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BLACKBOX: A New Approach to the Application of Theorem Proving to Problem Solving

BLACKBOX: A New Approach to the Application of Theorem Proving to Problem Solving

Bart SelmanCornell University

Joint work with Henry Kautz

AT&T Labs

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Planning as (Specialized) DeductionPlanning as (Specialized) Deduction

Situation calculus (McCarthy & Hayes 1969)

• Conceptually elegant

Planning as first-order theorem proving (Green 1969)

• Computationally infeasible

STRIPS: A New Approach to the Application of Theorem Proving to Problem Solving (Fikes & Nilsson 1971)

• More efficient, but still poor scaling

Partial-order planning (Tate 1977, Chapman 1985, McAllester 1991)

• Planning as specialized implementation of a particular theorem - the modal truth criteria

• Can be more efficient, but still hard (Minton, Bresina, & Drummond 1994)

SATPLAN (Kautz and Selman 1992, 1996)

• Planning as (simply) propositional reasoning

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SATPLANSATPLAN

axiomschemas instantiated

propositionalclauses

satisfyingmodelplan

mapping

length

problemdescription

SATengine(s)

instantiate

interpret

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What SATPLAN ShowedWhat SATPLAN Showed

A general propositional theorem prover can be competitive with specialized planning (specialized theorem proving) systems

• Good representations– propositional, not first-order

– can combine expressiveness with good computational properties

• Fast SAT engines– stochastic search - walksat

– large SAT/CSP community sharing ideas and code+ compare: general CPU's versus Lisp-chip

+ specialized engines can catch up, but by then, new general techniques

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GraphplanGraphplan

Planning as graph search (Blum & Furst 1995)

Set new paradigm for planning

Like SATPLAN...

• Two phases: instantiation of propositional structure, followed by search

• Possible to translate planning graph into CNF (Kautz & Selman 1996)

Unlike SATPLAN...

• Interleaves instantiation and pruning of plan graph

• Employs specialized search engine

• Graphplan - better instantiation

• SATPLAN - better search

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Bridging ParadigmsBridging Paradigms

Both SATPLAN and Graphplan are disjunctive planners (Kambhampati 1996)

avoid splitting - more compact representations, faster search

IJCAI Challenge in Bridging Plan Synthesis Paradigms (Kambhampati 1997)

• study tradeoffs between refinement (instantiation) and extraction (SAT) techniques

Our response: blackbox

efficient planning system and highly-configurable testbed

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Blackbox ArchitectureBlackbox Architecture

1. Front end: turn STRIPS input into planning graph

• interleaves instantiation and pruning (specialized limited deduction)

2. Translation to CNF

3. Simplification (general limited deduction)

• critical for hardest problems– unit propagation, failed literal, binary failed literal, ...

4. SAT solvers

• can schedule series of different solvers: walksat, satz, rel_sat, etc.

• new: randomized systematic solvers (more later)

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Specialized InferenceSpecialized Inference

SATPLAN was proof of concept that general deduction is feasible

When do specialized inference techniques have the greatest impact?

Two kinds of specialized deduction• Planning specific - graphplan's pruning rules

• General polytime inference - simplification performed after CNF generation– apply to all CNF formulas, may or may not be

designed with planning in mind

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Graph PruningGraph Pruning

Graphplan instantiates in a forward direction, pruning nodes with incompatible preconditions

• incomplete

• pair-wise mutex relations computed by incremental constraint propagation

– (see Kambhampati 1997, Euro CP)

In logical terms: limited application of negative binary propagation

• given: P V Q, P V R V S V ...

• infer: Q V R V S V ...

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Translation to CNFTranslation to CNF

Fact Act1 V Act2

Act1 Pre1 & Pre2

Wff is entailed by original SATPLAN hand encodings, but more compact, easier to solve

• compare with Medic (Weld 1997), which translated STRIPS CNF directly

Act1

Act2

Fact

Pre1

Pre2

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SimplificationSimplification

Generated wff can be further simplified by additional consistency propagation techniques

Because is in CNF, can take advantage of many fast, powerful simplifiers

Compact (Crawford & Auton 1996)

• failed literal rule: is Wff + { P } unsat by unit propagation?

• binary failed literal rule: is Wff + { P V Q } unsat by unit propagation?

Generally reduces number of variables and clauses by 30%

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Randomized Sytematic SolversRandomized Sytematic Solvers

Stochastic local search solvers (walksat)

• when they work, scale well

• cannot show unsat

• fail on some domains

• must use very simple (fast) heuristics

Systematic solvers (Davis Putnam)

• complete

• fail on (often different) domains

• might use more sophisticated (costly) heuristics

• seem to scale badly

Can we combine best features of each approach?

