1. Classical Planning
Chapter 10

2. Outline The challenges in planning with standard search algorithm
Representing Plans – the PDDL language
Planning as state-space search
Planning Graphs and the GRAPHPLAN Algorithm
Other Approaches

3. Classical Planning What is planning:
The task of coming up with a sequence of actions to achieve a given goal is called planning
Classical planning: When the environment is
Fully observable
Deterministic
Static
Discrete

4. Challenges in Planning with Standard Search Method (A*, etc)
Why real-world planning problem with state-space search is hard:
Too many irrelevant actions – goal oriented search would help
Difficult to find a good heuristic function
So the language for planning should make it easy for algorithms to exploit the logical structure of the problem

5. Challenges in Planning – too many choices!

6. PDDL Language
State: represented by a conjunction of ground atoms without function symbols
Action: represented by a set of action schemas

7. A specific planning problem
Goal state – a conjunction of literals (positive + negative) that may contain (existentially quantified) variables
A set of action schemas
Clearly a planning problem can be seen as a search problem

8. Example – the blocks-world

9. Complexity of classical planning
PlanSAT: Is there a plan for a planning problem
Both are decidable if the planning language is function-free – finitely many states
Both in the class of PSPACE – more difficult than NP!
But for many domains:
Bounded PlanSAT: NP-complete
PlanSAT: P – thus finding A (possibly suboptimal) plan may be not so hard!

10. Algorithms for planning as state-space search
Backward (regression) relevant-states search

11. Heuristics for planning
Adding edges in the state-space graph
Ignoring pre-conditions
Ignoring delete list
Key idea: decomposition – dividing the problem into parts (which may be independent) and solve each part independently, then combine the subplans
Subgoal independence assumption

12. Planning Graphs - motivation
A big source of inefficiency in search algorithms is the branching factor
One way to reduce branching factor:
First create a relaxed problem
Remove some restrictions of the original problem
Want the relaxed problem easy (polynomial time)
The solutions to the relaxed problem will include all solutions to the original problem
Then do a modified version of the original search (backward search)
Restrict the search space to include only those actions that occur in solutions to the relaxed problem

13. Planning Graphs
Search space for a relaxed version of the planning problem
Alternating layers of states (ground literals) and actions
Nodes at action-level i: actions that might be possible to execute at time i
Nodes at state-level i: literals that might possibly be true at time i
Edges: preconditions and effects
Mutual exclusion links

14. Sample Planning Graph

15. Mutual Exclusions Two actions at the same action-level are mutex if
Inconsistent effects: an effect of one negates an effect of the other
Interference: one deletes a precondition of the other
Competing needs: they have mutually exclusive preconditions
Two literals at the same state-level are mutex if
Inconsistent support: one is the negation of the other,
or all ways of achieving them are pairwise mutex

16. The GRAPHPLAN Algorithm
graph  Initial-planning-graph(problem)
goals  Conjuncts(Problem, GOAL)
nogoods  an empty hash table
For t = 0 to infinity do
if goals all non-mutex in St of graph then
solution  Extract-solution(graph, goals, num-lev(graph), nogoods)
if solution != failure, then return solution
if graph and nogoods have both leveled off then
return failure
graph  Expand-graph(graph, problem)

17. The spare tire problem

18. The spare-tire problem

19. Extract-solution as a search problem
Initial state: the last level of the planning graph Sn
Actions at state level Si: select any conflict-free subset of actions in Ai-1 that covers the goals of the state. The result state: level Si-1, the goals at the new state = the pre-conditions of the selected actions
Goal of the backward search: reach a state at S0 such that all goals are satisfied.
The cost of each action is 1
When extract-solution fails to find a solution for a set of goals at a level, a pair (level, goals) is added as a no-good.

20. Other Classical Planning Approaches
Classical Planning as Boolean satisfiability
Planning as a first-order logical deduction: Situation calculus
Planning as constraint satisfaction