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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 l.jpg

Planning as Satisfiability

Henry KautzUniversity of Rochester

in collaboration with Bart Selman and Jöerg Hoffmann

Ai planning l.jpg
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

Satplan l.jpg

  • 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 l.jpg
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 l.jpg
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:

Example l.jpg

  • 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

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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

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Plan Graph Based Instantiation

initial state: p

action a:

precondition: p

effect: p

action b:

precondition:  p

effect: p  q












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International Planning Competition

  • IPC-1998: Satplan (blackbox) is competitive

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International Planning Competition

  • IPC-2000: Satplan did poorly


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International Planning Competition

  • IPC-2002: we stayed home.

Jeb Bush

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International Planning Competition

  • IPC-2004: 1st place, Optimal Planning

    • Best on 5 of 7 domains

    • 2nd best on remaining 2 domains



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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]

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International Planning Competition

  • IPC-2006: Tied for 1st place, Optimal Planning

    • Other winner, MAXPLAN, is a variant of Satplan!

What changed l.jpg
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)

Objections l.jpg

  • 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: r=5

action a:

precondition: r>0

effect: r := r-1

  • Resource use represented as conditional effects









Large numeric domains l.jpg
Large Numeric Domains

Directly encode binary arithmetic

action: a

precondition: r  k

effect: r := r-k












Objections24 l.jpg

  • 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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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 l.jpg

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

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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

Summary l.jpg

  • 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