Quantified Boolean Formula (QBF) Reasoning Bart Selman, Carla Gomes, Ashish Sabharwal Cornell University Feb 13, 2007 Tutorial for Dr. Charles Holland

Quantified Boolean Formula (QBF) Reasoning

Bart Selman, Carla Gomes, Ashish Sabharwal

Cornell University

Feb 13, 2007

Tutorial forDr. Charles Holland and Dr. Tom Wagner

2

Tutorial Roadmap

1. Automated reasoning– The complexity challenge– State of the art in Boolean reasoning

2. SAT-based reasoning– Boolean logic– Search space, worst-case complexity– Hardness profiles, scaling in practice– Modeling problems as SAT

• Example domain: planning

3. QBF reasoning (extends SAT)– A new range of applications– Two motivating examples

• network planning, logistics planning

– Quantified Boolean logic– Modeling problems as QBF– Search space, worst-case complexity– Scaling in practice

4. High-Performance QBF reasoning– Key research advances– The technology behind QBF– A. New modeling techniques– B. Learning while reasoning– C. Structure discovery– Experimental Results

5. Summary

3

Tutorial Roadmap

1. Automated reasoning The complexity challenge– State of the art in Boolean reasoning







5. Summary

4

The Quest for Machine Reasoning

Objective:

Develop foundations and technology to enable effective, practical, large-scale automated reasoning.

Computational complexity of reasoning appears to severely limit real-world applications

Current reasoning technology

Revisiting the challenge:Significant progress with new ideas / tools for dealing with complexity (scale-up), uncertainty, and multi-agent reasoning

Machine Reasoning (1960-90s)

5

General Automated Reasoning

GeneralInferenceEngine

Solution

Domain-specific

Probleminstance

applicable to all domainswithin range of modeling language

ModelGenerator(Encoder)

Research objective

Better reasoning and modeling technology

Impact

Faster solutionsin several domains

e.g. logistics, war games,chess, space missions, planning, scheduling, ...

Generic

6

• EXPONENTIAL COMPLEXITY: INHERENT AN worst case N= No. of Variables/Objects A= Object states

• TIME/SPACE Granularity Object states

• Current implementations trade time with soundness

Question: Given: X1= true; X2 = false; X7=true. What is X4 = ?

Answer Development: Inference Chain

Step 1: X7 X8 (rule 4)Step 2: X8 X5 (rule 6)Step 3: X5 X3 or X6 (rule 3)

Case A: X6 = trueStep 4: X6 not X9Step 5: X9 not X8Step 6: Contradiction Backtrack to M

Case B: X3 = trueX1 & (not X2) & X3 X4Step 7: X4 = true (Rule 1)

M

Search for rules to apply

Check Contradictions

For N variables: 2N cases drive complexity!

Simple Example:

Variables (binary)X1 = email_ receivedX2 = in_ meetingX3 = urgentX4 = respond_to_email

X5 = near_deadlineX6 = postpone

X7 = air_ticket_info_requestX8 = travel_ requestX9 = info_request

Rules:1. X1 & (not X2) & X3 X42. X2 not X4

3. X5 X3 or X64. X7 X85. X8 X96. X8 X57. X6 not X9

Knowledge Base

Reasoning Complexity

7

Exponential Complexity Growth: The Challenge of Complex Domains

100 200

10K 50K

20K 100K

0.5M 1M

1M5M

Variables

1030

10301,020

10150,500

106020

103010

Cas

e co

mp

lexi

ty

Car repair diagnosis

Deep space mission control

Chess (20 steps deep)

VLSIVerification

War Gaming

100K 450K

Military Logistics

Seconds until heat death of sun

Protein foldingCalculation (petaflop-year)

No. of atomson the earth

1047

100 10K 20K 100K 1MRules (Constraints)

Exponential

Compl

exity

Note: rough estimates, for propositional reasoning

[Credit: Kumar, DARPA; Cited in Computer World magazine]

8

Tutorial Roadmap

1. Automated reasoning The complexity challenge State of the art in Boolean reasoning







5. Summary

9

Focus: Combinatorial Search Spaces

Specifically, the Boolean satisfiability problem, SAT

Significant progress since the 1990’s.

How much?

