1. GraphPlan ● (you don't have to know anything about this, but I put it here to point out that there are many clever approaches to try to find plans more efficiently; another would be OBDD, ordered binary decision diagrams)

2. SatPlan ● Encode as propositional Sat ● Add time index – At(p1,SFO) 0 &At(p2,JFK) 0 ● Axioms: one for each plane X city X time ● Actions: Fly(p1,SFO,JFK) 1, Fly(p1,JFK,SFO) 1, Fly(p1,JFK) 2... ● Guess a time and assert goal – at(p1,JFK) t &at(p2,SFO) t ● Try for t=0..Tmax ● Use DPLL to find a model – Look at truth values for actions Is satisfiable if... Pre-cond axioms: Effect/successor-state axioms: Exclusion axioms: -Fly(p1,JFK,SFO) 0 v -Fly(p1,JFK,LAX) 0 one destination at a time -Fly(p1,a,b) 0 v -Fly(p2,c,d) 0 one at a time?

3. Scheduling ● Find plan that minimizes total time ● Assume each operator has a known duration ● Strategy – Construct partial-order plan first – Then find an optimal linearization ● CPM – critical path method – identify max-duration sub-sequence; schedule actions without any delay/slack – More difficult with resource constraints – e.g. 1 drill press in shop, 1 inspector, 1 data-bus

4. – ES: earliest start time, LS: latest start time, slack = LS-ES – Forward pass: ES(B)=max(A<B) ES(A)+duration(A) – Backward pass: LS(A)=min(A<B) LS(B)-duration(A) CPM

5. Hierarchical Task Networks ● Plan libraries – Make pizza={roll dough,apply sauce,add toppings,put in oven} – Standard operation procedures, alternative methods ● Abstraction (leave out pre-conditions) – Example: fly to destination city, take “transport” to hotel, expand it later ● Opportunities for interleaving/sharing? ● Example: building a house ● Can use POP to expand nodes in graph with sub- graphs

7. Conditional Planning ● non-determinism – actions that could fail; external events could change state of world ● uncertainty, partial observability – can’t see in rooms not yet visited (e.g. Wumpus) – don’t know what resources will be available, or used ● approaches – build “robust” plans that will work regardless – use sensors and conditional/contingent actions ● keep chopping tree “until it falls” ● effect of sensing action is to change “belief state” – use probabilities to find plan most likely to succeed – Markov Decision Problems (MDP’s) ● maximize reward accumulated over time, subject to probability distribution over successor states

8. Plan Monitoring and Repair ● Non-determinism: what if actions fail, or unanticipated external events happen? ● Check pre-conds of each step ● If not sat, re-plan from new current state to goal? ● Make plan to get back to nearest state in original plan? ● Serenditiy – check pre-conds of every step ● Futility – re-executing same action might not help (e.g. Prometheus)

9. Additional topics ● Plan optimization - Goal is not only to find any plan, but to find one that minimizes cost, probability of success... ● Dealing with uncertainty – Probability, Markov Decision Problems, reward – Partial observability – keep track of belief states ● Multi-agent planning – e.g. 2 cooperating agents in BlocksWorld ● Reactive planning – build a state/action table (quick rules to eval) to handle all situations and lead to goal