Panini’s Grammar and Computer Science
The enterprise of computer science has two fundamental elements. The first element is to develop techniques that make the elucidation of the computational structure of nature and the mind easier. The second element is the creation of new computing algorithms and machines that have powerful cognitive and computational abilities: this includes development of new techniques of representing and manipulating knowledge, inference and deduction.
The tasks of representing and processing knowledge with a somewhat different emphasis has parallels in many ancient disciplines. Thus grammarians have long considered questions of relating facts about the physical world and cognition to linguistic expressions. Likewise logicians have developed formal structures to relate events and draw inferences from them. This is seen best in the work of ancient Indian logicians and grammarians. It has been argued by Ingalls, Staal, Matilal, Briggs, Kak and others that many contemporary developments in formal logic, linguistics, and computer science are a rediscovery of the work of these ancient masters.
But apart from the question of a correct history of ideas it raises the following important question of significance to Sanskritists as well as cognitive and computer scientists: Are there other rules in ancient Indian logic and grammar that may be of use in making further advance in cognitive and computer sciences? A little bit of history shows why this is a valid question. Nineteenth century Western linguists did not see the significance of the context-sensitive rules of Panini’s grammar. In fact their fundamental importance was seen only when Paninian style structures were first introduced by Western linguists such as Chomsky about thirty years ago.
According to the distinguished linguist Frits Staal:
“We can now assert, with the power of hindsight, that Indian linguists in the fifth century B.C. knew and understood more than Western linguists in the nineteenth century A.D. Can one not extend this conclusion and claim that it is probable that Indian linguists are still ahead of their Western colleagues and may continue to be so in the next century? Quite possible; all we can say is that it is difficult to detect something that we have not already discovered ourselves.” 
Computationally, grammars of natural language are as powerful as any computing machine. But since the setting of a grammar is so different from the typical purpose of a computer, this fact is often obscured. The formal structure of a grammar can be easily adapted so as to perform numerical processing. In this paper we discuss formal aspects of certain rules of Panini’s grammar- Astadhyayi (A), which is traditionally studied together with the dhatupatha, which is a list of verbal roots arranged into sublists, and the ganapatha, which is a list of various classes of morphs, one class being the dhatupatha.
It is now becoming clear that A does not merely deal with analysis of words (sabdansasana) but in fact provides a structure for the analysisof sentences. Due to its algebraic nature and its comprehensiveness, the structure has been described as a machine generating words and sentences of Sanskrit.
Composed in the succinct sutra style, A consists of nearly 4000 sutras that capture the fundamentals of Sanskrit language in terms of its phonology, morphology and syntax. As in any formal system, the structure consists of definitions, theorems (linguistic facts), and meta-theorems (rules regarding rules). The rules are of different kinds: some are universal and context-sensitive transformations, others operate sequentially or recursively.
Generally these rules are expressed in three groups: (i) rules of interpretation or meta-rules, samjna and paribhasa rules, (ii) rules of axation,rules prescribing axes after two kinds of basic dhatu and pratipadika roots, and (iii) rules of transformation for the stems and the suxes , the morpho-phonemic rules. Note that a computer program has exactly the same general features of context-sensitive rules, recursion, and sequential rule application.
It is not surprising, therefore, that these sutras have been compared to a computer program that generates Sanskrit sentences. Panini’s grammar is algebraic where a nite set of rules generates an in nite number of words and sentences.
It is generally agreed that the Paninian system is based on a principle of economy, an Occam’s razor. This makes the structure to be of special interest to cognitive scientists. Furthermore, development of logic has been seen as emerging from the background of grammatical categories both in India and Greece. Considering the preeminent position of the Paninian system in the Indian intellectual tradition, its significance for students of logic and history of science becomes clear.
It is also important to place Panini’s grammar in the context of a continuing development of mathematics and science in India. Seidenberg has shown  that the rise of the earliest mathematics should be seen in the Vedic literature. Furthermore, Kak has established  that the Brahmi script of Panini’s time is to be derived from the Indus script of the third millennium B.C. This means that Panini himself was heir to a very long and rich tradition of learning.
