- ...table
- The concepts of
*final*position and*tower-deadlock*will be defined formally later on.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ...strategy
- If we assume the existence
of an initial situation 9#9, the
problem of determining
what sequences of actions may be selected
according to a particular strategy can be
seen as the problem of determining what
situations are selectable for that strategy.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ...4#4
- The situational argument 4#4 in the
predicates
*Good, Bad*and*Better*allows reasoning about multiple goals. For example, by substituting the variable 4#4 by two different constants 5#5 and 6#6 we can express the fact that performing action*a*at situation*s*is good for the purpose of achieving the goal described by situation 5#5, but bad for achieving the goal described by situation 6#6.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ...situation
- The idea here is to
select always the best possible action.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ...achieved
- We define 10#10
and
*Prec*(*a*,*s*) formally later on. In general, a situation*s**achieves*the goal described by another situation 4#4 if all the conditions (propositional fluents) that*hold*at 4#4*hold*at*s*as well.*Prec*(*a*,*s*) is true if action*a*can be performed at situation*s*.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ...)
- In the paper, we use the
expression ``propositional fluent'' and
the word ``proposition'' to refer to the
corresponding reified formula.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ...symbols
- The symbols
*h*and*g*are meta-variables ranging over distinct function symbols; the expressions 16#16 and 17#17 represent tuples of variables.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ...that
- We use
the following notation to abbreviate the
description of situations 19#19,
and 20#20, where
*l*is a sequence of actions (i.e., sequences of actions are applied from left to right).. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ...values
- The symbols 23#23 and
24#24 are syntactical variables ranging
over block constants.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ...30#30)
- Notice
the distinction between 31#31
(axiom ) and
*reachable*32#32 (axiom ). It is crucial for understanding the role of the goal situation 33#33 in the formalization. The fact that axiom implies that the goal situation 33#33 is not*reachable*from the initial situation 34#34 does not imply that the planning problem is not solvable. When the program selects an action (see axiom defining the fluent*selectable*), it tries to find a situation*reachable*from the initial situation at which the goal is*achieved*. That is, a situation that satisfies the conditions imposed on the goal situation. In this sense, we could say that the role of the goal situation 33#33 is purely descriptive, as far as this paper is concerned.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ...partially
- Axioms and
are not explicit definitions of
*final*and*above*, because these symbols occur both on the left and right hand sides. But these formulas are strong enough for deriving both positive and negative ground instances of*Holds(Above(x,y),s)*and*Holds(Final(x,S36#36),s)*from the positive and negative ground instances of*Holds(On(x,y),s)*that can be derived from axioms to . [Davis1990] points out that it is possible to define*on*in terms of*beneath*(37#37), but it is not possible to fully define*beneath*in terms of*on*in a first order theory.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ...sets
- In the
following, we denote the universal
closure of a formula
*A*by*A*'.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ...logic
- Notice that the only second
order axiom in 41#41 is an axiom of
induction for situations, which we do not
need to decide the selectability of a
single situation.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ...descriptions
- The expression
*(strategy)*can also be parameterized with respect to the problem description (axioms and ), the theory of action (axioms to ), or the mechanism for action selection (axiom ).. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ...strategies
- A
strategy is
*state-based*if whether an action is good or bad for a given goal at a particular situation depends only on what holds at that situation (i.e. the*state*associated with that situation), and not on what situations or actions have been selected so far. All the strategies considered in the paper are state-based.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ...HREF="node1.html#safe2">
*Strategy-2*is defined, in fact, by two action selection rules. The first one is axiom , the second is as follows 57#57. This rule guarantees that the strategy terminates as shown in fig. . The rest of the strategies that terminate do not need this rule, because it is subsumed by axiom .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ...actions
- It
compares the maximum length of the solutions
generated by strategies 4 and 5.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ...bad
- Notice that
axiom implies
*Better(Move(A,F),Move(A,T),S36#36,S66#66)*.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Tue Jul 21 09:54:27 PDT 1998