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Planning as Satisfiability Henry Kautz University of Rochester in collaboration with Bart Selman and J ö erg Hoffmann AI Planning Two traditions of research in planning: Planning as general inference (McCarthy 1969) Important task is modeling

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planning as satisfiability

Planning as Satisfiability

Henry KautzUniversity of Rochester

in collaboration with Bart Selman and Jöerg Hoffmann

ai planning
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
  • 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
translating strips
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)

clause schemas
Clause Schemas
  • A large set of clauses can be represented by a schema
satplan in 15 seconds
Satplan in 15 Seconds
  • Time = bounded sequence of integers
  • Translate planning operators to propositional schemas that assert:
  • 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 i+1
sat encoding
SAT Encoding
  • If a fluent changes its truth value from time i to time i+1, one of the actions with the new value as an effect must have occurred at time i

Like “for”, but connects propositions with OR

plan graph based instantiation
Plan Graph Based Instantiation

initial state: p

action a:

precondition: p

effect: p

action b:

precondition:  p

effect: p  q












international planning competition
International Planning Competition
  • IPC-1998: Satplan (blackbox) is competitive
international planning competition11
International Planning Competition
  • IPC-2000: Satplan did poorly


international planning competition12
International Planning Competition
  • IPC-2002: we stayed home.

Jeb Bush

international planning competition13
International Planning Competition
  • IPC-2004: 1st place, Optimal Planning
    • Best on 5 of 7 domains
    • 2nd best on remaining 2 domains



the ipc 4 domains
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]
international planning competition15
International Planning Competition
  • IPC-2006: Tied for 1st place, Optimal Planning
    • Other winner, MAXPLAN, is a variant of Satplan!
what changed
What Changed?
  • Small change in modeling
    • Modest improvement from 2004 to 2006
  • Significant change in SAT solvers!
what changed17
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)
why care about optimal planning
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)
  • 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)
projecting variable domains
Projecting Variable Domains

initial state: r=5

action a:

precondition: r>0

effect: r := r-1

  • Resource use represented as conditional effects









large numeric domains
Large Numeric Domains

Directly encode binary arithmetic

action: a

precondition: r  k

effect: r := r-k












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

Encoding Bounded Failure Free Probabilistic Planning
one more objection
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
  • 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