sat planning for mprimes

SAT Planning Bounded Planning Problem

MPrimes (AIPS 1998)

Fariz Dararifadirra@gmail.com

FU Bolzano

Formulas

1. Initial State2. Actions (Preconditions and Effects)3. Explanatory Frame Axiom4. Complete Exclusion Axiom5. Goal

Formulas

• To satisfy:initial state & all possible action descriptions & goal

1 & {2, 3, 4} & 5

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))

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!).

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!

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!

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)))

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

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!

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!

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

Precondition

:precondition (and (at ?v ?l1) (conn ?l1 ?l2) (has-fuel ?l1 ?f1)

(fuel-neighbor ?f2 ?f1))

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))

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

Effects

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

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)))

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

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!

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))

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)))

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))

DEMO

Example Solution:---PLAN---68 of load_c0_v0_l0_s1_s0_045 of move_v0_l0_l1_f2_f1_196 of unload_c0_v0_l1_s0_s1_2

References

• Mprime: http://www.loria.fr/~hoffmanj/ff-domains.html

• Course page: http://teaching.case.unibz.it/mod/assignment/view.php?id=13400