Grammatical categories serve to express knowledge about the world. Panini’s system of knowledge representation is based in the karaka theory. The karaka are deep structure relations that mediate mappings from semantic relations (such as agent, goal, location) to phonological representations (in terms of case-endings that may express voices) via surface structures (in terms of morphological categories such as nominal cases, prepositions, and verbal voices) .
II The sutra style and the nature of the rules
The sutra style represents a genre of Sanskrit literature. Traditionally, a sutra is defined as the most concise of statements which uses as few letters as possible. Although many books have been written in the sutra style, Panini’s grammar or Paninisutra,(Ps), is unanimously regarded by tradition as a model of the sutra style of composition. Words and sentences constitute the data as well as the rules for grammar. Language is thus both a means and an end. Panini’s grammar deals with Sanskrit. But its end language (the object language) and the means language (the metalanguage) are distinctly different from each other.
Panini’s matalanguage has its own vocabulary, syntax, and grammar although it is basically Sanskrit. An extensive use of abbreviated expressions and other devices has given it an appearance of a code language. It is this feature of the Paninisutra that has inspired comparisons with a computer programme.
A few prominent aspects of this code language will be described later. A striking feature of the language of the sutra is the use of abbreviated expressions. Economy of expression is Panini’s primary concern and he has achieved it by employing several algebraic devices. Use of technical terms in place of lengthy expressions is one of them. He uses symbols like ti, ghu, gha and bha.
Further, a code representation, technically known as pratyahara, enables him to save words and even letters in a rule. For instance, instead of directly mentioning the letters y, v, r, l,Panini makes use of the pratyaharayan; for vowels he uses the term ac ; for consonants, hal and so on.
Following is an example of a rule containing all code words:
P.6.1.74: iko yan aci
i, u, r and l are replaced by y, v, r and l respectively when a vowel follows.
Use of words without adding endings to them traditionally known as (avibhaktikanirdesa) is another striking feature of the language of the Ps. However, it will not be described here.
A Ps is a single clause proposition consisting of a subject, a predicate, and an environment. It is a statement about grammatical features such as a sux, an augment, a substitute, accent, reduplication, elision, and compounding. It is usually of the form A is B in the environment C. This can be written in the following formula: Ps : A => B (C)
Here => stands for is or becomes , and ( ) stands for when, A stands for the subject, B represents predicate, and C stands for environment. While A and B are the necessary components of a sutra, C is optional. A unique feature of the Ps is the absence of a finite verb predicate. Tradition holds that the finite verb asti (is) or bhavati (becomes) is taken to be present in each rule.
A Paninian rule is thus a statement about something being or becoming something else. Panini’s marked predilection for nominalization is clearly re ected in his attempt to reduce all statements to those on being or becoming. Thus for instance, instead of saying, tat lupyate,
That (code letter) is dropped, Panini says: tasya lopah (P.1.3.9), Its (i.e. the code letter’s) elision (takes place).
Another interesting feature of the Paninian proposition is the total absence of syllogization or any other kind of logical argumentation. Other sutra works such as the Nyayasutra and the Brahmasutra often contain, in addition to A, B, and C, a cause (hetu) and an illustration (dristanta). In his grammar, Panini never poses the question, Why? His sutras are statements of linguistic facts in reply to the question, What? In other words, he describes facts of language without accounting for them. To sum up, the language of the Ps consists of three types A, B, and C as shown above.
The relation between A and B on the one hand, and that of C with A and B on the other, is expressed by Panini by the use of certain cases. While the predicate item is always used in the nominative, the case of the subject item is determined by its speci c relation with the predicate. For instance, if the predicate is a substitute (adesa) the subject is used in the genitive. This has been directly stated by Panini in a rule .
If, on the other hand, the predicate is a sux (pratyaya), the subject is put in the ablative . Environment is expressed in the locative if it follows the subject . These statements can be put in the following formula:
For substitute: A (gen) =>B (nom) (C(loc) )
For suffix: A (abl) =>B (nom) (C(loc) )
A few more formulas can be formed on the basis of other observations. The following rule is an illustration of the first formula:
P.6.1.74: iko yan aci yan is substituted in place of ik when ac follows.
