Intelligent Systems

    Planning

Agenda

  • Motivation

  • Technical Solution

  • Illustration by a Larger Example

  • Extensions

  • Summary

  • References

    Motivation

Motivation

  • What is Planning?
    • “Planning is the process of thinking about the activities required to create a desired goal on some scale” [Wikipedia]
  • We take a more pragmatic view – planning is a flexible approach for taking complex decisions:
    • decide about the schedule of a production line;
    • decide about the movements of an elevator;
    • decide about the flow of paper through a copy machine;
    • decide about robot actions.
  • By “flexible” we mean:
    • the problem is described to the planning system in some generic language;
    • a (good) solution is found fully automatically;
    • if the problem changes, all that needs to be done is to change the description.
  • We look at methods to solve any problem that can be described in the language chosen for the particular planning system.

An ‘easy’ planning example

  • Goal: Buy 1 liter of milk

  • It may sound like a simple task, but if you break it down, there are many small tasks involved:

      • obtain car keys,

      • obtain wallet,

      • exit home,

      • start car,

      • drive to store,

      • find and obtain milk,

      • purchase milk,

Logistics

  • During the 1991 Gulf War, US forces deployed an AI logistics planning and scheduling program ( Dynamic Analysis and Replanning Tool ) that involved up to 50,000 vehicles, cargo, and people
  • This one application alone reportedly more than offset all the money DARPA had funneled into AI research in the last 30 years.

Space Exploration

  • Deep Space 1
  • NASA's on-board autonomous planning program controlled the scheduling of operations for a spacecraft

Space Exploration (cont’d)

  • Mars Exploration Rover (MER)
  • Autonomous planning, scheduling, control for Mars Exploration Rover.
  • MER uses a combination of planning techniques including:
    • Mixed-initiative planning that involves significant human participation
    • Constraint-based - quantitative reasoning with metric time and other numerical quantities

    TECHNICAL SOLUTIONS

Planning as State Space Search

  • This lecture will consider planning as state-space search, for which there are several options:
    • Forward from I (initial state) until G (goal state) is satisfied;
    • Backward from G until I is reached;
    • Use a Heuristic function to estimate G- or I-distance, respectively – prefer states that have a lower heuristic value.

  • It will also introduce partial-order planning as a more flexible approach

  • In all cases there are three implicit concerns to consider:
    • Representation – how the problem/state space and goal is defined;
    • Algorithm – e.g. which (combination) of these approaches is used;
    • Complexity and decidability – the feasibility of the approach.

    REPRESENTATION

Planning – basic terminology

  • A problem to be solved by a planning system is called a planning problem

  • The world in which planning takes place is often called the domain or application domain.

  • A state is a point in the search space of the application

  • A planning problem is characterized by an initial state and a goal state .

  • The initial state is the state of the world just before the plan is executed

  • The goal state is the desired state of the world that we want to reach

  • after the plan has been executed. A goal state is also refer as simply goal

STRIPS

  • Stanford Research Institute Problem Solver (STRIPS) is an automated planner developed by Richard Fikes and Nils Nilsson in 1971 [Nilsson & Fikes, 1970]

  • STRIPS is also the name of the planning language used for the STRIPS planner

STRIPS

  • A state (excluding the goal state) is represented in STRIPS as a conjunction of function-free ground literals

    e.g. on(A, Table) ^ on (B, A) ^ ¬ clear(A) ^ clear(B)

    (block A is on the Table, block B is on block A, block A has something on top, block B has nothing on top)

  • A conjunction of function-free ground literals is also know as a fact

  • A goal is represented in STRIPS as a conjunction of literals that may contain variables

    e.g. on(B, Table) ^ on (A, B) ^ clear(A) ^ ¬ clear(B)

    (block B is on the Table, block A is on block B, block B has something on top, block A has nothing on top)

STRIPS

    Definition – Action

      Let P be a set of facts. An action a is a triple a = ( pre(a ), add(a ), del(a )) of subsets of P , where add(a ) \ del(a ) = Ø

      pre(a ) , add(a ) , and del(a ) are called the action precondition , add list , and delete list , respectively

