1. SAT Planning
Bounded Planning Problem
MPrimes (AIPS 1998)
Fariz Darari
fadirra@gmail.com
FU Bolzano

2. Formulas
1. Initial State
2. Actions (Preconditions and Effects)
3. Explanatory Frame Axiom
4. Complete Exclusion Axiom
5. Goal

3. Formulas
• To satisfy:
initial state & all possible action descriptions & goal
1 & {2, 3, 4} & 5

4. 1 - Initial State
• Specify what is true at the beginning (t = 0).
– Example: (at v0 l0 0)
• Specify what is not true at the beginning (t = 0).
– Example: (not (at v0 l1 0))

5. Dynamic vs Static
• We need to specify only the falsehood of
dynamic states, but not of static states!
• Why?
Possible actions will always satisfy the
precondition for corresponding static states, since
they are generated using the facts about static
states themselves!
• In other words, it is safe to give an interpretation
for any static state wrt. possible actions, to be
true at a specific time (well since they are static!).

6. OWA
• Propositional logic has no CWA!
– Therefore, what are not specified can be interpreted
as True (xor False).
– Example: (random-predicate random-constant) can be
interpreted as true!
• Therefore, if we ask for the satisfiability of:
(at v0 l0 0) & -(at v0 l1 0) & (asdf zxcv 0)
The answer will still be YES!
• Do we need to add -(asdf zxcv 0)? No since we
will never care about the value of (asdf zxcv 0)
and never put it in the formula!

7. 2 - Actions
• We want to represent the actions as compact as
possible!
• Characteristics:
– Action name
– Preconditions
– Effects
– Typing
• All these characteristics define:
Possible actions!
PS: We can think these possible actions as a
substitution!

8. Example
(:action move
:params (?v - vehicle ?l1 ?l2 - location ?f1 ?f2 - fuel)
:precondition (and (at ?v ?l1)
(conn ?l1 ?l2)
(has-fuel ?l1 ?f1)
(fuel-neighbor ?f2 ?f1))
:effect (and (not (at ?v ?l1))
(at ?v ?l2)
(not (has-fuel ?l1 ?f1))
(has-fuel ?l1 ?f2)))

9. Move
• move (?v, ?l1, ?l2, ?f1, ?f2)
• ?v is of type Vehicle
• (at ?v ?l1) is in precondition, but do you think
we can use this knowledge benefit? No, since
at is dynamic (time dependent)!
• Therefore, ?v ranges over objects defined as a
vehicle without any other restrictions

10. Move
• move (?v, ?l1, ?l2 , ?f1, ?f2)
• ?l1 and ?l2 are of type Location
• (conn ?l1 ?l2) is in precondition, but do you think
we can use this knowledge benefit? Yes, since
conn is static (time independent)!
• Therefore, ?l1 and ?l2 ranges over object
locations defined in conn!
• Another benefit: the precondition of conn is
always satisfied!

11. Move
• move (?v, ?l1, ?l2,?f1, ?f2 )
• ?f1 and ?f2 are of type Fuel
• (fuel-neighbor ?f2 ?f1), beware of the order, is in
precondition, but do you think we can use this
knowledge benefit? Yes, since fuel-neighbor is
static!
• Therefore, ?f1 and ?f2 ranges over object fuels
defined in fuel-neighbor!
• Another benefit: the precondition of fuel-
neighbor is always satisfied!

12. How much reduction do we get?
• Suppose |v| = 10, |l| = 10, |f| = 10,
|conn| = 2 * 10, |fuel-neighbor| = 10
• Naive encoding (typing) =
n ^ 5 = 10 ^ 5 = 100.000
• Improved encoding =
2 * (n ^ 3) = 10 * 2 * 10 * 10 = 2000
• Even worse, suppose n = 100, then:
100 ^ 5 = 10.000.000.000 (☠)
• But with the improved encoding:
100 * 2 * 100 * 100 = 2.000.000

13. Precondition
:precondition (and (at ?v ?l1)
(conn ?l1 ?l2)
(has-fuel ?l1 ?f1)
(fuel-neighbor ?f2 ?f1))

14. Relation between Precond and
Possible Actions
• Action -> Precond
• Example: move (?v, ?l1, ?l2, ?f1, ?f2) implies
(and (at ?v ?l1)
(conn ?l1 ?l2)
(has-fuel ?l1 ?f1)
(fuel-neighbor ?f2 ?f1))

15. Relation between Precond and
Possible Actions
• Substitution: (v0, l1, l2, f1, f0)
• Example: move (v0, l1, l2, f1, f0) implies
(and (at v0 l1) //dynamic
(conn l1 l2) //static
(has-fuel l1 f1) //dynamic
(fuel-neighbor f0 f1)) //static

16. Effects
:effect (and (not (at ?v ?l1))
(at ?v ?l2)
(not (has-fuel ?l1 ?f1))
(has-fuel ?l1 ?f2)))

17. Relation between Effect and Possible
Actions
• Action -> Effect
• Example: move (?v, ?l1, ?l2, ?f1, ?f2) implies
(and (not (at ?v ?l1))
(at ?v ?l2)
(not (has-fuel ?l1 ?f1))
(has-fuel ?l1 ?f2)))

18. Relation between Precond and
Possible Actions
• Substitution: (v0, l1, l2, f1, f0)
• Example: move (v0, l1, l2, f1, f0) implies
(and (not (at v0 l1)) //dynamic
(at v0 l2) //dynamic
(not (has-fuel l1 f1)) //dynamic
(has-fuel l1 f0))) //dynamic

19. 3 - Explanatory Frame Axiom
• Describe what doesn’t change between steps i and i +
1.
• Two axioms for every possible dynamic state at every
time step i
• Say that if s changes truth value between i and i+1 then
the action at step i must be responsible:
– not (s, i) & (s, i + 1) -> BIG OR {(a, i)| e in EFF(s)} OR False
– (s, i) & not (s, i + 1) -> BIG OR {(a, i)| e in EFF(-s)} OR False
• If s became true then some action must have added it,
if s became false then some action must have deleted
it!

20. Example
• at = (n ^ 2) * t * 2
• (and (at v0 l0 2) (not (at v0 l0 3)) )
IMPLY
(or (move v0 l0 l1 f1 f0 2) (false) )
• move (?v, ?l1, ?l2, ?f1, ?f2)
Precondition = (at ?v ?l1)
Effecct = (not (at ?v ?l1))

21. 4 - Complete Exclusion Axiom
• Very simple but huge!
• For all actions a and b and time steps i
include the formula
¬ (a, i) OR ¬ (b, i)
• This guaranteed that there could be only one
action at a time
• Example:
(or (not (unload c0 v0 l0 s0 s1 0))
(not (unload c0 v0 l1 s0 s1 0)))

22. 5 - Goal
• Very simple!
• Just a conjunction of your goals at the time n
(aka the bound).
• Example (n = 20):
(and (at c0 l0 20) (at c1 l1 20) (at c2 l2 20))

23. DEMO
Example Solution:
---PLAN---
68 of load_c0_v0_l0_s1_s0_0
45 of move_v0_l0_l1_f2_f1_1
96 of unload_c0_v0_l1_s0_s1_2

24. References
• Mprime: http://www.loria.fr/~hoffmanj/ff-
domains.html
• Course page:
http://teaching.case.unibz.it/mod/assignment
/view.php?id=13400