The genitive form ikah, the nominative form yan and the locative form aci are in accordance with the statement made above and the mutual relationship among the three items is conveyed by the case-endings.
Science may be viewed as a body of generalizations followed by statements of exception when necessary. Paninian science of grammar also consists of a set of general rules followed by statements of exception. A Ps can be thus either a generalized statement or a specific statement which stands as an exception to a generalization.
In a generalized as well as particular statement the subject or the predicate can be a multi-member category. A single predicate may be shared by many or all subjects.
For instance consider-
P.3.2.1: (dhatoh) karmany an
The suffix an is added to a root in the sense of object.
The word dhatoh, which is put inside the brackets (the reason will be explained later), is the subject and an is the predicate. Here dhatu stands for any root in general. The statement of the suffix an thus is applicable to all roots in general. The predicate an is thus shared by all subjects. Here the subject, which is a multi-member category, is represented by a class term (i.e.dhatu). This type of Ps can be represented as:
A(1-n)=>B (C). On the other hand, sometimes many predicates are linked with one subject. For example,
P.5.2.32: ner bidacbirisacau
P.5.3.33: (ner)inacpitaccikaci ca
The suffixes bidac, birisac, inac and pitac are added to ni (in the sense of at nose) and in case of the later two suxes ni is replaced by cika and ci , respectively.
Here as many as four predicates are shared by one single subject,ni .This statement could be represented as: A=>B (1-n) (C)
A third type in which both A and B are simultaneously multi-member categories is also occasionally met with. For instance,
The suffixes nvul and trc are added to any root. This statement is of the type
A(n-1)=>B (1-n) (C)
Just as is true for A and B, C also can be a multi-member category. For instance,P.1.3.13: (dhatoh) bhavakarmanoh (atmanepadam).
Atmanepada endings are added (to a root) in the sense of bhava (state) or karman (object). Bhavakarmanoh expresses the environment in terms of meaning. The two meanings bhava and karman are mentioned here. This can be represented as,
A=>B (C 1-n)
Finally, combination of all the three multi-member categories is also met with in certain sutra. For instance,
P.3.4.70:(dhatoh) tayoreva krtyaktakhalarthah
The suffixes krtya, kta and those conveying the same meaning as that of khal are added to any root in the sense of bhava (state) or karman (object).
The term tayoh is to be interpreted as bhavakarmanoh.We thus have a statement of the type
A(1-n)=>B 1-n (C 1-n)
Observations made above hold true of both the general and special rules in Panini’s grammar. To sum up, the three categories A, B and C may be either single-member or multi-member categories. They appear in all permutations and combinations in Ps.
Now it remains to be seen whether Panini has provided any clarification regarding the application of the multi-member categories. Two questions arise when a statement contains multi-member categories:
- Are the members in a category linked to each other disjunctively or conjunctively?
- What are the mutual relations between the members of two multi-member categories?
Panini has employed three linking devices in the A, namely, juxtaposition, dvandva compound, and the particle ca. They link either items or statements. We are at present concerned with linking of items. These devices work in terms of disjunction or conjunction. Disjunction implies application of all the items separately, whereas conjunction implies their application together. Items put together in a dvandva compound are disjunctively connected. For instance,
The suffixes nvul and trc are added to any root.
Here the compound nvultrcau is a multi-member predicate. The items nvul and trc are disjunctively connected with each other. Therefore, they are separately and not simultaneously added to a root. Thus we can derive two separate forms such as pacaka and paktr from root pac. On the contrary, if the items are put in juxtaposition they are conjunctively connected with each other and are, therefore, simultaneously applicable. For instance,
P.7.4.49: sah syardhadhatuke (tah), s is replaced by t when an ardhadhatuka ending beginning with s follows.
Here two items, si and ardhadhatuke, which belong to the category C are not put together in a compound, but are mentioned separately in juxtaposition. Therefore they are conjunctively connected.
In other words si and ardhadhatuke are co-referential. Whenever items belonging to one category are put in juxtaposition in a rule they hold a head-modifier (or adjective-substantive) relationship. Two or more items belonging to the same category and yet not connected by a head-modifier relation never occur in juxtaposition in a single sutra.