  • In other words, an action is defined in terms of:
    • Preconditions - conjunction of literals that need to be satisfiable before the action is performed
    • Effects - conjunction of literals that need to satisfiable after the action is performed. An effect description includes:
      • Add List: list of formulas to be added to the description of the new state resulting after the action is performed
      • Delete List: list of formulas to be deleted from the description of state before the action is performed

STRIPS

  • The action pickup in the Blocks World domain can be modeled as follows in STRIPS


      pickup(x)

          Preconditions : on(x, Table) ^ handEmpty ^ clear(x)

          Add List : holding(x)

          Delete List : on(x, Table) ^ handEmpty ^ clear(x)

STRIPS

    Definition – Task

    A task is a quadruple (P,A,I,G) where P is a (finite) set of facts, A is a (finite) set of actions, and I and G are subsets of P, I being the initial state and G the goal state.



  • In other words, a task is defined in terms of:
    • A (finite) set of facts
    • A (finite) set of actions that the planner can perform. Each action is defined in terms of preconditions and effects (i.e. add list and delete list)
    • An initial state i.e. the state of the world before the execution of the task
    • A goal state i.e. the desired state of the world that we want to reach

STRIPS

    Definition – Result
      Let P be a set of facts, s P a state, and a an action. The result result(s , ‹a› ) of applying a to s is:
      result(s , ‹a› ) = ( s add(a)) \ del(a) iff pre(a) s ; undefined otherwise
      In the first case, the action a is said to be applicable in s . The result of applying an action sequence a 1 , ... , a n of arbitrary length to s is defined by result(s , a 1 , . . . , a n ) = result(result(s , a 1 , . . . , a n-1 ), a n ) and result(s , ‹› ) = s .

STRIPS

  • In other words, the result of applying an action a in a given state s is a new state of the world that is described by the initial set of formulas describing state s to which the formulas from the add list are added and the formulas from delete list are removed. Action a can be performed only if action’s preconditions are fulfilled in state s
  • e.g.
      state s = on(A, B) ^ handEmpty ^ clear(A)
      action a = unstack (x,y)
          Preconditions : on(x, y) ^ handEmpty ^ clear(x)
          Add List : holding(x) ^ clear(y)
          Delete List : on(x, y) ^ handEmpty ^ clear(x)

STRIPS

      result(s , ‹a› ) = ( s add(a)) \ del(a)
          = ( on(A , B) ^ handEmpty ^ clear(A ) holding(x ) ^ clear(y )) \ on(x , y ) ^ handEmpty ^ clear(x )
          = holding(A ) ^ clear(B )
      Precondition on(x , y ) ^ handEmpty ^ clear(x ) is fulfilled in state s = on(A , B) ^ handEmpty ^ clear(A )

    For convenience let’s denote result(s , ‹a› ) with s1

    Given a second action a1
    action a1 = stack (x,y)
          Preconditions : hoding(x) ^ clear(y)
          Add List : on(x,y) ^ handEmpty ^ clear(x)
          Delete List : holding(x) ^ clear(y)

STRIPS

    result(s , ‹a , a1› ) = result(result(s , a ), ‹a1› ) = result(s1, ‹a1› ) = s1 add(a1)) \ del(a1)

        = ((holding (A) ^ clear(B ) holding(x ) ^ on(x,y ) ^ handEmpty ^ clear(x )) \ holding(x ) ^ clear(y )

        = on(A,B ) ^ handEmpty ^ clear(A )

    Precondition hoding(x ) ^ clear(y ) is fulfilled in state s1 = holding(A ) ^ clear(B )

STRIPS

  • Definition – Plan
      Let (P,A,I,G) be a task. An action sequence ‹a1, ... ,an› is a plan for the task iff G ⊆ result(I, ‹a1, ... ,an›).
    • In other words, a plan is an organized collection of actions.
    • A plan is said to be a solution to a given problem if the plan is applicable in the problem’s initial state, and if after plan execution, the goal is true.
    • Assume that there is some action in the plan that must be executed first. The plan is applicable if all the preconditions for the execution of this first action hold in the initial state.
    • A task is called solvable if there is a plan, unsolvable otherwise

STRIPS

    Definition – Minimal Plan

      Let (P,A,I,G) be a task. A plan ‹a1, . . . ,an› for the task is minimal if ‹a1, . . . , ai-1, ai+1, . . . ,an› is not a plan for any i.

