Pāṇini's Grammar and Modern Computation

Abstract

Pāṇini's fourth (?) century BC Sanskrit grammar uses rewrite rules utilizing an explicit formal language defined through a semi-formal metalanguage. The grammar is generative, meaning that it is capable of expressing a potential infinity of well-formed Sanskrit sentences starting from a finite symbolic inventory. The grammar's operational rules involve extensive use of auxiliary markers, in the form of Sanskrit phonemes, to control grammatical derivations. Pāṇini's rules often utilize a generic context-sensitive format to identify terms used in replacement, modification or deletion operations. The context-sensitive rule format is itself defined using Pāṇini's more general method of auxiliary markers, the latter used to define many dozens of linguistic categories and rules controlling derivations of Sanskrit sentences through the manipulation of ‘non-terminal’ and ‘terminal’ symbols. This technique for controlling formal derivations was rediscovered by Emil Post in the 1920s and later shown by him to be capable of representing universal computation. The same implicit computational strength of Pāṇini's formalism follows as a consequence: while Pāṇini's Sanskrit grammar is computationally limited, the metalanguage through which his formalism is defined can be directly used to define any rule-based system by mimicking standard formal language definitions as an extension of the grammatical system proper. Pāṇini's formal achievement is historically distinctive, as derivations of grammatically correct, spoken Sanskrit, are designed for oral recitation, with the grammar itself constructed as an organic extension of the spoken object language. Pāṇini's formulation of what amounts to an orally realized symbolic calculus stands in contrast to the implicit inscriptional methods of contemporary formalisms, such as Gottlob Frege's appropriately named Begriffsschrift and the early computing paradigms of Post and Alan Turing. Nonetheless, contemporary views on the cognitive status of phonemic recognition and historical writing systems support the conjecture that, in spite of Pāṇini's rigorous oral formulation, construction of the grammar almost surely relied on alphabetic writing.