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Also note that we never change a and b to differ on any variables (other than X,· and X j ) where they already agree, and each time we repeat 2 the loop of step 1 there has been a change, so we cannot repeat steps l ( a ) i or l ( a ) i i more than η times. Identifying //-Formula Decision Trees with Queries Variables appearing in a formula of φ* that is on the evaluation path of neither a nor αχίΛ χιί can be changed arbitrarily in a while X{ remains in the sensitive set. A n d variables not on the evaluation paths of b or bxj„xj can be changed in b.

4. Return a. Figure 2: Make assignment a as close to b as possible while sensitive to Xi. 2 more of the variables besides X , and X j , which gives the η bounds. < ,xi, and thus can be immediately changed to agree with b. (Once a agrees with 6 on some variable, that value will never change). If a and b induce the same value on a formula φ in φ* that contains neither X,· nor X j , then when we reach step 2 those assignments will agree on all variables in φ. That's because on each iteration of step 1 either both a and b reach the 6(Xjk) subtree of φ for some variable Xk where they disagree, in which case we can switch α(Χ&) to agree with 6, or else the set of variables where a and 6 already agree completely determines ^'s value, so remaining variables in φ can be switched in a.

ABOVE_FORK(X t,^y,Z>,0). 3. Otherwise (Φ §f doesn't exist), (a) Let φ^ (b) If φ^ be the deepest formula containing a variable dependent with Xi (if such exists). exists and contains some variable Xj independent with Xi (say D[i,j] = (independent,^)), i. Replace φ^ιη φ with a single formula testing Xj. Add a formula testing Xk immediately above Xj, subtree. Make X*'s other subtree be a single variable formula testing with Xj as A V s D[k,j](Xk) Xi, with trivial 0- and 1-subtrees labeled with t]x^o(^*) and D[i,t]x-«_i(0*) respectively.

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