By Mark A. Fulk (Eds.)
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Extra resources for Colt Proceedings 1990
Now consider the case of variables where a and 6 differ that appear in the same formula as Xi or Xj. Suppose at some iteration of the loop Xk is such a variable, with the property that no other such variable has a deeper gate above both it and Xi or Xj. Let G be the deepest gate to which there are inputs leading from Xk and Xi (without loss of generality); so no variables where a and 6 differ appear in the same sub-formula of G as X , , and Xj does not appear in the same sub-formula of G as Xk.
Since a's evaluation path in φ* must reach X , ' s node, it cannot reach X j ' s , and thus Xj £ S(a). Conversely suppose X , and Xj are dependent in 0*. If X , and Xj appear in the same node, one can easily come up with a setting of the variables in that node's μ-formula, for which changing either Xi or Xj changes the value of the formula. Construct a from a justifying assignment for X , by giving the variables in X,-'s formula the values of this setting. Since changing Xi will change a's value on that formula, we still have Xi G S(a).
T h e value of α on this subtree cannot be changed (as this would remove X,· from its sensitive set). So α now induces the same value on both subtrees of this formula of divergence, and hence can have variables from that formula all changed to their values in 6. A t this point all the remaining differing variables are not reached by a, and may be changed. In the case where X,· is an ancestor of X j , the value α induces on X,· 's subtree containing X j must be the same as b or δχ,—-,χ,'s, and hence a will be made to agree with that assignment on all variables appearing there.