Planning as Satisfiability with Blackbox - PowerPoint PPT Presentation

View by Category
Title:

Planning as Satisfiability with Blackbox

Description:

A disjunctive formula is a collection of one or more (positive and negative) ... eliminates in one shot all that can be eliminated and returns a smaller formula. ... – PowerPoint PPT presentation

Number of Views:40
Avg rating:3.0/5.0
Slides: 39
Provided by: cat5151
Category:
Transcript and Presenter's Notes

Title: Planning as Satisfiability with Blackbox

1
Planning as Satisfiabilitywith Blackbox
• Presented By Catie Welsh
• Sources
• Chapter 7
• Henry Kautz and Bart Selman (1999) Unifying
SAT-based and Graph-based Planning, Proc.
IJCAI-99, Stockholm
• Henry Kautz and Bart Selman (1992) Planning as
Satisfiability Proc. ECAI-92.

2
Introduction
• Classical planning has often been observed as a
form of logical deduction.
• Planning as satisfiability offers a new solution
to planning problems.
• Formulates a planning problem as a propositional
satisfiability problem.

3
Overview
• A planning problem is encoded as a propositional
formula.
• A satisfiability decision procedure determines
whether the formula is satisfiable by assigning
truth values to the propositional variables.
• A plan is extracted from the assignments
determined by the satisfiability decision
procedure.

4
Definitions
• Model an assignment of truth values to its
variables for which the formula evaluates to
true.
• Satisfiable a sentence or formula is
satisfiable if it is true in some model
• Satisfiability problem problem of determining
whether a formula has a model.

5
States as Propositional Formulas
• Use propositional formulas to represent facts
that hold in states.
• Ex at(r1,l1) ? loaded(r1) assuming there is a
robot r1, and a location l1
• A model for the above example is one that assigns
true to the propositional variable at(r1,l1) and
• Above example still allows for at(r1, l2) to be
true however.

6
States as Propositional Formulas (contd)
• A propositional formula can represent sets of
states rather than a single state.
• Ex (at(r1,l1) ? at(r1,l2)) V (at(r1,l1) ?
• The above example represents both the states
where the robot is at l1 and one where it is at
l2.
• As opposed to STRIPS planning where a state is a
conjunction of propositions

7
State Transitions as Propositional Formulas
• Behavior of deterministic actions is described by
the transition formula? S x A ? S
• So, Figure 7.1 could be said ?(s1,
move(r1,l1,l2)) s2
• If s1 is represented as at(r1,l1) ? at(r1,l2)
and s2 is represented as at(r1,l1) ? at(r1,l2),
this does not assert that the propositional
formulas correspond to their appropriate states.
• This again is opposed to STRIPS planning where
states are maintained at every step.

8
State Transitions as Propositional Formulas
(contd)
• To assert the states for each propositional
formula, we write it as suchat(r1,l1,s1) ?
at(r1,l2,s1) ? at(r1,l1,s2) ? at(r1,l2,s2)
• We can now represent that action move(r1,l1,l2)
causes this transition move(r1,l1,l2,s1) ?
at(r1,l1,s1) ? at(r1,l2,s1) ? at(r1,l1,s2) ?
at(r1,l2,s2)

9
Planning Problems as Propositional Formulas
• Encoding a planning problem to a propositional
formula is based on two main ideas
• Restrict the planning problem to a problem of
finding a plan of known length n for some fixed
n. This is called a bounded planning problem.
• Transform the bounded planning problem into a
satisfiability problem. Each state and action of
the bounded planning problem is mapped to
propositions that describe states and actions at
each step, from step 0 (initial state) to step n
(goal state).

10
Planning Problems as Propositional Formulas
(contd)
• Since planning as satisfiability can only deal
with bounded planning problems, the algorithm is
iteratively run for different tentative lengths
until a plan is found. (plan length is fixed at
2,4,8, etc)
• Uses iterative deepening.
• Fluents instantiation of a predicate
• States are sets of fluents.

11
Planning Problems as Propositional Formulas
(contd)
• Constructing formulas to encode bounded planning
problems into satisfiability problems
• If f is a fluent at(r1,l1), we write at(r1,l1,i)
as fi.
• If a is this action move(r1,l1,l2), we write
move(r1,l1,l2,i) as ai.