    Definition – Optimal Plan

      Let (P,A,I,G) be a task. A plan ‹a1, . . . ,an› for the task is optimal if there is no plan with less than n actions

STRIPS

  • Example: “The Block World”
    • Domain:
      • Set of cubic blocks sitting on a table
    • Actions:
      • Blocks can be stacked
      • Can pick up a block and move it to another location
      • Can only pick up one block at a time
    • Goal:
      • To build a specified stack of blocks

STRIPS

    Representing states and goals
      Initial State:
        on(A, Table) ^
        on(B, Table) ^
        on(C, B) ^
        clear(A) ^
        clear(C) ^
        handEmpty
      Goal:
        on(B, Table) ^
        on(C, B) ^
        on(A, C) ^
        clear(A) ^
        handEmpty

STRIPS

  • Actions in “Block World”
    • pickup(x)
      • picks up block ‘ x ’ from table
    • putdown(x )
      • if holding block ‘ x ’, puts it down on table
    • stack(x,y )
      • if holding block ‘ x ’, puts it on top of block ‘ y
    • unstack(x,y )
      • picks up block ‘ x ’ that is currently on block ‘ y
  • Action stack(x,y ) in STRIPS
        stack (x,y)
            Preconditions : hoding(x) ^ clear(y)
            Add List : on(x,y) ^ handEmpty ^ clear(x)
            Delete List : holding(x) ^ clear(y)

PDDL

  • Planning Domain Description Language (PDDL)
  • Based on STRIPS with various extensions
  • Created, in its first version, by Drew McDermott and others
  • Used in the biennial International Planning Competition (IPC) series
  • The representation is lifted, i.e., makes use of variables these are instantiated with objects
  • Actions are instantiated operators
  • Facts are instantiated predicates
  • A task is specified via two files: the domain file and the problem file
  • The problem file gives the objects, the initial state, and the goal state
  • The domain file gives the predicates and the operators; these may be re-used for different problem files
  • The domain file corresponds to the transition system, the problem files constitute instances in that system

PDDL

    Blocks World Example domain file :

    (define (domain blocksworld)
    (:predicates (clear ?x)
    (holding ?x)
    (on ?x ?y))
    (
        :action stack
        :parameters (?ob ?underob)
        :precondition (and (clear ?underob) (holding ?ob))
        :effect (and (holding nil) (on ?ob ?underob)
            (not (clear ?underob)) (not (holding ?ob)))
    )

PDDL

    Blocks World Example problem file :

    (define (problem bw-xy)
      (:domain blocksworld)
      (:objects nil table a b c d e)
      (:init (on a table) (clear a)
        (on b table) (clear b)
        (on e table) (clear e)
        (on c table) (on d c) (clear d)
        (holding nil)
      )
      (:goal (and (on e c) (on c a) (on b d))))

    ALGORITHMS

Algorithms

  • We now present particular algorithms related to (direct) state-space search:

    • Progression -based search also know as forward search

    • Regression -based search also know as backward search

    • Heuristic Functions and Informed Search Algorithms

Algorithms

    Definition – Search Scheme

    A search scheme is a tuple (S, s0, succ, Sol):

  1. the set S of all states s ∈ S,

  2. the start state s0 ∈ S,

  3. the successor state function succ : S → 2S, and

  4. the solution states Sol ⊆ S.

