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.

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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
  false to loaded(r1)
• Above example still allows for at(r1, l2) to be
  true however.

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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) ?
  at(r1,l2)) ? loaded(r1)
• 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

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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)

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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).

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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.

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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.

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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

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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))

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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
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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.

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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.

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Davis-Putnam Algorithm

• Where ? is the CNF propositional
  formula, and ? is a model.

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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)

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Break
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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.

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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.

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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

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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.

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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

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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)

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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.

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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
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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.

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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.

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Blackbox (contd)
Plan Graph
Mutex computation
STRIPS
Translator
CNF
Simplifier
General Stochastic / Systematic SAT engines
Solution
CNF
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Graphplan-based Encodings

• Translation begins at goal-layer of the graph,
  and works backwards.
• Example using the rocket domain
• Load(A,R,L,i) means load A into R at Location L
  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.
  Load(A,R,L,2) ? (At(A,L,1) ? At(R,L,1)
• Each fact at level i implies the disjunction of
  all the operators at level i-1 that have it as an
  add-effect, e.g. In(A,R,3) ? (Load(A,R,L,2) v
  Load(A,R,P,2) v Maintain(In(A,R),2))
• Conflicting actions are mutually exclusive, e.g.
  Load(A,R,L,2) v Move(R,L,P,2)

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Translation of Plan Graph
Act1
Pre1
Fact
Pre2
Act2

• Fact ? Act1 ? Act2
• Act1 ? Pre1 ? Pre2
• Act1 ? Act2

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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
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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.

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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

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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
• More information on Blackbox located here
  http//www.cs.rochester.edu/u/kautz/satplan/blackb
  ox/index.html

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Contact Info

• Problems or Questions about Blackbox?
• Email me at cew305_at_lehigh.edu