12
Planning Problems as Propositional Formulas
(contd)
• Formula is built with these five kinds of sets of
formulas
• (1) Initial state is encoded
• (2) The set of goal states is encoded
• (3) An action, when applicable, has some effects
is encoded

13
Planning Problems as Propositional Formulas
(contd)
• (4) An action changes only the fluents that are
in its effects.
• Explanatory frame axioms set of propositions
that enumerate the set of actions that could have
occurred in order to account for a state change.
• (5) Complete exclusion axiom only one action
occurs at each step.
• The propositional formula encoding the bounded
planning problem into a satisfiability problem is
the conjunction of all five.
• ((1) ? (2) ? (3) ? (4) ? (5))

14
Planning as Satisfiability
• Once the bounded planning problem is encoded to a
satisfiability problem, a model for the resulting
formula can be constructed by a satisfiability
decision procedure.
• Two most commonly used procedures
• Davis-Putnam
• Stochastic procedures based on the idea of
randomized search

15
Conjunction Normal Form
• What is CNF?
• A conjunctive normal form (CNF) is a Boolean
expression consisting of one or more disjunctive
formulas connected by an AND symbol (?). A
disjunctive formula is a collection of one or
more (positive and negative) literals connected
by an OR symbol (?).
• Example
• (a) ? ( a ? b ? c ? d) ? (c ? d) ? (d)
• CNF-Satisfaction Give an algorithm that receives
as input a CNF form and returns Boolean
assignments for each literal in form such that
form is true
• Example (above)
• a ? true, b ? false, c ? true, d ?
false

Prof. Munoz-Avila
16
Davis-Putnam Procedure
• Sound and complete
• Sound every input formula on which it returns a
model is satisfiable
• Complete returns a model on every satisfiable
input formula.
• Soundness and completeness only hold for bounded
planning problems.
• These definitions differ from the sound and
complete we learned so far in class as the
standard definitions for STRIPS. However, based
on the encodings, the STRIPS definitions will
still hold for these bounded planning problems.

17
Davis-Putnam Procedure (contd)
• Literal propositional variable (positive
literal) or its negation (negative literal)
• Unit Clause clause with only one literal
• Performs a depth-first search through the space
of all possible assignments until it finds a
model or explores the entire search space without
finding any.
• Unit-Propagate eliminates in one shot all that
can be eliminated and returns a smaller formula.
• For any unit clause, if the literal is positive
(negative), Unit-Propagate sets the literal to
true (false), and simplifies called unit
propagation.

18
Davis-Putnam Algorithm
• Where ? is the CNF propositional
formula, and ? is a model.

19
Davis-Putnam Example
• Given the following propositional formula in CNF
? D ? (D v A v B) ? (D v A v B) ? (D v
A v B) ? (D v A)

20
Break
21
Stochastic Procedures
• Unlike Davis-Putnam, stochastic procedures start
with total assignments of all variables.
• Randomly selects an initial total assignment
until a model is found or all possible
assignments are exhausted.
• Sound and complete but of no practical use.

22
Stochastic Procedures (contd)
• Basic idea underlying a set of incomplete
satisfiability decision procedures.
• Blackbox includes the local-search SAT solver
Walksat.
• Walksat is built off of the Iterative-Repair
algorithm.

23
Iterative-Repair
• Iterative repair approach iteratively modifies
the truth assignment such that it satisfies one
of the unsatisfied clauses selected.
• Iterative-Repair is sound and incomplete

24
Iterative Repair (contd)
• Example
• Given ? D ? (D v A v B) ? (D v A v B) ?
(D v A v B) ? (D v A)
• Initial guess ? D,A,B
• Have to repair clause (D v A v B).
• Modify ? to be D, A, B.
• Now ? satisfies the propositional formula.

25
Different Encodings
• Encodings determine the number of propositional
variables and the number of clauses in the
formula that is generated.
• Efficiency of satisfiability procedure depends on
the number of variables and clauses.
• Two main choices for encodings
• Encoding of actions
• Encoding of the frame problem

26
Operator Splitting
• Simple Operator Splitting replace each n-ary
action ground proposition with n unary ground
propositions.
• Based on the idea that since only a single action
can occur at a given step, an m-place action can
be represented at the conjunction of m 1-place
actions.
• Example Encoding the action move(a,b,c) occurs
at time 3 as a single proposition move(a,b,c,3),
could be written as
• source(a,3) ? object(b,3) ? destination(c,3)

27
SATPLAN
• First planning as satisfiability planner
• Showed that a general propositional theorem
prover could be competitive with some of the best
specialized planning systems.
• First creates a CNF wff (Conjunctive Normal Form
Well Formed Formula), then performs a search
that is constrained by that structure.
• Propositional structure corresponds to fixed plan
length, and search reveals whether or not a plan
of that length exists.