      Note:
    • Solution paths s0 →, ... , → sn ∈ Sol correspond to solutions to our problem

Progression

  • Definition Progression
      Let (P,A,I,G) be a task. Progression is the quadruple (S, s0, succ, Sol):
    • S = 2P is the set of all subsets of P,
    • s0 = I,
    • succ : S → 2S is defined by succ(s) = {s0 ∈ S | ∃a ∈ A: pre(a) ⊆ s, s0 = result(s, ‹a›)}
    • Sol = {s ∈ S | G ⊆ s}
  • Progression algorithm:
  1. Start from initial state
  2. Find all actions whose preconditions are true in the initial state (applicable actions)
  3. Compute effects of actions to generate successor states
  4. Repeat steps 2-3, until a new state satisfies the goal
      Progression explores only states that are reachable from I ; they may not be relevant for G

Progression

    Progression in “Blocks World“ – applying the progression algorithm

    Step 1:
      Initial state s0 = I = on(A, Table) ^ on(B, Table) ^ on(C, B) ^ clear(A) ^ clear(C) ^ handEmpty
    Step 2:
      Applicable actions: unstack(C,B), pickup(A)
    Step 3:
      result(s0, ‹unstack(C,B)›) = on(A, Table) ^ on(B, Table) ^ on(C, B) ^ clear(A) ^ clear(C) ^ handEmpty holding(x) ^ clear(y) \ on(x, y) ^ handEmpty ^ clear(x) = on(A, Table) ^ on(B, Table) ^ clear(A) ^ clear(B) ^ holding(C)
      result(s0, ‹pickup(A)›) = on(A, Table) ^ on(B, Table) ^ on(C, B) ^ clear(A) ^ clear(C) ^ handEmpty holding(x) \ on(x, Table) ^ handEmpty ^ clear(x) = on(B, Table) ^ on(C, B) ^ clear(C) ^ holding(A)

Regression

    Definition – Regression
    Let P be a set of facts, s P , and a an action. The regression regress(s , a) of s through a is:

    regress(s , a) = ( s \ add(a )) pre(a ) if add(a) s Ø, del(a ) s = Ø ; undefined otherwise

    In the first case, s is said to be regressable through a.
  • If s \ add(a ) = Ø ; then a contributes nothing
  • If s \ del(a ) Ø ; then we can’t use a to achieve p s p

Regression

    Regression algorithm:

  1. Start with the goal

  2. Choose an action that will result in the goal

  3. Replace that goal with the action’s preconditions

  4. Repeat steps 2-3 until you have reached the initial state

  • Regression explores only states that are relevant for G; they may not be reachable from I
  • Regression is not as non-natural as you might think: if you plan a trip to Thailand, do you start with thinking about the train to Frankfurt?

Regression

    Regression in “Blocks World“ – applying the regression algorithm
    Step 1:
      Goal state s n = G = on(B, Table) ^ on(C, B) ^ on(A, C) ^ clear(A) ^ handEmpty
    Step 2:
      Applicable actions: stack(A,C)
    Step 3:
      regress(G, stack(A,C)) = G \ add(stack ) pre(stack ) = on(B, Table) ^ on(C, B) ^ on(A, C) ^ clear(A) ^ handEmpty \ on(A,C) ^ handEmpty ^ clear(A) hoding(A) ^ clear(C) = on(B, Table) ^ on(C, B) ^ holding(A) ^ clear(C)

Heuristic Functions

    Definition – Heuristic Function

      Let (S, s 0 , succ , Sol) be a search scheme. A heuristic function is a function h : S → N 0 { } from states into the natural numbers including 0 and the symbol. Heuristic functions , or heuristics , are efficiently computable functions that estimate a state’s “solution distance”.  

Heuristic Functions

    A heuristic function for Blocks World domain could be:

    h1(s) = n sum(b(i )) , i =1… n , where

      b(i )=0 in state s if the block i is resting on a wrong thing

      b(i )=1 in state s if the block i is resting on the thing it is supposed to be resting on;

      n is the total number of blocks

    h1(I) = 3 – 1 (because of block B) – 0 (because of block C) – 0 (because of block A) = 2

    h1(G) = 3 – 1 (because of block B) – 1 (because of block C) – 1 (because of block A) = 0

Heuristic Functions

    Definition – Solution Distance

      Let (S, s 0 , succ , Sol) be a search scheme. The solution distance sd(s ) of a state s ∈ S is the length of a shortest succ path from s to a state in Sol , or 1 if there is no such path. Computing the real solution distance is as hard as solving the problem itself.

