Formal Definition
Chomsky's advisor, Zellig Harris, took transformations to be relations between sentences such as "I finally met this talkshow host you always detested" and simpler (kernel) sentences "I finally met this talkshow host" and "You always detested this talkshow host". Chomsky developed a formal theory of grammar where transformations manipulated not just the surface strings, but the parse tree associated to them, making transformational grammar a system of tree automata. This definition proved adequate for subsequent versions including the `extended', `revised extended', and `Government-Binding' (GB) versions of generative grammar, but may no longer be sufficient for the current minimalist grammar in that merge may require a formal definition that goes beyond the tree manipulation characteristic of Move α.
Read more about this topic: Transformational Grammar
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