28
Graphplan Review
P0
A1
P1
B
B
C
A
Mutex list for Move(B,C,table) -Move(A,table,B) -
Move(B,C,A)
C
A
Clear(C) On(B, table) On(B, C) On(A,
B) On(A,table) Clear(B) On(B, A) Clear(A) On(C,t
able)
Move(B,C,table)
A
Clear(B) On(B, C) Clear(A) On(A,
table) On(C,table)
Mutex list for Move(A,table,B) -Move(B,C,table) -
Move(B,C,A)
B
?
C
Move(A,table,B)
Mutex list for Move(B,C,A) -Move(B,C,table) -Move
(A,table,B)
?
?
?
?
Move(B,C,A)
B
C
A
29
Blackbox
• Blackbox Graphplan SATPLAN
• Converts problems specified in STRIPS notation
into Boolean satisfiability problems, then uses
state-of-the-art satisfiability engines to solve
the problems.
• Very versatile you can specify which SAT engine
to use for particular amounts of time
• Ex you can tell it to use Graphplan for 30
seconds, Walksat for 2 minutes, and if still no
solution is found, then Satz for 5 minutes.

30
Blackbox (contd)
1. A planning problem (specified in STRIPS) is
converted to a plan graph of length k, and
mutexes are computed as in Graphplan.
2. The plan graph is converted to CNF wff.
3. The wff is simplified by a general CNF
simplification algorithm.
4. The wff is solved by any variety of fast SAT
engines.
5. If a model of the wff is found, then the model is
converted to a plan otherwise k is incremented
and the process repeats.

31
Blackbox (contd)
Plan Graph
Mutex computation
STRIPS
Translator
CNF
Simplifier
General Stochastic / Systematic SAT engines
Solution
CNF
32
Graphplan-based Encodings
• Translation begins at goal-layer of the graph,
and works backwards.
• Example using the rocket domain
at time i
• Move(R,L,P,i) means move R from L to P at time i.
• Translation process
• Initial state holds at layer 1 and goals hold at
the highest layer.
• Operators imply their preconditions, e.g.
• Each fact at level i implies the disjunction of
all the operators at level i-1 that have it as an
• Conflicting actions are mutually exclusive, e.g.

33
Translation of Plan Graph
Act1
Pre1
Fact
Pre2
Act2
• Fact ? Act1 ? Act2
• Act1 ? Pre1 ? Pre2
• Act1 ? Act2

34
P0
A1
P1

B
B
C
A
Mutex list for Move(B,C,table) -Move(A,table,B) -
Move(B,C,A)
C
A
Clear(C) On(B, table) On(B, C) On(A,
B) On(A,table) Clear(B) On(B, A) Clear(A) On(C,t
able)
Move(B,C,table)
A
Clear(B) On(B, C) Clear(A) On(A,
table) On(C,table)
Mutex list for Move(A,table,B) -Move(B,C,table) -
Move(B,C,A)
B
?
C
Move(A,table,B)
Mutex list for Move(B,C,A) -Move(B,C,table) -Move
(A,table,B)
?
?
?
?
Move(B,C,A)
B
C
A
35
Blackbox Conclusions
• The name blackbox refers to the fact that the
plan generator knows nothing about the SAT
solvers, and the SAT solvers know nothing about
plans each is a "black box" to the other.
• Blackbox is an evolving system.
• The general goal is to unify many different
threads of research in planning and inference by
using propositional satisfiability as a common
foundation.

36
Blackbox Instructions
• Blackbox is installed on a pc in PL 250.
• Its the computer in the middle along the back
wall.
• To run blackbox
• Open a command prompt
• cd to C\Blackbox
• Type gtblackbox o domain.pddl f prob.pddl
• Use blackbox help for more options

37
Blackbox Instructions (contd)
• Blackbox takes domain and problem files in pddl
format. (no changes should be required from the
original STRIPS encoded files you get for your
domains).
• Examples files are located at C\Blackbox\Blackbox
43\Examples