• Problem size: We went from 100 variables, 200 constraints (early 90’s) to 1,000,000 vars. and 5,000,000 constraints in 15 years.

Search space: from 10^15 to 10^300,000.[Aside: “one can encode quite a bit in 1M variables.”]

• Tools: 50+ competitive SAT solvers available

Overview of state of the art: Plenary talk at IJCAI-05 (Selman); Gomes and Selman, Nature ’05

Progress in Last 15 Years

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How Large are the Problems?

A bounded model checking problem:

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i.e., ((not x1) or x7) ((not x1) or x6)

etc.

x1, x2, x3, etc. are our Boolean variables(to be set to True or False)

Should x1 be set to False??

SAT Encoding(automatically generated from problem specification)

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i.e., (x177 or x169 or x161 or x153 …x33 or x25 or x17 or x9 or x1 or (not x185))

clauses / constraints are getting more interesting…

…

Note x1 …

10 Pages Later:

13

…

4,000 Pages Later:

14

Current SAT solvers solve this instance in under 30 seconds!

Search space of truth assignments:

Finally, 15,000 Pages Later:

15

SAT Solver Progress

Instance Posit' 94 Grasp' 96 Sato' 98 Chaff' 01

ssa2670-136 40.66s 1.20s 0.95s 0.02s

bf1355-638 1805.21s 0.11s 0.04s 0.01s

pret150_25 >3000s 0.21s 0.09s 0.01s

dubois100 >3000s 11.85s 0.08s 0.01s

aim200-2_0-no-1 >3000s 0.01s < 0.01s < 0.01s

2dlx_..._bug005 >3000s >3000s >3000s 2.90s

c6288 >3000s >3000s >3000s >3000s

Source: Marques-Silva 2002

Solvers have continually improved over time

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How do SAT Solvers Keep Improving?

From academically interesting to practically relevant.

We now have regular SAT solver competitions.

(Germany ’89, Dimacs ’93, China ’96, SAT-02, SAT-03, SAT-04, SAT-05, SAT-06)

E.g. at SAT-2006 (Seattle, Aug ’06):

• 35+ solvers submitted, most of them open source

• 500+ industrial benchmarks

• 50,000+ benchmark instances available on the www

This constant improvement in SAT solvers is the key to making, e.g.,SAT-based planning very successful.

17

Current Automated Reasoning Tools

Most-successful fully automated methods: based on Boolean Satisfiability (SAT) / Propositional Reasoning

– Problems modeled as rules / constraints over Boolean variables– “SAT solver” used as the inference engine

Applications: single-agent search

• AI planning SATPLAN-06, fastest optimal planner; ICAPS-06 competition (Kautz & Selman ’06)

• Verification – hardware and softwareMajor groups at Intel, IBM, Microsoft, and universitiessuch as CMU, Cornell, and Princeton.SAT has become the dominant technology.

• Many other domains: Test pattern generation, Scheduling,Optimal Control, Protocol Design, Routers, Multi-agent systems,E-Commerce (E-auctions and electronic trading agents), etc.

18

Tutorial Roadmap

Automated reasoning The complexity challenge State of the art in Boolean reasoning

2. SAT-based reasoning Boolean logic– Search space, worst-case complexity– Hardness profiles, scaling in practice– Modeling problems as SAT






5. Summary

19

Boolean Logic

Defined over Boolean (binary) variables a, b, c, …

Each of these can be True (1) or False (0)

Variables connected together with logic operators: and, or, not (denoted )

E.g. (a or b) is True iff at least one of a and b is True

((c and d)) or f) is True iff either c is True and d is False, or f is True

Fact: All other Boolean logic operators can be expressed with and, or, not E.g. (a b) same as (a or b)

Boolean formula, e.g. F = (a or b) and (a and (b or c))

(Truth) Assignment: any setting of the variables to True or False

Satisfying assignment: assignment where the formula evaluates to True

20

Boolean Logic: Example

F = (a or b) and (a and (b or c))

Note: True often written as 1, False as 0

• There are 23 = 8 possible truth assignments to a, b, c– (a=0,b=1,c=0) representing (a=False, b=True, c=False)

– (a=0,b=0,c=1)

