1) How many levels will GraphPlan need to expand in order to solve the
   "Sussman Anomaly" shown in Figure 11.16?  Explain your answer.

   For the set of actions, consider actions that correspond to the
   blocks-world STRIPS schemas given in the book.  

2) 11.9 (only the first part/bullet)

   (here "final level of the graph" refers to the level after which no
    more changes occur)

3) 11.17 (Keep answers concise.)  
   
   For part b you can interpret "spurious" as a solution that either
   does not achieve the goal or one that achieves the goal but then
   moves away from the goal.

4) A standard propositional STRIPS planning problem (as described in
   class) specifies: 1) a goal condition, 2) an initial state, and 3)
   a set of propositional STRIPS actions with positive
   preconditions. We will refer to the set of all standard STRIPS
   problems as STANDARD.
  
   Consider the class of planning problems POS-ONLY, which contains
   any STRIPS planning problem that contains only positive
   preconditions and positive effects (i.e. there are no delete
   lists). 

   Give an efficient algorithm (polynomial time in the problem size)
   for deciding PlanSat for this class of problems (i.e. determining
   whether a problem has a solution). Prove your answer for full
   credit. 

   HINT: It may help to think about Graphplan.

5) Consider the CNF formula constructed in class for the simple 2
   location robot world. Call this formula F. We showed that this
   formula was constructed from four distinct subformulas F1, F2, F3,
   F4 which respectively described the "initial state", "goal",
   "action effects/preconditions", and "explanatory frame
   axioms". This exercise is intended to convince you that each of the
   subformulas are necessary.

   For each of the subformulas, consider removing it from F and give a
   model of the resulting formula that does not correspond to a valid
   plan. That is, for each formula, F-F1, F-F2, F-F3, F-F4 give a
   model that does not correspond to a legal plan. (F-Fi is simply the
   formula F with the subformula Fi removed.) 

   By legal plan we mean the following. Recall that a model assigns a
   truth value to each state and action fluent at each level. From a
   model we can extract a totally ordered or layered plan, given by
   the sequence of action fluents that are true in the model at each
   level. We can also extract a sequence of states, where the state
   corresponding to level i contains all facts that correspond to true
   fluents at level i. We say that a model corresponds to a legal plan
   iff it satisfies the following three conditions: 
     1) The first state of the extracted state sequence is equal to
        the initial state of the planning problem.  
     2) The final state of the extracted state sequence satisfies the
        goal of the planning problem.  
     3) The plan and state sequence are consistent in that actions at
        level i can be executed in the level i state and produce 
        exactly the state at level i+1.