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FORMALIZATION OF STRIPS IN

SITUATION CALCULUS

John McCarthy

Computer Science Department

Stanford University

Stanford, CA 94305

jmc@cs.stanford.edu

2002 Oct 26, 3:00 p.m.

Abstract

[This is a 1985 note with modernized typography.]

STRIPS [FN71] is a planning system that uses logical formulas to repre-

sent information about a state. Each action has a precondition, an add list

and a delete list. When an action is considered, it is ﬁrst determined whether

its precondition is satisﬁed. This can be done by a theorem prover, but my

understanding is that the preconditions actually used are simple enough that

whether one is true doesn’t require substantial theorem proving. If the pre-

condition isn’t met, then another action must be tried. If the precondition

is met, then then sentences on the delete list are deleted from the database

and sentences on the add list are added to it. STRIPS was considered to

be an improvement on earlier systems [Gre69] using the situation calculus,

because these earlier systems ran too slowly.

It was often said that STRIPS was just a specialization of the situation

calculus that ran faster. However, it wasn’t easy to see how to model a

system with delete lists in the situation calculus. In this paper we present

a situation calculus formalization of a version of STRIPS. Whether it is the

full STRIPS isn’t entirely clear. The reader will notice that a large amount

of reiﬁcation is done.

We have situations denoted by s sometimes with subscripts. The initial

situation is often called s0. We also have proposition variables p and q

sometimes with subscripts. Associated with each situation is a database

of propositions, and this gives us the wﬀ db(p, s) asserting that p is in the

database associated with s. We action variables a, possibly subscripted, and

have the function result, where s(cid:48) = result(a, s) is the situation that results

when action a is performed in situation s.

STRIPS is characterized by three predicates.

precondition(a, s) is true provided action a can be performed in situation

deleted(p, a, s) is true if proposition p is to be deleted when action a is

add(p, a, s) is true if proposition p is to be added when action a is per-

s.

performed in situation s.

formed in situation s.

STRIPS has the single axiom

∀p a s.db(p, result(a, s))

≡ if ¬precondition(a, s) then db(p, s)

else db(p, s) ∧ ¬deleted(p, a, s)add(p, a, s).

We now give an example of this use of STRIPS in the blocks world. Our

individual variables x, y and z range over blocks. The constant blocks used

in the example are a, b, c, d and T able. The one kind of proposition is on(x, y)

asserting that block x is on block y. The one kind of action is move(x, y)

denoting the act of moving block x on top of block y. We assume the blocks

are all diﬀerent, i.e.

U N A(a, b, c, d, T able)

We will be interested in an initial situation s0 characterized by

∀xy.db(on(x, y), s0) ≡ (x = a∧y = b)∨(x = b∧y = c)∨(x = c∧y = T able)∨(x = d∧y = T able).Next we characterize the predicates precondition, delete and add. We have

∀xys.precondition(move(x, y), s)

≡ x (cid:54)= T able ∧ x (cid:54)= y ∧ ∀z.¬db(on(z, x), s) ∧ (y (cid:54)= T able ⊃ ∀z.¬db(on(z, y), s)),∀xys.deleted(on(w, z), move(x, y), s) ≡ w = x ∧ z (cid:54)= y

and

∀xys.add(on(w, z), move(x, y), s) ≡ w = x ∧ z = y.

From these axioms and the initial conditions given above we can prove

∀xy.db(on(x, y), result(move(a, b),

result(move(b, c),

result(move(c, d),

≡ (x = a ∧ y = b) ∨ (x = b ∧ y = c) ∨ (x = c ∧ y = d) ∨ (x = d ∧ y = T able).result(move(b, T able),

result(move(a, T able), s0))))))

Remarks:

1. In order to handle deleting we made the database explicit and reiﬁed propositions. This allowed us to quantify over propositions.
2. Since the precondition was expressible conveniently in terms of the presence of sentences in the database, we didn’t require a logic of propo-

sitions. Such a logic can be given by introducing an additional predicate

holds(p, s) and the axiom

db(p, s)holds(p, s)

together with axioms like

holds(pandq, s)holds(p, s)holds(q, s)

and

holds(not p, s) ≡ ¬holds(p, s).