– …Truth Table for F

a b c F

0 0 0 0

0 0 1 0

0 1 0 1

0 1 1 1

1 0 0 1

1 0 1 0

1 1 0 0

1 1 1 0

• Exactly 3 truth assignments satisfy F– (a=0,b=1,c=0)

– (a=0,b=1,c=1)

– (a=1,b=0,c=0)

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Rules:1. X1 & (not X2) & X3 X42. X2 not X4

3. X5 X3 or X64. X7 X85. X8 X96. X8 X57. X6 not X9

VariablesX1 = email_ receivedX2 = in_ meetingX3 = urgentX4 = respond_to_email

X5 = near_deadlineX6 = postpone

X7 = air_ticket_info_requestX8 = travel_ requestX9 = info_request

Boolean Logic: Expressivity

All discrete single-agent search problems can be cast as a Boolean formula!

Variables a, b, c, … often represent “states” of the system, “events”, “actions”, etc.(more on this later, using Planning as a example)

Very general encoding language. E.g. can handle

• Numbers (k-bit binary representation)

• Floating-point numbers

• Arithmetic operators like +, x, exp(), log()

• …

SAT encodings (generated automatically from high level languages) routinely used in domains like planning, scheduling, verification, e-commerce, network design, …

Recall Example:

“state”

“action”

constraint

“event”

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Boolean Logic: Standard Representations

Each problem constraint typically specified as (a set of) clauses:

E.g. (a or b), (c or d or f), (a or c or d), …

Formula in conjunctive normal form, or CNF: a conjunction of clauses

E.g. F = (a or b) and (a and (b or c)) changes to

FCNF = (a or b) and (a or b) and (b or c)

Alternative [useful for QBF]: specify each constraint as a term (only “and”, “not”):

E.g. (a and d), (b and a and f), (b and d and e), …

Formula in disjunctive normal form, or DNF: a disjunction of terms

E.g. FDNF = (a and b) or (a and b and c)

clauses (only “or”, “not”)

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Boolean Satisfiability Testing

• A wide range of applications• Relatively easy to test for small formulas (e.g. with a Truth Table)• However, very quickly becomes hard to solve

– Search space grows exponentially with formula size (more on this next)

SAT technology has been very successful in taming this exponential blow up!

The Boolean Satisfiability Problem, or SAT:

Given a Boolean formula F,

• find a satisfying assignment for F

• or prove that no such assignment exists.

24

Tutorial Roadmap


2. SAT-based reasoning Boolean logic Search space, worst-case complexity– Hardness profiles, scaling in practice– Modeling problems as SAT






5. Summary

25

SAT Search Space

SAT Problem: Find a path to a True leaf node.

For N Boolean variables, the raw search space is of size 2N

• Grows very quickly with N• Brute-force exhaustive search unrealistic without efficient heuristics, etc.

All vars free

Fix one variable to True or False

Fix another var

Fix a 3rd var

TrueTrueFalse False

False

Fix a 4th var

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Worst-Case Complexity

SAT is an NP-complete problem

• Worst-case believed to be exponential(roughly 2N for N variables)

• 10,000+ problems in CS are NP-complete (e.g. planning, scheduling, protein folding, reasoning)

• P vs. NP --- $1M Clay Prize

However, real-world instances are usually not pathological and can often be solved very quickly with the latest technology!

Typical-case complexity provides a moredetailed understanding and a more positive picture.

exponential

polynomial

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Tutorial Roadmap


2. SAT-based reasoning Boolean logic Search space, worst-case complexity Hardness profiles, scaling in practice– Modeling problems as SAT






5. Summary

29

Typical-Case Complexity

A key hardness parameter for k-SAT: the ratio of clauses to variables

Add Constraints

Delete Constraints

Problems that are not critically constrained tend to be much easier in practicethan the relatively few critically constrained ones

[Mitchell, Selman, and Levesque ’92; Kirkpatrick and Selman – Science ’94]

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Typical-Case Complexity

Random 3-SAT as of 2004

Random Walk

DP

DP’

Walksat

SP

Linear time algs.

GSAT

Phase transition

SAT solvers continually getting close to tackling problems in the hardest region!

SP (survey propagation) now handles 1,000,000 variablesvery near the phase transition region

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Tractable Sub-Structure Can Dominate and Drastically Reduce Solution Cost!