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Heavy TailsHeavy Tails

Bad scaling of systematic solvers can be caused by heavy tailed distributions

Deterministic algorithms get stuck on particular instances

• but that same instance might be easy for a different deterministic algorithm!

Expected (mean) solution time increases without limit over large distributions

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Heavy Tailed Cost DistributionHeavy Tailed Cost Distribution

0.1

1

1 10 100 1000 10000 100000

log( Backtracks )

log

( 1

- F

(x)

)

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Randomized RestartsRandomized Restarts

Solution: randomize the systematic solver

• Add noise to the heuristic branching (variable choice) function

• Cutoff and restart search after a fixed number of backtracks

Eliminates heavy tails

In practice: rapid restarts with low cutoff can dramatically improve performance

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Rapid Restart SpeedupRapid Restart Speedup

1000

10000

100000

1000000

1 10 100 1000 10000 100000 1000000

log( cutoff )

log

( b

ackt

rack

s )

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Blackbox as Experimental TestbedBlackbox as Experimental Testbed

All components of blackbox are parameterized

Can experiment with different schedules for instantiating, simplifying, and solving problems

blackbox -solver -maxsec 20 graphplan

-then compact -l

-then satz -cutoff 20 -restart 100

-then walksat -cutoff 1000000 -restart 10

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Blackbox ResultsBlackbox Results

0.01

0.1

1

10

100

1000

10000

rocket.a rocket.b log.a log.b log.c log.d

Graphplan

BB-walksat

BB-rand-sys

Handcoded-walksat

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blackbox version 9Bcommand line: blackbox -o logistics.pddl -f logistics_prob_d_len.pddl -solver compact -l -then satz -cutoff 25 -restart 10

----------------------------------------------------Converting graph to wff6151 variables243652 clausesInvoking simplifier compactVariables undetermined: 4633Non-unary clauses output: 139866----------------------------------------------------Invoking solver satz version satz-rand-2.1Wff loaded[1] begin restart[1] reached cutoff 25 --- back to root[2] begin restart[2] reached cutoff 25 --- back to root [3] begin restart[3] reached cutoff 25 --- back to root [4] begin restart[4] reached cutoff 25 --- back to root [5] begin restart**** the instance is satisfiable ********* verification of solution is OK ****

total elapsed seconds = 25.930000----------------------------------------------------Begin plan1 drive-truck_ny-truck_ny-central_ny-po_ny

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Begin plan1 drive-truck_ny-truck_ny-central_ny-po_ny1 drive-truck_sf-truck_sf-airport_sf-po_sf1 load-truck_package5_bos-truck_bos-po1 drive-truck_pgh-truck_pgh-airport_pgh-central_pgh1 fly-airplane_airplane2_pgh-airport_sf-airport1 load-truck_package6_bos-truck_bos-po2 load-truck_package2_pgh-truck_pgh-central2 load-truck_package4_ny-truck_ny-po2 load-truck_package7_ny-truck_ny-po2 load-truck_package3_pgh-truck_pgh-central2 drive-truck_bos-truck_bos-po_bos-airport_bos2 load-airplane_package8_airplane2_sf-airport2 fly-airplane_airplane1_pgh-airport_sf-airport2 drive-truck_la-truck_la-po_la-airport_la3 fly-airplane_airplane2_sf-airport_bos-airport3 unload-truck_package6_bos-truck_bos-airport3 drive-truck_pgh-truck_pgh-central_pgh-airport_pgh3 fly-airplane_airplane1_sf-airport_pgh-airport3 unload-truck_package5_bos-truck_bos-airport3 drive-truck_ny-truck_ny-po_ny-airport_ny3 drive-truck_sf-truck_sf-po_sf-airport_sf4 unload-truck_package3_pgh-truck_pgh-airport4 unload-truck_package2_pgh-truck_pgh-airport4 unload-truck_package4_ny-truck_ny-airport4 load-airplane_package6_airplane2_bos-airport4 load-airplane_package5_airplane2_bos-airport4 drive-truck_la-truck_la-airport_la-po_la4 drive-truck_bos-truck_bos-airport_bos-central_bos4 unload-truck_package7_ny-truck_ny-airport5 drive-truck_ny-truck_ny-airport_ny-po_ny5 drive-truck_bos-truck_bos-central_bos-po_bos5 load-airplane_package2_airplane1_pgh-airport5 drive-truck_la-truck_la-po_la-central_la5 drive-truck_pgh-truck_pgh-airport_pgh-po_pgh5 load-airplane_package3_airplane1_pgh-airport5 fly-airplane_airplane2_bos-airport_ny-airport6 drive-truck_sf-truck_sf-airport_sf-central_sf6 unload-airplane_package6_airplane2_ny-airport6 load-airplane_package4_airplane2_ny-airport6 drive-truck_la-truck_la-central_la-po_la6 drive-truck_bos-truck_bos-po_bos-airport_bos6 load-airplane_package7_airplane2_ny-airport6 drive-truck_ny-truck_ny-po_ny-airport_ny6 unload-airplane_package8_airplane2_ny-airport6 fly-airplane_airplane1_pgh-airport_sf-airport6 load-truck_package1_pgh-truck_pgh-po7 fly-airplane_airplane2_ny-airport_la-airport7 fly-airplane_airplane1_sf-airport_bos-airport7 load-truck_package9_sf-truck_sf-central7 load-truck_package6_ny-truck_ny-airport