Juxtaposed occurrence of two heads or modifiers always indicates the existence of separate sutra. The particle ca is never used to link two or more items as it does in ordinary language . Items belonging to the same category in a rule are either put in a compound or are juxtaposed according to their relation with each other .
Panini accepts the principle of numerical correspondence for linking items in two multi-member categories. He states the rule as follows:P.1.3.10: yathasankhyam anudesah samanam
Items (in two categories) having the same (number) are connected (with each other) in their respective number (i.e. order).
Take, for instance, P.4.3.94: tudisalaturavarmmatikucavarad dhakchandhanyakah
The suffixes dhak, chan, dhan, yak are respectively added to (the stems) tudi, salatura,varmmat and kucavara (in the sense `it is the place where his ancestors lived’).
Here both A and B consist of four members each and the members of A are connected with the members of B in the same order in which they are put in the compound.
III Arrangement of the Rules
As stated earlier, a Paninian rule consists of three elements: A, B, and C, the last being optional. All these elements are not always explicitly present in the wording of a sutra. Just as a nite verb form is implicit, a certain element is understood to be present in a sutra from the context. While interpreting certain rules, commentators actually borrow the missing term from the preceding rule.
This borrowing or continuation of a word or words is technically called anuvrtti. The procedure of anuvrtti is nothing but ellipsis which is a regular feature of ordinary language. While ellipsis is optional and has an ad hoc character in ordinary language, anuvrtti is a systematic and mechanical device in A.
The sutras in A are arranged in such a manner that a rule borrows an item or items from the preceding context. By putting together such rules which share an item or items in common Panini has been able to achieve economy of expression to a large extent. A few examples of anuvrtti will show the working of this device:
P.1.4.14: suptinantam padam
That which ends in sup or tin endings is called pada.
P.1.4.15: nah kye (padam)
That which ends in n (is called pada when the suffix kya follows.)
The predicate item padam which is missing in P 1.4.15 is continued from P.1.4.14. (The missing item when borrowed from the preceding rule is put into brackets). P.3.3.114: napumsake bhave ktah (dhatoh).
The suffix kta is added (to the root in the sense of bhava state, and the form is used in neuter).
P.3.3.115: lyut ca (napumsake bhave ktah dhatoh).
And suffix lyut (is added to a root in the sense of bhava and the form is used in the neuter).
The environment expressed by the terms napumsake and bhave is carried forward in P.3.3.115 from P.3.3.114. The subject item dhatoh, which is carried forward in P.3.3.114, is also continued in a number of rules. For example, the term dhatoh which is mentioned in P.3.1.92 is continued throughout the following third chapter, nearly in 500 rules. Through the device of anuvrtti Panini has been thus able to avoid repetition of the word dhatoh in more than 500 rules.
Anuvrtti is thus intrinsic with the style of the Ps. Although Panini has arranged rules in his grammar mainly on thematic basis, the arrangement of rules within different sections is totally governed by the dictates of anuvrtti.
A very important difference between anuvrtti and ellipsis in ordinary language consists in the fact that while the latter is dependent upon expectancy and the listener’s (or rather receiver’s) intention, the former is obligatory. Items in the previous rules must continue in the subsequent rules.
Expectancy is not just sucient ground for continuing an item. An item is found to be continued even when there is no expectancy. For example,
A sarvadhatuka suffix, other than the one which is pit is nit.
P.1.2.5.: asamyogallit kit (apit)
A lit suffix other than the one which is pit added to (a root) not ending in a conjunct consonant is kit.
All the three elements, namely subject (lit), predicate (kit) and environment (asamyogat) are present in P.1.2.5. It presents no expectancy for any item in order to complete its meaning. Yet the item apit is continued in the rule.
There are, however, some constraints on this flow of anuvrtti.The major constraint is that an item is carried forward in the subsequent rules until it is blocked by an incompatible item. Thus compatibility and incompatibility play a major role in deciding anuvrtti of words.
