A set - theoretical investigation of Pān . ini ’ s

In Pān. ini’s grammar one finds the Śivasūtras, a table which defines the natural classes of phonological segments in Sanskrit by intervals. We present a formal argument which shows that, using his representation method, Pān. ini has chosen an optimal way of ordering the phonological segments to represent the natural classes. The argument is based on a strict settheoretical point of view depending only on the set of natural classes and does not explicitly take into account the phonological features of the segments, which are, however, implicitly given in the way a language clusters its phonological inventory. Moreover, the argument is so general that it allows one to decide for each set of sets whether it can be represented with Pān. ini’s method. Actually, Pān. ini had to modify the set of natural classes in order to define it by the Śivasūtras (the segment h plays a special role). We show that this modification was necessary and, in fact, the best possible modification. We discuss how every set of classes can be modified in such a way that it can be defined in a Śivasūtra-style representation.1 0.1 Pān. ini’s Śivasūtras Pān. ini’s grammar is recognized as a consistent theoretical analysis of spoken Sanskrit (bhās. ā) of the time of its origin (ca. 350 BC). The Śivasūtras form the first part of it (a short survey of the structure of Pān. ini’s grammar can be found in Kiparsky (1994)) and define the phonological segments of the language and their grouping in natural phonological classes, called pratyāhāras. In the As. t .ādhyāyı̄, a system of about 4000 grammatical rules or rule elements forming the central part of his grammar, Pān. ini refers to 42 of the pratyāhāras in hundreds of rules. The Śivasūtras state 42 phonological segments and consist of 14 sūtras (rows in table 1), each of which consists of a sequence of phonological segments (transcribed with small letters) bounded by a marker (transcribed with a capital letter), This approach fits naturally in the framework of Formal Concept Analysis, since the investigated graphs are formal concept lattices. The proofs of the presented propositions can be found in my thesis, which will appear in 2003, and are sketched in an the extended version of the present paper.