2+p-SAT model: mix 2-SAT (tractable) and 3-SAT (intractable) clauses

> 40% 3-SAT: exponential scaling

40% 3-SAT: linear scaling!

(Monasson, Selman et al. – Nature ’99; Achlioptas ’00)

Number of variables

Med

ian

runt

ime

32

Tutorial Roadmap


2. SAT-based reasoning Boolean logic Search space, worst-case complexity Hardness profiles, scaling in practice Modeling problems as SAT






5. Summary

33

SAT Encoding Example: Planning Domain

Planning Problem Propositional CNF formulaby axiom schemas

Discrete time, modeled by integers

• state predicates: indexed by time at which they hold

E.g. at_location(x,,loc,i), free(x,i+1), route(cityA,cityB,i)

• action predicates: indexed by time at which action begins

E.g. fly(cityA,cityB,i), pickup(x,loc,i), drive_truck(loc1,loc2,i)

– each action takes 1 time step

– many actions may occur at the same step

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Encoding Rules

• Actions imply preconditions and effects

fly(x,y,i) at(x,i) and route(x,y,i) and at(y,i+1)

• Conflicting actions cannot occur at same time (A deletes a precondition of B)

fly(x,y,i) and yz not fly(x,z,i)

• If something changes, an action must have caused it(Explanatory Frame Axioms)

at(x,i) and not at(x,i+1) y . route(x,y) and fly(x,y,i)

• Initial and final states hold

at(NY,0) and ... and at(LA,9) and ...

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Using SAT Solvers for Planning

axiomschemas instantiated

propositionalclauses

satisfyingmodelplan

mapping

length

Problem description inhigh level language

SATengine(s)

instantiate

interpret

Modeling and Solving a Planning Problem

(fully automatic)

(manual)

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Planning Benchmark Complexity

Logistics domain – a complex, highly-parallel transportation domain

E.g. logistics.d problem:

o 2,165 possible actions per time slot

o optimal solution contains 74 distinct actions over 14 time slots

(out of 5 x 10^46 possible sequential plans of length 14)

Satplan [Selman et al.] approach is currently fastest optimal planning approach. Winner ICAPS-05 & ICAPS-06 international planning competitions.

40

Tutorial Roadmap


SAT-based reasoning Boolean logic Search space, worst-case complexity Hardness profiles, scaling in practice Modeling problems as SAT

Example domain: planning

3. QBF reasoning (extends SAT) A new range of applications– Two motivating examples




5. Summary

41

The Next Challenge in Reasoning Technology

Multi-Agent Reasoning:Quantified Boolean Formulae (QBF)

– Allow use of Forall and Exists quantifiers over Boolean variables– QBF significantly more expressive than SAT:

from single-person puzzles to competitive games!

New application domains:• Unbounded length planning and verification• Multi-agent scenarios, strategic decision making• Adversarial settings, contingency situations• Incomplete / probabilistic information

But, computationally *much* harder (formally PSPACE-complete rather than NP-complete)

Key challenge: Can we do for QBF what was done for SAT solving in the last decade?

Would open up a tremendous range of advanced automated reasoning capabilities!

war gaming

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Multi-Agent Reasoning

A Wide Range of Applications

• Logistics planning and scheduling under adversarial / uncertain conditions– Multi-agent contingency planning with performance guarantees

• Analysis and validation of distributed agent strategies and coordination – Event logic

• Reasoning in rich multi-player (adversarial) settings– War gaming, designing secure data and communication networks

QBF Technology for Multi-Agent Reasoning offers:

• Performance guarantee / optimality / “worst-case” scenario analysis– NOTE: Worst-case may be rare but can have catastrophic consequences!

The Challenge: Overcoming the high computational complexity

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SAT Reasoning vs. QBF Reasoning

SAT Reasoning Combinatorial search

for optimal and near-optimal solutions

NP-complete(hard)

planning, scheduling, verification, model checking, …

From 200 vars in early ’90s to 1M vars. Now a commercially viable technology.

QBF Reasoning Combinatorial search

for optimal and near-optimal solutions in multi-agent, uncertain, orhostile environments

PSPACE-complete(harder)

adversarial planning, gaming, security protocols, contingency planning, …

From 200 vars in late 90’s to 100K vars currently. Still rapidly moving.