Heuristic Functions

    In Progression search:
      sd(I ) = 4, sd(result(I , pickup(C) )) = 3, sd(result(result(I , pickup(C) )), ‹stack (C,B) ) = 2, … for all other s ∈ succ(I)


    In Regression search:
      sd(G ) = 4, sd(regress(G , stack (A,C) )) = 3, sd(regress(regress(G , stack (A,C) )), pickup (A ) ) = 2, …

Heuristic Functions

  • Definition – Admissible Heuristic Function

      Let (S, s0 , succ , Sol) be a search scheme. A heuristic function h is admissible if h(s ) sd(s ) for all s S .

    • In other words, an admissible heuristic function is a heuristic function that never overestimates the actual cost i.e. the solution distance.

    • e.g. h1 heuristic function defined below is admissible

      h1(s) = n sum(b(i )) , i =1… n , where

      • b(i )=0 in state s if the block i is resting on a wrong thing

      • b(i )=1 in state s if the block i is resting on the thing it is supposed to be resting on;

      • n is the total number of blocks

    COMPLEXITY AND DECIDABILITY

Complexity and Decidability

  • Its time complexity : in the worst case, how many search states are expanded?

  • Its space complexity : in the worst case, how many search states are kept in the open list at any point in time?

  • Is it complete , i.e. is it guaranteed to find a solution if there is one?

  • Is it optimal , i.e. is it guaranteed to find an optimal solution?

Complexity and Decidability

    Definition - PLANSAT

      Let PLANSAT denote the following decision problem. Given a STRIPS task (P, A, I, G), is the task solvable?

    Theorem

      PLANSAT is PSPACE-complete. Proof in see [Bylander, 1994]

    Definition – Bounded- PLANSAT

      Let Bounded-PLANSAT denote the following decision problem. Given a STRIPS task (P, A, I, G) and an integer b . Is there a plan with at most b actions?

    ILLUSTRATION BY A LARGER EXAMPLE

    PROGRESSION

Progression

    Progression in “Blocks World“ – applying the progression algorithm
    Step1:
      Initial state s 0 = I = on(A, Table) ^ on(B, Table) ^ on(C, Table) ^ on(D, C) ^ clear(A) ^ clear(B) ^ clear(D) ^ handEmpty
    Step 2:
      Applicable actions: pickup(A), pickup(B), unstack(D,C)
    Step 3:
      s11 = result(s 0 , ‹pickup(A)›) = on(A, Table) ^ on(B, Table) ^ on(C, Table) ^ on(D, C) ^ clear(A) ^ clear(B) ^ clear(D) ^ handEmpty holding(x) \ on(x, Table) ^ handEmpty ^ clear(x) = on(B, Table) ^ on(C, Table) ^ on(D, C) ^ clear(B) ^ clear(D) ^ holding(A)

Progression

    s12 = result(s0, ‹pickup(B)›) = on(A, Table) ^ on(B, Table) ^ on(C, Table) ^ on(D, C) ^ clear(A) ^ clear(B) ^ clear(D) ^ handEmpty holding(x) \ on(x, Table) ^ handEmpty ^ clear(x) = on(A, Table) ^ on(C, Table) ^ on(D, C) ^ clear(A) ^ clear(D) ^ holding(B)



    s13 = result(s0, ‹unstack(D,C)›) = on(A, Table) ^ on(B, Table) ^ on(C, Table) ^ on(D, C) ^ clear(A) ^ clear(B) ^ clear(D) ^ handEmpty holding(x) ^ clear(y) \ on(x, y) ^ handEmpty ^ clear(x) = on(A, Table) ^ on(B, Table) ^ on(C, Table) ^ clear(A) ^ clear(B) ^ clear(C) ^ holding(D)

Progression

    2 nd iteration
    Step1:
      s11 = on(B, Table) ^ on(C, Table) ^ on(D, C) ^ clear(B) ^ clear(D) ^ holding(A)
    Step 2:
      Applicable actions: putdown (A) , stack(A,D)
    Step 3:
      s111 = result(s11, ‹putdown (A)›) = on(B, Table) ^ on(C, Table) ^ on(D, C) ^ clear(B) ^ clear(D) ^ holding(A) on(x, Table) ^ handEmpty ^ clear(x) \ holding(x) = on(A, Table) ^ on(B, Table) ^ on(C, Table) ^ on(D, C) ^ clear(A) ^ clear(B) ^ clear(D) ^ handEmpty
      we are back in the initial state!

