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T h e detailed proof of this theorem is in [8], but we will discuss it briefly. T h e forward direction is a straightforward and tedious induction. T h e reverse direction, usually done by contradiction, involves a careful analysis of the leftmost sections of derivation trees. It is these properties which can be used with the LL(k) property to derive a contradiction. A. 3 Let U{ G Σ*, 1 < ι< r r} for some alphabet Σ . = uxu2 · · · ur U i=1(u{) concatenation. Then iUT W e are now ready to proceed to our first iteration theorem for the L L languages.

This turns out to be possible but the concepts change in an interesting way. 2 Let Τ be a grammatical Let ( j / x , . . , y m ) be a left-to-right tree, and let m = l g ( f r ( T ) ) . sequence of all terminal nodes in Τ. /n} U {χ G Τ I there exists b G Τ such that b\7yn Also, let {°^T be the empty tree, and let also call a left n-part. , δΠ + = Τ if η > m. χ}. We contains all nodes in ^ T ' , and in addition contains all immediate descendants of nodes on the path from r t ( T ) to yn. shows a grammatical tree Τ and Figure 4 ^T.

2 / / = < ( n ) T = n + 1 > T if and only if^T ( n ) r then = w = T. V for any j < n. nodes in a tree. Then xVy means that y is an immediate descendent of x. Let a: Π y if ι and y are both children of the same node and χ is immediately to the left of y. We say that x—>* y if χ is to the left of y in the tree, though not necessarily at the same level. Providing Nonmembership 27 Our principal result about the structure of derivation trees of a strict deterministic grammar can be informally stated as follows.