Scope oftechnology

Worst-casecomplexity

Applicationareas

Researchstatus

44

Tutorial Roadmap




3. QBF reasoning (extends SAT) A new range of applications Two motivating examples




5. Summary

45

The Need for QBF Technology

SAT technology, while very successful for single-agent search, is not suitable for adversarial reasoning.

Must model the adversary and incorporate his actions into reasoning• SAT does not provide a framework for this• In fact, it cannot (more on this later)

Two examples next:

1. Network planning: create a data/communication network between N nodes which is robust under failures during and after network creation

2. Logistics planning: achieve a transportation goal in uncertain environments

46

Adversarial Planning: Motivating Example

Network Planning Problem:– Input: 5 nodes, 9 available edges that can be placed between any two nodes– Goal: all nodes finally connected to each other (directly or indirectly)– Requirement (A): final network must be robust against 2 node failures– Requirement (B): network creation process must be robust against 1 node failure

E.g. a sample robust final configuration:(uses only 8 edges)

Side note: Mathematical structure of the problem:

1. (A) implies every node must have degree ≥ 3(otherwise it can easily be “isolated”)

2. At least one node must have degree ≥ 4(follows from 1. and that not all 5 nodes can have odd degree in any graph)

3. Need at least 8 edges total (follows from 1. and 2.)

4. If one node fails during creation, the remaining 4 must be connected with 6 edges to satisfy (A)

5. Actually need 9 edges to guarantee construction (follows from 4. because a node may fail as soon as its degree becomes 3)

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Example: A SAT-Based Sequential Plan

Ideal situation: No failure during network creation

The plan goes smoothly and we end up with the target network, which is robust against any 2 node failures

Create edge

Next move if no failures

Final network robust against2 failures

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What if the leftnode fails?

Can still make the remaining 4 nodesrobust using 2 more edges (total 8 used)

• Feasible, but must re-plan to find a different final configuration


Node failures may render the original plan ineffective, but re-planning could help makethe remaining network robust.

Create edge

Node failure during network creation

Next move if a particular node fails

Next move if no failures

Final network robust against2 more failures

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What if the topnode fails?

Need to create 4 more edges tomake the remaining 4 nodes robust

• Stuck! Have already used up 6 of the 9 available edges!


Trouble! Can get stuck if

• Resources are limited(only 9 edges)

• Adversary is smart(takes out node with degree 4)

• Poor decisions were made early on in the network plan

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Example: A QBF-Based Contingency Plan

A QBF solver will return a robust contingency plan (a tree)• Will consider all relevant failure modes and responses

(only some “interesting” parts of the plan tree are shown here)

9 edgesneeded

only 8edgesused

9 edgesneeded

only 8edgesused

Create edge

Node failure during network creation

Next move if a particular node fails

Next move if no failures…

.

….

….

Final networks robust against2 more failures

….

….

….

….

….

….

….

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Another Example: Logistics Planning

Base 1

City-1

City-2 City-4

Base 2

City-3

• Blue nodes are cities, green nodes are military bases

• Blue edges are commercial transports, green edges are military

• Green edges (transports) have a capacity of 60 people, blue edges have a capacity of 100 people

• operator: “transport” t(who, amount, from, to, step)

• parallel actions can be taken at each step

• Goal: Send 60 personal from Base-1 to Base-2 in at most 3 steps

60p

• One player: military player, deterministic classic planning, SatPlan

• (1) Sat-Plan: t(m, 60, base-1, city-3, 1), t(m, 60, city-3, city-4, 2), t(m, 60, city-4, base-2, 3)

• Two players: deterministic adversarial planning QB Plan

• Military Player (m) is “white player”, Commercial Player (c) is “black player” (Chess analogy). Commercial player can move up to 80 civilians between cities. Commercial moves can not be invalidated. Goal can be read as: “send 60 personal from Base-1 to Base-2 in at most 3 steps whatever commercial needs (moves) are”

• If commercial player decides to move 80 civilians from city-3 to city-4 at the second step, we should replan (1). Indeed, the goal can not be achieved if we have already taken the first action of (1)

• (2) QB-Plan: t(m, 20, base-1, city-1, 1), t(m, 20, base-1, city-2, 1), t(m, 20, base-1, city-3, 1), t(m, 20, city-1, city-4, 2), t(m, 20, city-2, city-4, 2), t(m, 20, city-3, city-4, 2), t(m, 60, city-4, base-2, 3)

(1) SatPlan

(s1) (s2)

(s3)

*At any step commercial player can transport up to 80 civilians

(60p) + (80c) > 100 (civilian transport capacity) Re-planning needed !!!

