Automated Planning

Abstract

Automated planning deals with the tasks of finding out ordered sets of actions that allow a system to transform an initial state to a state satisfying goal specification. The set of actions is a plan, and it belongs to PSPACE-complete. Automatic planing or scheduling generates a set of actions automatically. This chapter presents the nature of automated planning, the classical planning problem, agent types that execute the problem, and worked examples. It also covers, the concepts and implementation aspects of forward planning, partial-order planning, planning languages, a case study of general planning language—STRIPS, and search strategies. Planning with propositional logic, planning graphs, and hierarchical network planning are demonstrated. The multiagent planning techniques are presented for goal and task refinement, decentralized planning, and on how to do coordination after it is planned. This is followed by the chapter summary, and a set of exercises.

Exercises

• 1. Consider the standard Towers of Hanoi problem with 3 pegs and 4 number of disks ($d_1, d_2, d_3, d_4$, with $d_1$ at the top). The disks are to be transferred from start-peg to end-peg, using intermediate peg, one at a time such that at no time larger disk comes over the smaller. The disk $d_1$ is the smallest and $d_4$ is the largest. Make use of only 3-predicates: unary predicate: clear, and binary predicates: on and smaller, and only one action: puton(x, y) needs to be used.

  Write the domain of the problem, and make use of forward planning to plan the solution to move all the 4 disks from start-peg to end-peg.

• 2. Given the 3-SAT problem:

  $$\begin{aligned} (\lnot p_1 \vee p_2 \vee p_3) \wedge (p_1 \vee \lnot p_2 \vee p_3) \wedge (p_1 \vee p_2 \vee \lnot p_3), \end{aligned}$$

  solve it using forward planning. (Hint: you need to assume some operators (i.e., actions) to assign the values to variables $p_1 \dots p_3$.)

• 3. Given the Table T, and blocks A, B, C, D, having different positions on the table, apply STRIPS to plan the solution of the following problem: Initial state $\mathbf {I}$ as

  $$\begin{aligned} clear(A), clear(B), clear(C), clear(D),\\ on(A, T), on(B, T), on(C, T), on(D, T), \end{aligned}$$

  i.e., the blocks $A \dots D$ are on the table, and their tops are clear. Final State $\mathbf {F:}$ on(A, B), on(B, C), on(C, D), on(D, T), clear(A).

  Use the action puton(X, Y), $x \ne y$, where X is a block $A \dots D$ and Y is either table T or block $A \dots D$. Give the forward planning to reach state $\mathbf {G}$ starting with state $\mathbf {I}$.

• 4. Use STRIPS for planning of the following problem: You are at home, and you have money, and you are required to buy milk. Assume the necessary start and goal states, actions, preconditions, and results for this planning job.

• 5. Give the STRIPS representations to actions: pick up mail and deliver mail (ref. Fig. 15.2).

• 6. Suppose the robot (in Fig. 15.2) cannot carry both coffee and mail at the same time. Make use of some constraints to provide the planning for this situation. Assume that the robot can carry a box in which it can place objects, so that it can carry the box and the box can hold the mail and coffee.

• 7. Modify the problem in Fig. 15.2, so that the robot has the work of cleaning the four rooms (mail room, office, coffee shop, lab). Assume that it will clean the room only when the room is unclean, and will not consume more than one rotation mcc or mc to reach any of these rooms.

• 8. Using the method of hierarchical task network planning, provide the automated planning for the following problems:

  • a. Shopping grocery items from market.

  • b. Deliver a lecture of AI.

  • c. Robot path planning to cover the diagonal in a room.

• 9. Assume that you have three operators:

  • $f_1:$ Precondition: a; effect: $\lnot a \wedge b$

  • $f_2:$ Precondition: $a \wedge c$; effect: $\lnot a \wedge b \wedge \lnot c$

  • $f_3:$ Precondition: $b \wedge c$; effect: $\lnot c \wedge d$

  Show the first three layers (proposition, action, and proposition) of the graph plan when the initial state is $a \wedge c$ (a and c both are true). Include the mutual exclusions and justify each of them.