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### Planning as Satisfiability

Henry KautzUniversity of Rochester

in collaboration with Bart Selman and Jöerg Hoffmann

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

- 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

- 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

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

Satplan in 15 Seconds

- Time = bounded sequence of integers
- Translate planning operators to propositional schemas that assert:

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 i+1

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

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

International Planning Competition

- IPC-1998: Satplan (blackbox) is competitive

International Planning Competition

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

PROLEMA /

philosophers

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 Competition

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

What Changed?

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

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?

- 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

- 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

initial state: r=5

action a:

precondition: r>0

effect: r := r-1

- Resource use represented as conditional effects

a0

a1

r=5

r=5

r=5

r=4

r=4

r=4

Large Numeric Domains

Directly encode binary arithmetic

action: a

precondition: r k

effect: r := r-k

a1

-k

r11

r12

r21

r22

+

r31

r32

r41

r42

Objections

- 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

- 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

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 PlanningOne 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

- 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

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