Progression

    3 rd iteration
    Step1:
      s11 = on(B, Table) ^ on(C, Table) ^ on(D, C) ^ clear(B) ^ clear(D) ^ holding(A)
    Step 2:
      Applicable actions: putdown (A), stack(A,D )
    Step 3:
      s112 = result(s11, ‹stack (A,D)›) = on(B, Table) ^ on(C, Table) ^ on(D, C) ^ clear(B) ^ clear(D) ^ holding(A) on(x, y) ^ handEmpty ^ clear(x) \ holding(x) ^ clear(y)= on(B, Table) ^ on(C, Table) ^ on(D, C) ^ on(A,D) ^ handEmpty ^ clear(A) ^ clear(B)

Progression

    4 th iteration
    Step1:
      s12 = on(A, Table) on(C, Table) on(D, C) ^ clear(A) ^ clear(D) ^ holding(B)
    Step 2:
      Applicable actions: putdown (B), stack(B,D)
    Step 3:
      s121 = result(s12, 〈putdown (B)〉) = on(A, Table) on(C, Table) on (D, C) ^ clear(A) ^ clear(D) ^ holding(B) ∪ on(x, Table) ^ handEmpty ^ clear(x) \ holding(x) = on(A, Table) on(B, Table) on (C, Table) on(D, C) ^ clear(A) ^ clear(B) ^ clear(D) ^ handEmpty

      we are back in the initial state!

Progression

    5 th iteration
    Step1:
      s12 = on(A, Table) on(C, Table) on(D, C) ^ clear(A) ^ clear(D) ^ holding(B)
    Step 2:
      Applicable actions: putdown (B), stack(B,D)
    Step 3:
      s122 = result(s11, 〈stack (B,D)〉) = on(A, Table) on(C, Table) on(D, C) ^ clear(A) ^ clear(D) ^ holding(B) ∪ on(x, y) ^ handEmpty ^ clear(x) \ holding(x) ^ clear(y)= on(A, Table) on(C, Table) on(D, C) on(B,D) ^ handEmpty ^ clear(A) ^ clear(B)

Progression

    6 th iteration
    Step1:
      s13 = on(A, Table) on(B, Table) on(C, Table) ^ clear(A) ^ clear(B) ^ clear(C) ^ holding(D)
    Step 2:
      Applicable actions: putdown(D), stack(D,A), stack(D,B), stack(D,C)
    Step 3:
      s131 = result(s13, 〈putdown(D)〉) = on(A, Table) on(B, Table) on(C, Table) ^ clear(A) ^ clear(B) ^ clear(C) ^ holding(D) ∪ on(x, Table) ^ handEmpty ^ clear(x) \ holding(x) = on(A, Table) on(B, Table) on (C, Table) on(D, Table) ^ clear(A) ^ clear(B) ^ clear(C) ^ clear(D) ^ handEmpty
    In iterations 7,8,9, new states are obtained from s13 by applying the actions stack(D,A), stack(D,B), stack(D,C)

Progression

    10 th iteration
    Step1:
      s131 = on(A, Table) on(B, Table) on(C, Table) on(D, Table) ^ clear(A) ^ clear(B) ^ clear(C) ^ clear(D) ^ handEmpty
    Step 2:
      Applicable actions: pickup(A), pickup(B), pickup(C) , pickup(D)
    Step 3:
      s 1311 = result(s 131 , 〈putdown(C)〉) = on(A, Table) on(B, Table) on (C, Table) on(D, Table) ^ clear(A) ^ clear(B) ^ clear(C) ^ clear(D) ^ handEmpty ∪ holding(x) \ on(x, Table) ^ clear(x) ^ handEmpty = on (A, Table) on(B, Table) on(D, Table) ^ clear(A) ^ clear(B) ^ clear (D) ^ holding(C)