For instance, in the example given above the item nit which is continued in P.1.2.4. is not further continued in P.1.2.5 because it contains the item kit which is incompatible with nit. Two incompatible items do not exist in a rule except under some special circumstances. The fundamental rule of anuvrtti can thus be stated as follows:
An item is continued in the subsequent rules unless it is blocked by an incompatible item. Two items are incompatible if they belong to the same category (i.e. subject, predicate, or environment). Again in the same example quoted above the items sarvadhatuka and lit are incompatible with each other. Therefore, the former is not continued in P.1.2.5 as it is blocked by the latter.
Items which are incompatible with each other usually appear in the same case-ending. However, appearance in the same case-ending is not the only identification mark of incompatible items. Their relative syntactic position has also to be taken into consideration. Turning back again to the above example, the two items sarvadhatukam and apit together form the subject category in P.1.2.4.
While sarvadhatukam is the head item, apit is its modifier (adjective). There is obviously no incompatibility between a head and a modifier. This is true not only when they belong to one and the same rule as in the above case, but also when they are mentioned in two different rules.
Thus modifier item apit mentioned in P.1.2.4 is compatible with the head item in the subsequent rule. Therefore, although the head item lit in P.1.2.5 blocks the incompatible item sarvadhatukam in the preceding rule, it does not block the modi er item apit, which is therefore, continued in P.1.2.5. Another rule of anuvrtti may be laid down on the basis of this observation as follows:
A head item blocks an incompatible head item, but it does not block a modifier if it is not incompatible. A modifier blocks an incompatible modi er, but it does not block a head item if it is not incompatible.
There are, however, cases when a head or a modi er is not continued since it is incompatible not on syntactic, but on semantic grounds. (These cases will not be discussed here as they have no direct bearing on the present topic.)
Arrangement of the sutra in the A is initially topic wise. Thus the Ps begins with de nitions of various technical terms and rules of interpretation and treats various types of derivations such as compounds, primary derivatives and secondary derivatives in separate sections. The last part of the A is devoted to morphophonemics including euphonic combination, accent and tone. Within a thematic group the sutra are arranged on the basis of the principles of anuvrtti. Although, a generalization is followed by specific or individual rules, this order is often violated due to exigencies of anuvrtti. Anuvrtti is thus a key-word for the arrangement of the Ps .
IV Techniques of Description
In addition to anuvrtti and artificial technical terminology including pratyaharas Panini employs the device called anubandha. An anubandha is a code-letter which indicates a grammatical function like elision and reduplication. Panini has made use of almost all vowels and consonants as symbols for various functions.
Anubandhas are added to various grammatical units such as suffix, an augment and a root. For example, the suffix a is mentioned as an where the code letter n suggests that the vowel (either initial or nal depending upon the type of derivation) of the stem to which the ax is added is lengthened . The anubandha k attached to an augment indicates that it is added at the end of an element .
Thus the augment t which is mentioned as tuk in the rule hrasvasya piti kr ti tuk (P.6.1.69) is added after an element, e.g. in the form adrtya it appears after root dr which ends in a short vowel. The anubandha n attached to a verbal root indicates that the root is conjugated in the middle voice. Anubandha is thus a powerful device.
A major aspect of Panini’s descriptive technique is the law of utsarga and apavada that relates exceptions and individual rules. Although Panini never explicitly states the law of utsarga and apavada it is part of the interpretative apparatus used with the Ps .
The law of utsarga and apavada states that an apavada `exception’ is more powerful than an utsarga ‘general rule’. Therefore before applying the utsarga one has to give check for its apavada(s). The utsarga thus occupies the area not occupied by its exceptions. Further, once an utsarga is barred from entering in to the area of its exception, it can never enter the area again. For example:
P.4.1.92: tasyapatyam (an)
(The suffix an is added to a noun in the sense) `his off spring.’
P.4.1.95: ata in tasyapatyam
The suffix in is added to a noun ending in a (in the sense `his off spring’).
P.4.1.95 is an exception to P.4.1.92. Therefore, the suffix an taught by
P.4.1.92 is barred from being applied to stems ending in a.
