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Overview

- Planning as a separate problem
- Planning formalism
- Example planning problems
- Planning in State Space
- Planning Graphs
- GraphPlan Algorithm
- Other approaches

Planning as a Separate Problem

- Planning = Determining a sequence of actions to take that will achieve some goal(s)
- Doesn’t that sound familiar?
- That sounds similar to a search algorithm
- Also, we could re-cast this idea using first-order logic
- Key insight: planning domains are much more carefully structured and constrained than the general search/resolution problems
- A customized algorithm will have better performance

Planning Domain Definition Language

- PDDL is a factored representation
- Entire world represented by variables
- In essence PDDL defines a search problem:
- Initial state
- Actions (with preconditions)
- Action results (post conditions)
- Goal test/State
- Similar to traditional language called STRIPS

Representing States

- Use a language similar to FOL:
- Poor ^ Unknown
- At(Truck1, Melbourne) ^ At(Truck2, Sydney)
- Difference #1 from FOL: Database semantics
- Anything not mentioned presumed false (no negations needed/allowed)
- Unique names certain to specify distinct objects
- Difference #2: all facts are ground, and functionless
- At(x,y) not permitted, nor is At(Father(Fred), Sydney)
- Can be specified using logic notation (common) or with set semantics where fluents are categorized into groups that are manipulated using set operators

Each fact is called a Fluent

Representing Actions

- Actions are specified in terms of what changes
- Anything not mentioned is presumed to stay the same
- Uses an action schema
- Limited FOL representation (all universally quantified)
- Action name
- Variable list
- Precondition
- Effect
- Taking an action constitutes successfully grounding all the variables to literals that are true (or not) in the state

At(here) ,Path(here,there)

Go(there)

At(there) , At(here)

Sample Action

Action Schema:

Action(Fly(p,from,to),

PRECOND: At(p,from) Plane(p) Airport(from) Airport(to)

Effect: At(p,from) At(p,to))

Actual (Grounded) Action:

Action(Fly(P1,SFO,JFK),

PRECOND: At(P1,SFO) Plane(P1) Airport(SFO) Airport(JFK)

Effect: At(P1,SFO) At(P1,JFK))

Number of grounded Actions: O(Vk) where V is the number of variables in the action, k is the number of literals defined in the state

Applying Actions

- An action is applicable if the preconditions are satisfied by some literals in the state S
- Applying an action a in state S, has the result S’:
- S’ = (S – Del(a)) Add(a)
- Add(a): Add-list comprised of positive literals in a’s effects list
- Del(a): Delete-list comprised of negative literals in a’seffects list
- To remain consistent, we require that any variable in the effects list also appear in the preconditions list
- Time is implicit in the language: Actions are taken at time T, and effects occur at time T+1

Planning Domains

- A set of action schemas completely specifies a Planning Domain
- A single Planning Problem includes all the schemas from the domain, plus an initial state and a goal
- Initial state: any conjunction of ground atoms such that each atom either appears, or can be bound to a variable, in at least one precondition item for at least one action
- Goal: Many ways of veiwing. Simplest: Action schema where goal test is the precondition and the effect is the ground literal: GoalAccomplished
- Multiple Goals are covered by making one more goal in which accomplishing each goal is part of the precondition

Example Planning Domains

- Cargo transport
- Spare tire problem
- Blocks World

Example: Cargo Transport

- Reality: Fed-Ex and UPS
- Simplification:
- Actions = Load, Unload, Fly
- Predicate: In(c,p): package c is in plane p
- Predicate: At(x,a): Item (plane or package) is At airport a
- Packages are no longer ‘at’ the airport, if they are ‘in’ the plane (to compensate for lack of universal quantifiers)
- In Class: define the action schema

Example: Spare Tire Problem

- Goal: restore a car to having 4 good tires
- Fluents: Tire(Flat), Tire(Spare), Trunk, Axle, Ground
- Predicate: At(x,y)
- Actions:
- Remove(obj, loc)
- PutOn(obj,loc)
- LeaveOvernight—all tires are stolen
- In Class: define actions and discuss whether domain is realistic enough to be useful

A

C

TABLE

Example: Blocks worldThe blocks world is a micro-world that consists of a table, a set of blocks and a robot hand.