7 drive-truck_bos-truck_bos-airport_bos-central_bos7 drive-truck_pgh-truck_pgh-po_pgh-airport_pgh7 load-truck_package8_ny-truck_ny-airport8 drive-truck_sf-truck_sf-central_sf-po_sf8 fly-airplane_airplane2_la-airport_pgh-airport8 unload-truck_package1_pgh-truck_pgh-airport8 drive-truck_bos-truck_bos-central_bos-po_bos8 drive-truck_ny-truck_ny-airport_ny-central_ny8 fly-airplane_airplane1_bos-airport_la-airport8 drive-truck_la-truck_la-po_la-airport_la9 unload-airplane_package7_airplane2_pgh-airport9 unload-truck_package8_ny-truck_ny-central9 unload-airplane_package5_airplane2_pgh-airport9 unload-truck_package9_sf-truck_sf-po9 unload-airplane_package3_airplane1_la-airport9 unload-truck_package6_ny-truck_ny-central9 drive-truck_pgh-truck_pgh-airport_pgh-po_pgh9 load-airplane_package1_airplane2_pgh-airport10 drive-truck_ny-truck_ny-central_ny-po_ny10 fly-airplane_airplane2_pgh-airport_bos-airport10 load-truck_package3_la-truck_la-airport10 fly-airplane_airplane1_la-airport_ny-airport10 drive-truck_pgh-truck_pgh-po_pgh-airport_pgh11 drive-truck_bos-truck_bos-po_bos-airport_bos11 drive-truck_ny-truck_ny-po_ny-airport_ny11 unload-airplane_package2_airplane1_ny-airport11 drive-truck_la-truck_la-airport_la-central_la11 drive-truck_sf-truck_sf-po_sf-airport_sf11 unload-airplane_package1_airplane2_bos-airport11 load-truck_package7_pgh-truck_pgh-airport11 load-truck_package5_pgh-truck_pgh-airport12 drive-truck_sf-truck_sf-airport_sf-po_sf12 load-truck_package1_bos-truck_bos-airport12 fly-airplane_airplane2_bos-airport_la-airport12 load-truck_package2_ny-truck_ny-airport12 fly-airplane_airplane1_ny-airport_pgh-airport12 drive-truck_pgh-truck_pgh-airport_pgh-po_pgh12 unload-truck_package3_la-truck_la-central13 drive-truck_ny-truck_ny-airport_ny-po_ny13 load-truck_package3_la-truck_la-central13 load-truck_package9_sf-truck_sf-po13 drive-truck_bos-truck_bos-airport_bos-po_bos13 unload-truck_package5_pgh-truck_pgh-po13 unload-airplane_package4_airplane2_la-airport14 unload-truck_package9_sf-truck_sf-po14 unload-truck_package1_bos-truck_bos-po14 unload-truck_package7_pgh-truck_pgh-po14 unload-truck_package2_ny-truck_ny-po14 unload-truck_package3_la-truck_la-centralEnd plan

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SummarySummary

Blackbox combines best features of Graphplan, SATPLAN, and new randomized systematic search engines

Automatic generation of wffs from standard STRIPS input

• No performance penalty over hand-encodings!

Testbed for bridging different planning paradigms

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Current Research IssuesCurrent Research Issues

Tradeoffs between planning-specific and more general limited inference (simplification) techniques

• More powerful / general rules than mutex computation?

Incorporating explicit domain knowledge (Kautz & Selman, 1998)

• state invariants

• optimal conditionsOther front-ends: causal encodings (McAllester, Selman, Kautz

1996), HTN encodings (Mali & Kambhampati 1998)

Download from www.research.att.com/~kautz/blackbox