Advanced Knowledge Representation & Reasoning

The Frame Problem

Last Update: 24 November 2008, 3:31 p.m.

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Adapted from:

using some of the notation of:

  1. Let s1 be a blocks-world situation, with blocks a,b,c,d and table "table1" described by:

    1. On(b,a,s1)
    2. Clear(b,s1)
    3. Clear(c,s1)
    4. Clear(d,s1)
    5. On(a,table1,s1)
    6. On(c,table1,s1)
    7. On(d,table1,s1)

    Suppose that there is a robot that can do the following actions on blocks:

    1. Stack(x,y)
    2. Unstack(x,y)


  2. Suppose that the following precondition/effect axioms apply:

    1. (∀s,x,y)[Clear(x,s) ∧ Clear(y,s) ∧ x ≠ y ⊃ On(x,y,Do(Stack(x,y),s))]
    2. (∀s,x,y)[On(x,y,s) ∧ clear(x,s) ⊃ Clear(y,Do(Unstack(x,y),s))]

  3. Let A =def the sequence of actions A1;...;A_n.

    Let Do(A,s) =def Do(A_n, Do(A1;...;A_n-1,s)).

    Let KB contain descriptions of initial situation s1, the effect axioms, the definition of Do, etc.

    Let φ(x,s) be a proposition expressing the goal situation.

    Then "the planning problem" is to find A such that KB ||- &phi(x, Do(A,s))

  4. E.g., suppose KB = I & II & III above,

    & suppose φ = On(a,c,s1).

    Then a plan is: A = Unstack(b,a); Stack(a,c).

    I.e., show KB ||- On(a,c, Do(A,s1)).

  5. But this can't be done!:

    From II(2), can infer: Clear(a, Do(Unstack(b,a),s1)).

    From I(3), the above "Clear" proposition, & II(1), we would like to infer:

    But to infer that, we need:

    But, from the fact that c is initially clear, we cannot infer that it remains clear!

    That is the frame problem!

Copyright © 2008 by William J. Rapaport (