Progression

    n th iteration
    Step1:
      s n-1 = on(A, Table) on(D, Table) on(B,C) ^ clear(A) ^ clear(B) ^ holding(C)
    Step 2:
      Applicable actions: putdown(C), stack(C,A) , stack(C,B)
    Step 3:
      s n = result(s n-1 , 〈stack(C,A)〉) = on(A, Table) on(D, Table) on (B,C) ^ clear(A) ^ clear(B) ^ holding(C) ∪ on(x,y) ^ clear(x) ^ handEmpty \ clear(y) ^ holding(x) = on(A, Table) on(D, Table) on (C, A) on(B,D) ^ clear(C) ^ clear(B) ^ handEmpty

      s n is the goal state G

    REGRESSION

Regression

    Regression in “Blocks World“ – applying the regression algorithm
    1 th iteration
    Step1:
      Goal state sn = G = on(A, Table) ôn(D, Table) ôn(C, A) ôn(B,D) ^ clear(C) ^ clear(B) ^ handEmpty
    Step 2:
      Applicable actions: stack(C,A), stack(B,D)

Regression

    Step 3:

      sn-1=regress(G, stack(C,A)) = G \ add(stack ) pre(stack ) = on(A, Table) ^ on(D, Table) ^ on(C, A) ^ on(B,D) ^ clear(C) ^ clear(B) ^ handEmpty \ on(C,A) ^ handEmpty ^ clear(C) holding(C) ^ clear(A) = on(A, Table) ^ on(D, Table) ^ on(B, D) ^ clear(B) ^ clear(A) ^ holding(C)


      sn-2=regress(G, stack(B,D)) = G \ add(stack ) pre(stack ) = on(A, Table) ^ on(D, Table) ^ on(C, A) ^ on(B,D) ^ clear(C) ^ clear(B) ^ handEmpty \ on(B,D) ^ handEmpty ^ clear(B) holding(B) ^ clear(D) = on(A, Table) ^ on(D, Table) ^ on(C, A) ^ clear(C) ^ clear(D) ^ holding(B)

Regression

    2 nd iteration
    Step1:
      s n-1 = on(A, Table) on(D, Table) on(B, D) ^ clear(B) ^ clear(A) ^ holding (C)
    Step 2:
      Applicable actions: pickup(C) , unstack(C,A)
    Step 3:
      s n-2 =regress(s n-1 , pickup(C)) = s n-1 \ add(pickup) ∪ pre(pickup) =on(A, Table) on(D, Table) on(B, D) ^ clear(B) ^ clear(A) ^ holding(C) \ holding (C) ∪ clear(C) on(C, Table) ^ handEmpty = on(A, Table) on(D, Table) ^ on(B, D) ^ clear(B) ^ clear(A) ^ clear (C) on(C, Table) ^ handEmpty

Regression

    3 rd iteration
    Step1:
      sn-1 = on(A, Table) on(D, Table) on(B, D) ^ clear(B) ^ clear(A) ^ holding (C)
    Step 2:
      Applicable actions: pickup(C), unstack(C,A)
    Step 3:
      sn-3=regress(sn-1, unstack(C,A)) = sn-1 \ add(unstack) ∪ pre(unstack) = on (A, Table) on(D, Table) on(B, D) ^ clear(B) ^ clear(A) ^ holding(C) \ holding(C) ^ clear(A) ∪ clear(C) on(C, A) ^ handEmpty = on(A, Table) ^ on(D, Table) on(B, D) ^ clear(B) ^ clear (C) on(C, A)
      we are back in the goal state!