Some domain constraints:

- Only one block can be on another block
- Any number of blocks can be on the table
- The hand can only hold one block

Typical representation:

ontable(a)

ontable(c)

on(b,a)

handempty

clear(b)

clear(c)

Blocks world operators

- Here are the classic basic operations for the blocks world:
- stack(X,Y): put block X on block Y
- unstack(X,Y): remove block X from block Y
- pickup(X): pickup block X
- putdown(X): put block X on the table
- Each will be represented by
- a list of preconditions
- optionally, a set of (simple) variable constraints
- The effects, split into ADD and DEL:
- a list of new facts to be added (add-effects)
- a list of facts to be removed (delete-effects)
- For example:

preconditions(stack(X,Y), [holding(X), clear(Y)])

deletes(stack(X,Y), [holding(X), clear(Y)]).

adds(stack(X,Y), [handempty, on(X,Y), clear(X)])

constraints(stack(X,Y), [XY, Ytable, Xtable])

Blocks world operators II

operator(stack(X,Y),

Precond [holding(X), clear(Y)],

Constr [XY, Ytable, Xtable],

Add [handempty, on(X,Y), clear(X)],

Delete [holding(X), clear(Y)]).

operator(pickup(X),

[ontable(X), clear(X), handempty],

[Xtable],

[holding(X)],

[ontable(X), clear(X), handempty]).

operator(unstack(X,Y),

[on(X,Y), clear(X), handempty],

[XY, Ytable, Xtable],

[holding(X), clear(Y)],

[handempty, clear(X), on(X,Y)]).

operator(putdown(X),

[holding(X)],

[Xtable],

[ontable(X), handempty, clear(X)],

[holding(X)]).

STRIPS planning

- STRIPS: first major planning system out of SRI
- STRIPS maintains two additional data structures:
- State List - all currently true predicates.
- Goal Stack - a push down stack of goals to be solved, with current goal on top of stack.
- If current goal is not satisfied by present state, examine add lists of operators, and push operator and preconditions list on stack. (Subgoals)
- When a current goal is satisfied, POP it from stack.
- When an operator is on top stack, record the application of that operator on the plan sequence and use the operator’s add and delete lists to update the current state.

C

B

A

B

C

Typical BW planning problemInitial state:

clear(a)

clear(b)

clear(c)

ontable(a)

ontable(b)

ontable(c)

handempty

Goal:

on(b,c)

on(a,b)

ontable(c)

- A plan:
- pickup(b)
- stack(b,c)
- pickup(a)
- stack(a,b)

C

B

A

B

C

Another BW planning problemInitial state:

clear(a)

clear(b)

clear(c)

ontable(a)

ontable(b)

ontable(c)

handempty

Goal:

on(a,b)

on(b,c)

ontable(c)

- A plan:
- pickup(a)
- stack(a,b)
- unstack(a,b)
- putdown(a)
- pickup(b)
- stack(b,c)
- pickup(a)
- stack(a,b)

B

C

C

A

B

Goal state

Goal interaction- Simple planning algorithms assume that the goals to be achieved are independent
- Each can be solved separately and then the solutions concatenated
- This planning problem, called the “Sussman Anomaly,” is the classic example of the goal interaction problem:
- Solving on(A,B) first (by doing unstack(C,A), stack(A,B) will be undone when solving the second goal on(B,C) (by doing unstack(A,B), stack(B,C)).
- Classic STRIPS could not handle this, although minor modifications can get it to do simple cases

Initial state

B

C

C

A

B

Sussman AnomalyAchieve on(a,b) via stack(a,b) with preconds: [holding(a),clear(b)]

|Achieve holding(a) via pickup(a) with preconds: [ontable(a),clear(a),handempty]

||Achieve clear(a) via unstack(_1584,a) with preconds: [on(_1584,a),clear(_1584),handempty]

||Applying unstack(c,a)

||Achieve handempty via putdown(_2691) with preconds: [holding(_2691)]

||Applying putdown(c)

|Applying pickup(a)

Applying stack(a,b)

Achieve on(b,c) via stack(b,c) with preconds: [holding(b),clear(c)]

|Achieve holding(b) via pickup(b) with preconds: [ontable(b),clear(b),handempty]

||Achieve clear(b) via unstack(_5625,b) with preconds: [on(_5625,b),clear(_5625),handempty]

||Applying unstack(a,b)

||Achieve handempty via putdown(_6648) with preconds: [holding(_6648)]

||Applying putdown(a)

|Applying pickup(b)

Applying stack(b,c)

Achieve on(a,b) via stack(a,b) with preconds: [holding(a),clear(b)]

|Achieve holding(a) via pickup(a) with preconds: [ontable(a),clear(a),handempty]

|Applying pickup(a)

Applying stack(a,b)

From [clear(b),clear(c),ontable(a),ontable(b),on(c,a),handempty]