Thus from the stem daksa is derived the patronymic daks (daksa+in) and not daksa+an). Sometimes application of the utsarga, even in the domain of apavada ,is desired. In such cases, Panini announces that the apavada operates optionally. For instance,
P.4.1.121: dvyacah (dhak striyah)
(The suffix dhak is added to a feminine noun) consisting of two vowels (in sense `his off spring’)
P.4.1.118: pilaya va (an striyah)
(The suffix an is added) optionally to pila(in the sense `his offspring’).
The option marker va in P.4.1.118 suggests that the exceptional suffix an operates optionally. Therefore, the utsarga suffix dhak taught by P.4.1.121 is also applied and two alternate forms,paileya (pila + dhak) and paila (pila +an) are derived.
An extremely important principle is the siddha principle. Even though Panini does not directly mention it, his statement of the asiddha principle (P.8.2.1) implies it.
Traditionally, the whole A is divided into two parts on the basis of P.8.2.1: (1) the siddhakanda (P.1.1.1. to the end of the rst section of the eight Chapter) and (2) the asiddhakanda or tripadi (P.8.2.1 to the end of the fourth section of the eighth Chapter).
Tripadi begins with the adhikara, `chapter heading’, 8.2.1: purvatrasiddham (From now on every rule is regarded as) not having taken e ect with reference to preceding ones’. The term siddhakanda implies that any rule in this part of A is siddha `having taken effect’ for any other rule in the whole of A. In other words,before being effective, a rule takes into consideration possibility of application of other rules. The sequence of rules in the book does not matter in the derivational process. What matters is the siddha relation among the rules. The finite verb form bhavati is, for instance, derived as follows:
bho+a+ti P.7.2.115; P.6.1.78
It will be clear from the derivational stages given above that the rule in the rst section of the third Chapter applies after the rule in the fourth section of the same chapter and the rule in the sixth Chapter applies after the rule in the seventh Chapter.
These rules are siddha for each other so that they can feed each other (the application of P.6.1.78 is dependent on the application of P.7.2.115 in the present example.) This free movement of rules in all directions is implied by the siddha principle.
Yet this arbitrary application of rules within 1.1-8.1 is restricted somewhat by a category of rules that are ordered pairs. In each pair, the rule that is applied first is called antaranga and the rule that is applied next is called bahiranga.
On the contrary the rules in the asiddhakanda are operative only in one direction.P.8.2.1. purvatrasiddham states that all the rules stated subsequently are asiddha, not effective for the rules stated earlier, that is for the rules in the siddhakanda. Similarly for each rule in the asiddhakanda, all subsequent rules are asiddha.In other words, rules in the asiddhakanda operate in the same order in which they are arranged. For example, the form pakva is derived from root pac as follows:
pac + ta P.3.2.102
pak + ta P.8.2.30
pak + va P.8.2.52
It is clear from the procedure given above that P.8.2.30 precedes the application of P.8.2.52. In fact, if P.8.2.52 is applied rst P.8.2.30 cannot be applied since the environment favorable for its application does not exist. The rules in the asiddhakanda must therefore apply in the same sequence in which they are stated by Panini.Both the siddha and asiddha principles have been recently studied carefully, leading to important new insights .
V Concluding Remarks
Our analysis was meant to highlight several formal features of Panini’s grammar that have direct parallels in computer science. What might be other features of the grammar that have not yet been rediscovered in computer science remains to be seen. But the very success of A suggests that aspects of its structure will have implications for further advance in computer science, knowledge representation, and linguistics. In particular we can hope for significant applications in natural language processing. The ongoing analysis of the structures of Panini and those of the later grammarians and logicians will be aided by the development of software to implement A on a digital computer.
The specific issues of immediate interest to the computer scientist include analysis of the arrangement of the rules and search for other arrangements that are equivalent in terms of their generative power. The formal aspects of these arrangements and their relationships is likely to help defie the notion of distance between grammars. Such a notion is of immediate relevance for machine translation. Given two languages with grammars that are close in structure, as in the Indo-Aryan family of languages, one would expect the translation across the languages to be relatively easy. A formaliztion of the notion of closeness is also likely to give pointers regarding how an automatic translation might proceed.