We can then express preconditions in terms of what holds and not merely

what is in the database. However, holds(p, s) will only be very powerful if the

propositions can contain quantiﬁers. This raises some diﬃculties which we

don’t want to treat in this note but which are discussed in [McC79]. Keep-

ing holds(p, s) conceptually separate from db(p, s) means that what holds is

updated each time the situation changes.

3. The database in the blocks world example consists of independent atomic constant propositions. These can be added and deleted independently.

For this reason there is no diﬃculty in determining what remains true when

a proposition is deleted (or rather what propositions are true in result(a, s)).

4. Nilsson points out (conversation 1985 February 4) that we might want more complex propositions to persist rather than be recomputed. For exam-

ple, we might have clear(x) deﬁned by

∀xs.db(clear x, s) ≡ ∀y.¬db(on(y, x), s)

and when move(w, z) occurs with w (cid:54)= x and z (cid:54)= x, we would like to avoid

recomputing clear x. Obviously this can be done.

5. This formulation does no non-monotonic reasoning. The use of ≡ enables us to completely characterize the set of blocks in s0 and to pre-

serve complete knowledge about the situation resulting from moving blocks.

The role of non-monotonic reasoning in these simple situations needs to be

studied. Compare [McC86].

6. Because this formalism reiﬁes propositions, there need not be an iden- tity between logical consistency and consistency of the propositions in their

intended meaning. For example, holds(p, s) and holds(not p, s) are consistent

unless we have an axiom forbidding it, e.g. ∀ps.holds(p, s) ≡ ¬holds(notp, s).

This axiom may be stronger than we want, because it may be convenient to

use an interpretation of holds in which neither holds(p, s) nor holds(notp, s)

is true.

7. As Vladimir Lifschitz [Lif87] has pointed out, a working STRIPS sys- tem may be unstable with regard to adding true propositions to the database,

because unless the proposition is deleted when its truth value changes, it can

lead to inconsistency when an action is performed. This instability is a de-

fect of STRIPS and other systems in which updating is performed by other

means than deduction.

8. As long as much of the knowledge is encoded in the updating rules, the information encoded in sentences may be minimal.
9. Many of the considerations discussed here concerning STRIPS apply to other semi-logical formalisms, e.g. EMYCIN and ART. [1997 note: They

also apply to linear logic and Wolfgang Bibel’s formalism TR.]

References

[FN71] R. E. Fikes and N. J. Nilsson. Strips: A new approach to the appli-

cation of theorem proving to problem solving. Artiﬁcial Intelligence,

2:189–208, 1971.

[Gre69] C. Green. Applications of theorem proving to problem solving. In

Proceedings IJCAI 69, pages 219–240, 1969.

[Lif87] Vladimir Lifschitz. On the semantics of strips. In Michael Georgeﬀ

and Amy Lansky, editors, Reasoning about Actions and Plans.

Morgan-Kaufmann, 1987. reprinted in Readings in Planning Allen,

Handler and Tate (eds.), Morgan-Kaufmann, 1990.

[McC79] John McCarthy. First Order Theories of Individual Concepts and

Propositions1. In Donald Michie, editor, Machine Intelligence, vol-

ume 9. Edinburgh University Press, Edinburgh, 1979. Reprinted in

[McC90].

[McC86] John McCarthy. Applications of Circumscription to Formaliz-

ing Common Sense Knowledge2. Artiﬁcial Intelligence, 28:89–116,

1986. Reprinted in [McC90].

[McC90] John McCarthy. Formalizing Common Sense: Papers by John Mc-

Carthy. Ablex Publishing Corporation, 1990.

/@steam.stanford.edu:/u/jmc/w85/strips.tex: begun Thu Oct 16 11:35:32 1997, latexed October 26, 2002 at 3:00 p.m.

1http://www-formal.stanford.edu/jmc/concepts.html

2http://www-formal.stanford.edu/jmc/applications.html