To [on(a,b),on(b,c),ontable(c)]

Do:

unstack(c,a)

putdown(c)

pickup(a)

stack(a,b)

unstack(a,b)

putdown(a)

pickup(b)

stack(b,c)

pickup(a)

stack(a,b)

Goal state

Initial state

Algorithms for Finding Plans in State Space

- STRIPS is intuitive, but has problems because it is ‘working as it thinks’
- Let’s think of another method…
- Planning algorithms have a start-state
- Actions, if applicable, will change the state
- Goal: easily tested by analyzing the current state for the given conditions
- Sounds like a general search problem!
- Why we wasted the time with PDDL: better heuristics

Forward Search

- Start with initial state, apply any operators for which the preconditions are satisfied
- Repeat on frontier nodes until you reach the goal
- Problem: Lots of wasted time exploring irrelevant actions
- Problem: Schema may be small, but adding more fluents exponentially increases the size of the state space
- Not all hope is lost: heuristics will be very helpful!

Backward (Regression) Relevant-States Search

- Similar in philosophy to STRIPS
- Start with goal, search backward to start state
- ONLY explore actions that are relevant to satisfying the preconditions of the current state
- Difference: search algorithms and heuristics can prevent ‘sussman’s anomaly’
- Backward search requires reversible actions, but PDDL is good for that
- Also needs to allow state SETS (since the goal may have non-ground fluents)
- Unification of variables allows backward search to dramatically reduce branching factor vs. forward search
- Set-based heuristics are harder to design.

Planning Heuristics

- Key advantage of PDDL and planning as its own topic: effective problem-independent heuristics
- Insight: PDDL problem descriptions all have a similar structure, independent of problem area
- Makes forward search for many planning problems feasible
- Common choices
- Ignore preconditions
- Ignore delete lists
- Subgoal independence
- Pattern Databases

A new view on heuristics

- We’ve already discussed viewing search in a state space as a graph search problem:
- One way of thinking about designing a heuristic is to add edges to the state space graph
- Shorter paths mean shorter (admissible) estimates for cost to the goal

Ignoring Preconditions

- Apply actions regardless of whether the state has satisfied the preconditions
- Value of heuristic: minimum number of actions such that the union of their effects-sets equals the set of conditions in the goal
- Sound Familiar?
- Approximations are effective, but lose A* admissibility requirement
- May also ignore only selected preconditions
- But this is a domain-dependent heuristic

Ignoring Delete Lists

- Assume goals and preconditions only have positive literals (easy to augment any problem—how?)
- What happens then?
- Remove delete lists
- Executing one action never ‘undoes’ another
- Applying any action monotonically increases toward the goal
- FF Planner (Hoffman 2005) uses this heuristic with a modified hill climbing search that keeps track of solution path

Subgoal Independence

- Assume that solving each of K items in the goal is independent of the others
- Run K planning searches, one for each goal then sum the cost of each sub-plan to estimate cost of total plan
- Optimistic Heuristic: when negative interactions occur between subplans (one undoes the other)
- Admissible
- Pessimistic Heuristic: when subplans contain redundant actions
- Not admissible

Pattern Databases

- Frequently Called Plan Libraries
- Many planners are used repeatedly using new start states
- Action schemas make this possible
- Record exact solutions to plans and sub-plans as you solve them
- Research questions:
- Detecting frequently recurring and useful sub-plans
- Using plan libraries to solve problems in a different domain
- based on qualitative graph structures (Bulka, 2009)

Planning Graphs

- Planning Graph: secondary data structure generated from a planning problem
- Analyzing this data structure leads to a very effective heuristic
- Approximates a fully expanded search tree using polynomial space
- Estimates the steps to reach the goal
- Always correct if goal not reachable
- Always underestimates (so what?)

Planning Graph Basics

- A planning Graph is divided into levels
- Two types of alternating levels:
- State levels Si fluents that MIGHT be true at level i
- Always underestimates the time at which it will actually be true
- Action levels AiActions that MIGHT be applicable at step i
- Takes into consideration some but not all negative interactions between actions
- Negative interaction = Performing one action violates the preconditions of the other
- “The level j at which a literal first appears is a good estimate of how difficult it is to achieve the literal from the initial state [in the final plan].”

