Homework 1 ( Written Portion )  Max : 75  Min : 38  Avg : 57.6  Median : 58 (77%)

Planning Graph  A planning graph consists in a sequence of levels that correspond to time steps  Level 0 is the initial state  Each level contains a set of literals that could be true at this time step  Each level contains a set of actions that could be applied at this time step

Have Cake and Eat it Too Init ( Have ( Cake )) Goal ( Have ( Cake ) ∧ Eaten ( Cake )) Action ( Eat ( Cake ) PRECOND : Have ( Cake ) EFFECT : ￢ Have ( Cake ) ∧ Eaten ( Cake )) Action ( Bake ( Cake ) PRECOND : ￢ Have ( Cake ) EFFECT : Have ( Cake ))

Planning Graph  Level A 0 contains all the actions that could occur in state S 0.  Persistence actions ( small boxes ) represent the fact that one literal is not modified.  Mutual exclusions ( mutexes, gray lines ) represent conflicts between actions.  To go from level 0 to the level 1, you pick a set of non exclusives actions ( for instance, action Eat ( Cake ))  Level S 1 contains all the literals that could result from picking any subset of actions in A 0.  Mutexes represent conflicts between literals.

How to build the planning graph

Mutexes  A mutex between two actions indicates that it is impossible to perform these actions in parallel.  A mutex between two literals indicates that it is impossible to have these both literals true at this stage.

How to compute mutexes  Actions  Inconsistent effects : two actions that lead to inconsistent effects  Interference : an effect of the first action negates the precondition of the other action  Competing needs : a precondition of the first action is mutually exclusive with a precondition of the second action.  Literals  one literal is the negation of the other one  Inconsistency support : each pair of action achieving the two literals are mutually exclusive.

 Inconsistent effects : Eat ( Cake ) & noop of Have ( Cake ) disagree on effect Have ( Cake )  Interference : Eat ( Cake ) negates precondition of the noop of Have ( Cake )  Competing needs : Bake ( Cake ) & Eat ( Cake ): compete on Have ( Cake ) precondition

 In S 1, Have ( Cake ) & Eaten ( Cake ) are mutex  In S 2, they are not because Bake ( Cake ) & the noop of Eaten ( Cake ) are not mutex

Plan Graph Summary  Continue until two consecutive levels are identical.  Graph indicates which actions are not executable in parallel  Construction polynomial  No choice which action to take, only indicate which are forbidden to occur in parallel

Planning graph for heuristic search  Using the planning graph to estimate the number of actions to reach a goal  If a literal does not appear in the final level of the planning graph, then there is no plan that achieve this literal !  h = ∞

Heuristics  max - level : take the maximum level where any literal of the goal first appears  admissible  level - sum : take the sum of the levels where any literal of the goal first appears  not admissible, but generally efficient ( specially for independent subplans )  set - level : take the minimum level where all the literals of the goal appear and are free of mutex  admissible

Graphplan Algorithm  Extracts a plan directly from the plan graph G RAPH P LAN (problem) returns solution or failure graph  I NITIAL P LANNING G RAPH (problem) goals  G OALS [problem] loop do if goals all non-mutex in last level of graph then do solution  E XTRACT S OLUTION (graph,goals,L ENGTH (graph)) if solution  failure then return solution else if N O S OLUTION P OSSIBLE (graph) then return failure graph  E XPAND G RAPH (graph,problem)

Questions ?

Partially Ordered Plans  Least Commitment Principle :  One should make choices only about things that you currently care about, leaving the others to be worked out later.  Total Order Planner  Planner in which plans consist of a simple lists of steps ( this is what we ’ ve seen so far )  Partial Order Planner  A planner that can represent plans in which some steps are ordered ( before or after ) w. r. t each other and other steps are unordered.

Left Sock Start Finish Right Shoe Left Shoe Right Sock Start Right Sock Finish Left Shoe Right Shoe Left Sock Start Right Sock Right Sock Right Sock Right Sock Right Sock Left Sock Left Sock Left Sock Left Sock Left Sock Left Sock Right Shoe Right Shoe Right Shoe Right Shoe Right Shoe Left Shoe Left Shoe Left Shoe Left Shoe Finish Left Shoe on Right Shoe on Left Sock onRight Sock on Partial Order Plan: Total Order Plans:

Planning Languages  STRIPS  Stanford Research Institute Problem Solver  Historically important  ADL  Action Description Languages  See Table 11.1 for STRIPS versus ADL  PDDL  Planning Domain Definition Language  Revised & enhanced for the needs of the International Planning Competition

PLANNING IN REAL WORLD

Planning in Ideal and Real Worlds  Classical planners assume  Fully observable, static and deterministic domains  Correct and complete action descriptions  … allowing a “ plan - first - then - act ” planning approach  … but in the real world  The world is dynamic, and time cannot be ignored  Information on the world is incomplete and incorrect  … the agent must be prepared for unexpected events  Plus - scaling up to real - world problem size !

Time, Schedules and Resources  The PDDL language allows events ( actions ) and ordering of events, but not time duration  In real - life planning, we must take duration, delays, etc. into account ( not just ordering )  Example : Job shop scheduling :  The problem is to complete a set of jobs  Each job consists of a set of actions, with given duration and resource requirements  Determine a schedule that minimizes total time needed while respecting resource constraints  Must extend representation language to express duration and resource constraints

Job - Shop Scheduling Problem

Scheduling – no resource constraints 1. Create a partially - ordered plan 2. Use the critical - path method to determine the schedule.  Critical path = the path whose total duration is longest.  This solution assumes no resource constraints. Note that 2 engines are being added simultaneously.  Shortest duration schedule, given partial - order plan :  85 minutes [earliest start, latest start]

Scheduling – with resource constraints  Takes longer, since same actions ( on different cars ) cannot overlap due to resource constraints (115 mins )  Critical Path = AddEngine 1, AddEngine 2, AddWheels 2, Inspect 2  Remaining actions have considerable slack time : they can begin much later without affecting the total plan time.

Planning + Scheduling  Serial : Plan, then Schedule.  Interleaved : Mix planning and scheduling. For example, include resource constraints during partial planning.  Note : We are still working in classic planning environments ( i. e., observable, deterministic, static and discrete ), but have added absolute start times and durations.

Complexity of Temporal Planning  Finding critical path O ( Nb )  N is number of actions, b is branching factor  Introducing resource constraints that restrict which actions can overlap turns this into an NP - hard problem !  This is a really big area in AI right now.  Recent work out of MIT, in collaboration with Boeing and NASA, can perform human - robot teaming tasks of 15+ actions.

Hierarchical Planning  Hierarchical Problem Decomposition - express actions at one level as small sets of actions at the next lower level.  Solve relatively small problems at each level of abstraction.  Hierarchical Task Network ( HTN ) Planning - begin with abstract plan and continue expanding each activity ( i. e., replacing it with its lower - level realization ) until plan consists of only low - level actions. Most real - world planners use HTN variants

The formalism of HTN planning  Retain the basic environmental assumptions as for classical planning  What we previously simply called actions are now " primitive actions “  Add HLAs : High Level Actions ( like Build House )  each has 1 or more possible refinements into sequences of actions

Example

What To Remember  Basics of classical planning  State, action, preconditions, effects, goals  Assumptions made by classical planning  How forward and backward search works  What a planning graph is and it tells us  What a partially ordered plan is  Temporal planning  Critical path is key to planning when actions have durations  Adding resource constraints makes planning really hard  Hierarchical Task Networks help to simplify all types of planning