū}, then tum(x) = x tum(x) sū svi so svitum sotum svṛ svari svar svaritum svartum case ii: if x has two syllables such that x(1)= ṝ, then tum(x) = x tum(x) v vari varī varitum varītum k kari karī karitum karītum case iii: if x є {gup}, then tum(x) = 2 an exhaustive list of cases for tumun and san pratyayas including the multivalued cases are given in the appendix in devanagari script. word occurrence usage vā 136 times preferable vibhaṣā 258 times marginal anyatarasyām 161 times simple options ## conclusion according to the mathematical definition of a function, it generates a unique output for every input. however, while mathematically modelling pratyayas in sanskrit we came across several instances where a single input was generating multiple outputs, which have been represented by multivalued functions. to ensure brevity, pāṇini has used several tools which have been compared with their equivalent tools in our functional approach. what we are essentially denoting as x(2) in our functions i.e. the penultimate term is nothing but upadhā. pāṇini by convention treats x(1) as the end and calls it antya. this is clear from the definition of upadhā given by pāṇini in aṣṭādhyāyī sūtra '1.1.65 alontyāt pūrva upadhā', which means 'the letter immediately preceding the last letter of a word is called penultimate (upadhā) (creative commons, 2020)'. as stated before in the paper, the words vā, vibhaṣā, and, anyatarasyam are used by pāṇini to denote optional forms that we have demoted by multivalued functions. another important feature of pāṇinian grammar is anuvṛtti, which is a technique of carrying some parts of the previous sūtras to the next sūtras. due to anuvṛtti, the order in which various elements appear in the sūtra itself are very important. however, we do not need to define any such equivalent tool in our modeling as long as we define some global functions and operators such as p(x) and the '+' operator. by mathematically modeling pratyayas, the reason behind use of these techniques employed by pāṇini to ensure brevity becomes very clear. mathematical modelling of pāṇinian grammar in this way helps identify some general patterns, each of which is grouped separately as a case in the functions. these patterns are mainly dependent upon the occurrence of certain specific syllables at certain places. however, we observed that there are some dhātus which even after fulfilling the conditions given in the cases, give an output which is different from what is observed in the literature. all such cases needed a separate approach. hence the for the treatment of such cases input sets for those particular cases have been defined. the knowledge of pāṇinian rules also helps us reduce the number of individual cases that have been constructed for each function. it helps group certain cases together into a single generalized case. for example: instead of writing three individual functions for @xmath12, @xmath13, and ṛ→ar, the knowledge of the rules in aṣṭādhyāyī helps to write a general case of the form i u ṛ → e o ar. writing such functions for all other pratyaya functions may lead us towards a global function for pratyayas and for other grammatical tools as well. this technique of mathematical modelling is extremely helpful to understand sanskrit grammar for people who are non-linguists or do not understand the technicalities of sanskrit grammar. this mathematical model can also form a base for further processing of the grammatical rules for natural language processing of the language with the help of well-defined input and output sets.