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1
Planning as Satisfiability

• Henry KautzUniversity of Rochester
• in collaboration with Bart Selman and Jöerg
  Hoffmann

2
AI Planning

• Two traditions of research in planning
• Planning as general inference (McCarthy 1969)
• Important task is modeling
• Planning as human behavior (Newell Simon 1972)
• Important task is to develop search strategies

3
Satplan

• Model planning as Boolean satisfiability
• (Kautz Selman 1992) Hard structured benchmarks
  for SAT solvers
• Pushing the envelope planning, propositional
  logic, and stochastic search (1996)
• Can outperform best current planning systems

4
Translating STRIPS

• Ground action a STRIPS operator with constants
  assigned to all of its parameters
• Ground fluent a precondition or effect of a
  ground action
• operator Fly(a,b)
• precondition At(a), Fueled
• effect At(b), At(a), Fueled
• constants NY, Boston, Seattle
• Ground actions Fly(NY,Boston), Fly(NY,Seattle),
  Fly(Boston,NY), Fly(Boston,Seattle),
  Fly(Seattle,NY), Fly(Seattle,Boston)
• Ground fluents Fueled, At(NY), At(Boston),
  At(Seattle)

5
Clause Schemas

• A large set of clauses can be represented by a
  schema

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Satplan in 15 Seconds

• Time bounded sequence of integers
• Translate planning operators to propositional
  schemas that assert

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Example

• If an action occurs at time i, then its
  preconditions must hold at time i
• If an action occurs at time i, then its effects
  must hold at time i1

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SAT Encoding

• If a fluent changes its truth value from time i
  to time i1, one of the actions with the new
  value as an effect must have occurred at time i

Like for, but connects propositions with OR
9
Plan Graph Based Instantiation

• initial state p
• action a
• precondition p
• effect ?p
• action b
• precondition ? p
• effect p ? q

m0
m1


p0
p1
p2
a0
a1
b1
q2
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International Planning Competition

• IPC-1998 Satplan (blackbox) is competitive

11
International Planning Competition

• IPC-2000 Satplan did poorly

Satplan
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International Planning Competition

• IPC-2002 we stayed home.

Jeb Bush
13
International Planning Competition

• IPC-2004 1st place, Optimal Planning
• Best on 5 of 7 domains
• 2nd best on remaining 2 domains

PROLEMA / philosophers
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The IPC-4 Domains

• Airport control the ground traffic Hoffmann
  Trüg
• Pipesworld control oil product flow in a
  pipeline network Liporace Hoffmann
• Promela find deadlocks in communication
  protocols Edelkamp
• PSR resupply lines in a faulty electricity
  network Thiebaux Hoffmann
• Satellite Settlers Fox Long, additional
  Satellite versions with time windows for sending
  data Hoffmann
• UMTS set up applications for mobile terminals
  Edelkamp Englert

15
International Planning Competition

• IPC-2006 Tied for 1st place, Optimal Planning
• Other winner, MAXPLAN, is a variant of Satplan!

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What Changed?

• Small change in modeling
• Modest improvement from 2004 to 2006
• Significant change in SAT solvers!

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What Changed?

• In 2004, competition introduced the optimal
  planning track
• Optimal planning is a very different beast from
  non-optimal planning!
• In many domains, it is almost trivial to find
  poor-quality solutions by backtrack-free search!
• E.g. solutions to multi-airplane logistics
  planning problems found by heuristic state-space
  planners typically used only a single airplane!
• See Local Search Topology in Planning
  Benchmarks A Theoretical Analysis (Hoffmann 2002)

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Why Care About Optimal Planning?

• Real users want (near)-optimal plans!
• Industrial applications assembly planning,
  resource planning, logistics planning
• Difference between (near)-optimal and merely
  feasible solutions can be worth millions of
  dollars
• Alternative fast domain-specific optimizing
  algorithms
• Approximation algorithms for job shop scheduling
• Blocks World Tamed Ten Thousand Blocks in Under
  a Second (Slaney Thiébaux 1995)

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Domain-Independent Feasible Planning Considered
Harmful
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Objections

• Real-world planning cares about optimizing
  resources, not just make-span, and Satplan cannot
  handle numeric resources
• We can extend Satplan to handle numeric
  constraints
• One approach use hybrid SAT/LP solver (Wolfman
  Weld 1999)
• Modeling as ordinary Boolean SAT is often
  surprisingly efficient! (Hoffmann, Kautz, Gomes,
  Selman, under review)

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Projecting Variable Domains

• initial state r5
• action a
• precondition rgt0
• effect r r-1
• Resource use represented as conditional effects

a1
a0
r5
r5
r5
r4
r4
r4
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2002 ICAPS Benchmarks
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Large Numeric Domains
a1

• Directly encode binary arithmetic
• action a
• precondition r ? k
• effect r r-k

-k
r11
r12
r21
r22

r31
r32
r41
r42
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Objections

• If speed is crucial, you still must use feasible
  planners
• For highly constrained planning problems, optimal
  planners can be faster than feasible planners!

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Constrainedness Run Time
26
Constrainedness Percent Solved
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Further Extensions to Satplan

• Probabilistic planning
• Translation to stochastic satisfiability
  (Majercik Littman 1998)
• Alternative untested idea
• Encode action failure as conditional resource
  consumption
• Can find solutions with specified probability of
  failure-free execution
• (Much) less general than full probabilistic
  planning (no fortuitous accidents), but useful in
  practice

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Encoding Bounded Failure Free Probabilistic
Planning

• plan failure free probability ? 0.90
• action a
• failure probability 0.01
• preconditions p
• effects q

• action a
• precondition p ?
• s ? log(0.89)
• effect q ?
• s s log(0.99)

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One More Objection!

• Satplan-like approaches cannot handle domains
  that are too large to fully instantiate
• Solution SAT solvers with lazy instantiation
• Lazy Walksat (Singla Domingos 2006)
• Nearly all instantiated propositions are false
• Nearly all instantiated clauses are true
• Modify Walksat to only keep false clauses and a
  list of true propositions in memory

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Summary

• Satisfiability testing is a vital line of
  research in AI planning
• Dramatic progress in SAT solvers
• Recognition of distinct and important nature of
  optimizing planning versus feasible planning
• SATPLAN not restricted to STRIPS any more!
• Numeric constraints
• Probabilistic planning
• Large domains