Regression

    4 th iteration
    Step1:
      sn-2 = on(A, Table) on(D, Table) on(B, D) ^ clear(B) ^ clear(A) ^ clear (C) on(C, Table) ^ handEmpty
    Step 2:
      Applicable actions: putdown(C), putdown(A), stack(B,D)
    Step 3:
      sn-4=regress(sn-2, stack(B,D)) = sn-2 \ add(stack) ∪ pre(stack) = on(A, Table) ^ on(D, Table) on(B, D) ^ clear(B) ^ clear(A) ^ clear (C) on(C, Table) ^ handEmpty \ on(B,D) ^ clear(B) ^ handEmpty ∪ clear(D) ^ holding(B) = on (A, Table) on(D, Table) on(C, Table) ^ clear(A) ^ clear (C) ^ clear(D) ^ holding(B)

Regression

    n th iteration
    Step1:
      s1 = on(A, Table) on(B, Table) on(C, Table) ^ clear(A) ^ clear(B) ^ clear (C) ^ holding(D)
    Step 2:
      Applicable actions: pickup(D), unstack(D,C)
    Step 3:
      s0=regress(s1, unstack(D,C)) = s1 \ add(unstack) ∪ pre(unstack) =on(A, Table) on(B, Table) on(C, Table) ^ clear(A) ^ clear(B) ^ clear (C) ^ holding(D) \ holding(D) ^ clear(C) ∪ on(D,C) ^ clear(D) ^ handEmpty = on (A, Table) on(B, Table) on(C, Table) on(D, C) ^ clear(A) ^ clear(B) ^ clear(D) ^ handEmpty

    EXTENSIONS

Partial-Order Planning

  • Forward and backward state-space search are both forms of totally-ordered plan search – explore linear sequences of actions from initial state or goal.

  • As such they cannot accomodate problem decomposition and work on subgraphs separately.

  • A partial-order planner depends on:
    • A set of ordering constraints , wherein A ≺ B means that A is executed some time before B (not necessarily directly before)
    • A set of causal links , A –p-> B, meaning A achieves p for B
    • A set of open preconditions not achieved by any action in the current plan

Partial-Order Plans

   
 
Partial Order Plan:
Total Order Plans:
 

Constraint-Based Search

  • Compared with a classical search procedure, viewing a problem as one of constraint satisfaction can reduce substantially the amount of search.

  • Constrain-based search involved complex numerical or symbolic variable and state constraints

  • Constraint-Based Search:

    1. Constraints are discovered and propagated as far as possible.

    2. If there is still not a solution, then search begins, adding new constraints.

Constraint-Based Search

  • Constrain the task by a time horizon, b

  • Formulate this as a constraint satisfaction problem and use backtracking methods to solve it: “do x at time t yes/no”, “do y at time t yes/no”,

  • If there is no solution, increment b



  • Inside backtracking : constraint propagation, conflict analysis are used

  • Constraint-Based Searcg is an “undirected” search i.e. there is no fixed time-order to the decisions done by backtracking

    SUMMARY

Planning Summary

  • Planning is a flexible approach for taking complex decisions:
    • the problem is described to the planning system in some generic language;
    • a (good) solution is found fully automatically;
    • if the problem changes, all that needs to be done is to change the description.
  • Following concerns must be addressed:
    • Syntax - we have examined representations in PDDL and STRIPS
    • Semantics – we have examined progression, regression and the use of heuristics in algorithms
    • Complexity and decidability

    REFERENCES

References

  • Mandatory reading:
    • D. McDermott. “PDDL – the planning domain definition language”, Technical Report CVC TR-98-003/DCS TR-1165, Yale Center for Computational Vision and Control, 1998.
    • N. Nilsson and R. Fikes. “STRIPS: A New Approach to the Application of Theorem Proving to Problem Solving”, SRI Technical Note 43, 1970. http://www.ai.sri.com/pubs/files/tn043r-fikes71.pdf
  • Further reading:
    • S. Russell and P. Norvig. “AI: A Modern Approach” (2nd Edition), Prentice Hall, 2002
    • G. Malik; D.S. Nau, P. Traverso (2004), Automated Planning: Theory and Practice, Morgan Kaufmann, ISBN 1-55860-856-7
    • T .Bylander,The Computationa lComplexity of Propositional STRIPS Planning, Artificial Intelligence Journal, 1994
  • Wikipedia links:
    • http://en.wikipedia.org/wiki/Automated_planning_and_scheduling
    • http://en.wikipedia.org/wiki/STRIPS
    • http://en.wikipedia.org/wiki/Hierarchical_task_network
    • http://en.wikipedia.org/wiki/Planning_Domain_Definition_Language

    Questions?