1. [2 points] Let $h_1$ and $h_2$ be consistent heuristics. Define a new heuristic $h(s) = \max\{h_1(s), h_2(s)\}$. Prove that $h$ is consistent.
2. [2 points] Suppose a heuristic $h$ is consistent for a particular state space model. If new edges are added to the search space graph, does $h$ remain consistent for the new model? What if edges are removed? In both cases, give a proof or a counterexample.
3. [2 points] Consider a search problem with start state $s_\text{start}$ and goal state $s_\text{goal}$, where each action is reversible with the same cost (i.e., if an action takes you from $s$ to $t$ with cost $c$, then some action can take you from $t$ to $s$ with cost $c$). Suppose you've run Uniform Cost Search (UCS) and found the correct answer. Now consider the problem of finding the minimum cost path from a different start state $s_\text{new}$ to a the goal state $s_\text{start}$. Define a consistent heuristic for A* for the new model.
4. [2 points] Assume all edge costs are strictly positive. Define a heuristic $h(s)$ based on breadth-first search and show that it is consistent. Now suppose you are given a new model with the same graph structure but different costs. How would you adapt the old heuristic so that it is consistent for the new model? Your adaptation should take constant time.
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