Example Problem and Planning Graph

Initial State: Have(Cake)

Goal State: Have(Cake) Eaten(Cake)

Action(Bake(Cake)

Pre: Have(Cake)

Effect: Have(Cake))

Action(Eat(Cake)

Pre: Have(Cake)

Effect: Have(Cake)Eaten(Cake))

Constructing a Planning Graph

- All action schemas must be propositionalized
- Generate all possible grounded actions so no variables are left.
- Start with S0 = all initially true fluents
- Construct Ai = all actions whose preconditions are satisfied by Si
- Construct Si = all fluents made true by the effects of the actions in Ai-1
- All levels in Aihave the NO-OPaction which passes all true fluents in Si to Si+1

Constructing a Planning Graph (Continued)

- Add links:
- Between levels
- From fluents in Si to preconditions of actions in Ai
- From effects of actions in Ai to fluents in Si+1
- Within Levels (mutual exclusion)
- Action Mutex Link: Two actions compete for resources
- State Mutex Link: Two fluents that cannot both be true at the same time

Rules for Constructing Mutex Links

- Action Mutex:
- Inconsistent Effects: One action negates the effect of the other (e.g., Eat(Cake) and Bake(Cake) )
- Interference: One of the effects of one action is the negation of a precondition of the other (e.g., Eat(Cake) and the persistence of Have(Cake))
- Competing Needs: One of the preconditions of one action is mutually exclusive with a precondition of the other
- Depends on State Mutex links in previous level
- State Mutex:
- One is the negation of the other
- All pairs of actions that could make both true have Acton mutex links between them

Example Problem and Planning Graph

Initial State: Have(Cake)

Goal State: Have(Cake) Eaten(Cake)

Action(Bake(Cake)

Pre: Have(Cake)

Effect: Have(Cake))

Action(Eat(Cake)

Pre: Have(Cake)

Effect: Have(Cake)Eaten(Cake))

Growth of the Planning Graph

- Levels are added until two consecutive levels are identical (graph has leveled off)
- How Big is this Graph?
- For a (propositionalized) planning problem with L literals and a actions,
- Each Si level: Max L nodes, L2mutex links
- Each Ai level: Max a + L nodes, (a + L)2mutex links
- Inter-level linkage: L(a + L) links
- Graph with n levels O(n(a+L)2)

Planning Graph Heuristics

- If any goal literal fails appear, then the plan is not solvable
- For a single goal:
- Level at which it appears is a simple estimate for cost of achieving goal
- Serial Planning Graphs insist that one action is performed at each level by adding mutex links between all non-NO-OP actions
- Leads to more effective heuristic
- For a conjunction of goals: next slide

Heuristics for Conjunction of Goals

- Max-Level: Take the max of all goal literals
- Admissible, but poorly conservative
- Level-Sum: sum all goal costs together
- not admissible but works OK much of the time
- Set-Level: First level where all goal literals appear without Mutex links
- Admissible, dominates Max-Level

Comments on Planning Graphs

- Planning Graphs do have shortcomings:
- Only guarantees failure in the obvious case
- If goal does appear at some level, we can only say that there is ‘possibly’ a plan
- In other words, there’s no obvious reason the plan should fail
- Obvious reason = mutex relations
- We could expand the process to consider higher-order mutexes
- More accurate heuristics
- Not usually worth the higher computational cost
- Similar to arc-consistency in CSP’s

Extracting a (Possible) Solution

- Implement as Backward-search
- States = A level of the graph, and a set of goals
- Start at level Sn, add all main goal fluents
- Select a set of non-mutex actions that achieve the goals
- Also, their preconditions cannot be mutex
- New state is Level Sn-1 with its goals being the preconditions of the chosen action set
- If we reach S0 with all goals satisfied, Success!
- Heuristic: Solve the highest-level fluents first, using the lowest cost action (based on preconditions)

About No-Goods

- no-goods is a hash table:
- Store (level,goals) pairs
- If Extract-solution fails it will:
- Record its current goals and level in no-goods
- If Extract-solution reaches the exact same state later, we can stop early and return FAILURE (or, backtrack to a different part of search)
- No-Goods is important for termination
- Can’t just say ‘planning graph leveled off so we’re done’
- Planning graphs grow vertically faster than horizontally
- Some plans need more horizontal space to be solved
- Extract Solutions is thorough
- If a plan exists, it will find it
- If it can’t find a new reason to fail at level Si then adding a level won’t help

Other Approaches

- Boolean satisfiability
- Propositionalize actions, goal
- Add successor-state axioms
- Add pre-condition axioms
- Use SATPLAN algorithm to find a solution
- Seems complex, but is very fast in practice
- Situation Calculus
- Similar to satisfiability above
- Incoporates all of FOL, more expressive than PDDL
- Partial Order Planners (Up Next!)

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