Pāṇini grammar is the earliest known computing language

Pāṇini’s fourth (?) century BCE Sanskrit grammar uses rewrite rules guided by an explicit and formal metalanguage. The metalanguage makes extensive use of auxiliary markers, in the form of Sanskrit phonemes, to control grammatical derivations. The method of auxiliary markers was rediscovered by Emil Post in the 1920s and shown capable of representing universal computation. The same potential computational strength of Pāṇini’s metalanguage follows as a consequence. Pāṇini’s formal achievement is philosophically distinctive as his grammar is constructed as an extension of spoken Sanskrit, in contrast to the implicit inscription of contemporary formalisms. 1 GRAMMAR AND COMPUTATION For purposes of this paper, ‘computing language’ means a formal calculus capable of representing universal computation according to the rules of some formal language whose rules are explicitly described through a metalanguage. In this sense, modern machine and high-level programming languages, by virtue of their formal (meta-)language rules, are computing languages. So too are the classical models of Post, Turing, Church, Kleene and others, including Gödel’s formalization of metamathematics as number theory. Though not ‘programming’ languages intended for machine implementation, the classical models all succeed by virtue of defining ‘effective procedure’ through a procedure-level formalism which can be then used to represent all such procedures. Frege’s first-order logic (as streamlined by Hilbert and Ackermann) may be included here just because, as recognized by Church and Turing, Gödel’s number theoretic coding may be translated into the language of first-order logic (and so showing the valid sentences of firstorder logic to be undecidable). We tend to think of such formal systems, capable of expressing arbitrary algorithms, as thoroughly modern, certainly as at least late 19 century creations. It’s also a modern idea to see how to describe the derivational rules of a formal language also through the language, so that objectand metalanguage are one. But the 19th and early 20th century formalisms for algorithmic expression are not the earliest such, by about two millennia. The first computing language – again, for our purposes, a generic formalism, described through a metalanguage for representing exact generative symbolic procedures of any kind – was devised circa 350 BCE by the Indian grammarian and linguist Pāṇini. The formalism is not identical with Pāṇini’s Sanskrit grammar, but is a significant part of it, constituting the grammar’s formidable formal methods. 1 Policy & Decision Science, Menlo Park, California. Email: [email protected] Those formal methods are allied with perhaps some centuries of Indian linguistic theory to define the grammar as a whole. Pāṇini, as put by the late Frits Staal, is the ‘Indian Euclid’ [1]. Parallel to Euclid’s codification of the earliest, if informal, deductive systems, including proof by contradiction, Pāṇini in his Sanskrit grammar formulated and applied the world’s first formal metalanguage for generic symbolic manipulation. Pāṇini’s basic method was rediscovered in the 1920s and 1930s by Emil Post through his production/rewrite systems. Post proved [2], as Pāṇini could not even conceive, that his systems were capable of universal computation. But then that fact has also to be true of Pāṇini’s grammar, even as the latter is meant for computationally modest linguistic derivations and not calculation nor computation generally. To a first approximation, Pāṇini’s formal methods, exclusive of his linguistics, are Post’s, or vice versa. Pāṇini’s grammar has the further remarkable property, relevant to contemporary debates in the philosophy of programming languages, that it is formulated for oral recitation, not inscription; indeed, Pāṇini’s formalism can be construed as a grammatical generalization of the spoken Sanskrit object language which the grammar describes. In this way, Pāṇini’s grammar is in effect the realization of a computing environment as formally recited human speech. This paper provides a sketch of Pāṇini’s grammar and its metalinguistic technology. By way of historical context, the grammar is motivated to construct ‘certificates of authenticity’, so to speak, for Sanskrit expressions, for both scientific and ideological reasons. Procedural exactness has deep roots in the habitus of Hindu culture, particularly through older traditions of ritual theory, through which the earliest Indian linguistic theories were conceived, including the characterization of grammar as representing continuous speech (saṃhitā) using artificial discrete simplifications (pada, [3]). As noted repeatedly by Staal, language and linguistics had a preeminent scientific role in ancient India, comparable to geometry and astronomy in Greece, but with a complementary prestige associated in India with algorithmic thinking of all kinds. The oldest theoretical formulations of the topic appear to be those of various so-called ritual ‘manuals’, guiding explicit ritual design and execution in the Vedas and elsewhere. Such were the procedural programming manuals of the time, so to speak.