One great virtue of the Paninian system is that it operates at the level of roots and suxes de ning a deeper level of analysis than a orded by recent approaches like generalized phrase structure grammars  that have been inspired by development of computer parsing techniques. This allows for one to include parts of the lexicon in the de nition of the grammatical structure. Closeness between languages that share a great deal of a lexicon will thus be represented better using a Paninian structure.
These fundamental investigations that have bearing on linguistics, knowledge representation, and natural language processing by computer require collaboration between computer scientists and Sanskritists. Computer oriented studies on A would also help to introduce AI (artificial intelligence), logic, and cognitive science as additional areas of study in the Sanskrit departments of universities. This would allow the Sanskrit departments to complement the programme of the computer science departments. With the incorporation of these additional areas, a graduate of Sanskrit could hope to make useful contributions to the computer software industry as well, particularly in the elds of natural language processing and artificial intelligence.
- D.H.H. Ingalls,Materials for the Study of Navya-Nyaya Logic.Cambridge: Harvard University Press, 1951. Frits Staal,Universals, Studies in Indian Logic and Linguistics. Chicago: University of Chicago Press, 1988.B.K. Matilal, The Navya-Nyaya Doctrine of Negation. Cambridge:Harvard University Press, 1968. Briggs, \Knowledge representation in Sanskrit and arti cial intelligence”, AI Magazine, vol. 6, 1985, pp. 22-38. Kak, “The Paninian approach to natural language processing”,International Journal of Approximate Reasoning, vol. 1, 1987, pp. 117-130.
- F. Staal, op. cit., page 47
- S.D. Joshi, “Sentence structure according to Panini”, in Glimpses of Veda and Vyakarana, edited by G.V. Devasthali. Bombay: Popular Prakashan, 1985. S. Kak, op. cit.
- A. Seidenberg, \The ritual origin of geometry,”Archive for History of Exact Science, vol. 1, 1962, pp. 488-527. Seidenberg, \The origin of mathematics,”Archive for History of Exact Science, vol. 18, 1978, pp. 301-342.
- S. Kak, “A frequency analysis of the Indus script”, Cryptologia,vol.12, 1988, pp. 129-143. Kak, “Indus writing”, Mankind Quarterly, vol. 30, 1989, pp. 113-118.
- P. Kiparsky and J.F. Staal in Staal,op. cit.
- S.D. Joshi,op. cit.
- P. Kiparsky and F. Staal,op. cit.
- It must be borne in mind that exceptions to such a structure for Ps do exist. But these exceptions do not de ne the general structure of the rules.
- For details, see Saroja Bhate, “Some aspects of Panini’s sutra style of composition,” B.R. Modak Felicitation Volume, March 1989, pp. 37-46.
- For example,ramasca krsnasca gacchatah.Here ca links two items disjunctively. The particle ca can also link two or more items conjunctively in ordinary language: for example `He eats curds and honey’.
- For details see S.D. Joshi and Saroja Bhate. The Role of the particle ca in the interpretation of the Astadhyayi. Publication of the Centre of Advanced Study of Sanskrit, University of Poona, 1983.
- For a discussion of the principles of anuvrtti, see S.D. Joshi and Saroja Bhate,The Fundamentals of Anuvrtti, Publication of the Centre of Advanced Study of Sanskrit, University of Poona, 1984.
- P.7.2.115 to 117.
- For example, the rules of interpretation (paribhasa) 57, 58, 59, 60, 62, 64, and 65 in the Paribhasendusekhara of Nagojibhatta edited byK.G. Abhayankar, Bhandarkar Oriental Research Institute, Pune, 1962.
- S.D. Joshi and Paul Kiparsky, “Siddha and asiddha in Paninian phonology,” in Current Approaches to Phonological Theory ,edited by D.Dinneen. Bloomington, 1979.
- Gazdar, E. Klien, G. Pullum, I. Sag,Generalized Phrase Structure Grammar. Cambridge: Harvard University Press, 1985. Gazdar et al. claim that context-sensitive rules can be replaced by a larger set of context-free rules for all natural languages, excepting Bambara (Mande family, West Africa) and a certain Swiss dialect of German. Context-free rules make parsing by computer easier.
This paper was originally published in the Annals of the Bhandarkar Oriental Research Institute, vol. 72,1993,
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