(2) QbPlan

20p20p 20p

(s1)

(s1)

(s1)

(s2)

(s2)

(s2)

60p

(s3)

(20p) + (up to 80c) <= 100

52

Tutorial Roadmap





network planning, logistics planning

Quantified Boolean logic– Modeling problems as QBF– Search space, worst-case complexity– Scaling in practice


5. Summary

53

Boolean logic extended with “quantifiers” on the variables

– “there exists a value of x in {True,False}”, represented by x

– “for every value of y in {True,False}”, represented by y

– The rest of the Boolean formula structure similar to SAT,usually specified in CNF form

E.g. QBF formula F(v,w,x,y) = v w x y : (v or w or x) and (v or w) and (v or y)

Quantified Boolean Logic

Quantified Boolean variables constraints (as before)

54

Quantified Boolean Logic: Semantics

F(v,w,x,y,z) = v w x y : (v or w or x) and (v or w) and (v or y)

What does this QBF formula mean?

Semantic interpretation:

F is True iff “There exists a value of v s.t.

for both values of w

there exists a value of x s.t.

for both values of y

(v or w or x) and (v or w) and (v or y) is

True”

55

Quantified Boolean Logic: Example

F(v,w,x,y,z) = v w x y : (v or w or x) and (v or w) and (v or y)

Truth Table for F as a SAT formula

v w x y F

0 0 0 0 0

0 0 0 1 1

0 0 1 0 0

0 0 1 1 1

0 1 0 0 0

0 1 0 1 0

0 1 1 0 0

0 1 1 1 0

1 0 0 0 0

1 0 0 1 0

1 0 1 0 1

1 0 1 1 1

1 1 0 0 1

1 1 0 1 1

1 1 1 0 1

1 1 1 1 1

Is F True as a QBF formula?

Without quantifiers (as SAT):have many satisfying assignmentse.g. (v=0, w=0, x=0, y=1)

With quantifiers (as QBF):many of these don’t worke.g. no solution with v=0

F does have a QBF solutionwith v=1 and x set depending on w

56

Tutorial Roadmap






Quantified Boolean logic Modeling problems as QBF– Search space, worst-case complexity– Scaling in practice


5. Summary

57

QBF Modeling: Building Blocks Similar to SAT

• Boolean Variables: states + actions

w-king_at_(2,1)_3 = “white king is at cell (1,2) after time step 3”

move_b-rook_(1,1)_(2,1)_2 = “move b-rook from (1,1) to (2,1) at step 2”

truck1_at_cityA_2 = “truck #1 is at city A after time step 2”

load_box20_truck1_4 = “load box #20 onto truck #1 at time step 4”

• Constraints: initial state + goal / query + rules of the domain

w-king_at_(4,1)_start = “w-king starts out at cell (4,1)”

(move_b-rook_(1,1)_(3,1)_2 implies (b-rook_at_(1,1)_2 and empty_(2,1)_2))

= “if b-rook moves from (1,1) to (3,1) at time step 2, then it must be at cell (1,1) then, and cell (2,1) must be empty ”

box12_at_cityC_end = “box12 must reach cityC at the end”

(drive_truck1_cityA_cityB_1 implies truck1_at_cityA_1)

chess

chess

logistics

logistics

58

QBF Modeling: The Semantics

Example 1: a 4-move chess game

There exists a move of the white s.t. for every move of the black there exists a move of the white s.t. for every move of the black the white player wins

Example 2: contingency planning for disaster relief

There exist preparatory steps s.t. for every disaster scenario within limits there exists a sequence of actions s.t. necessary food and shelter can be guaranteed within two days

59

QBF and Uncertainty

Adversarial planning requires reasoning about uncertainty in the actions of the adversary

(A) Uncertainty may be captured as probabilities over adversarial actions : probabilistic planning results good in expectation or with high probability

(B) Uncertainty may alternatively be modeled as a set of contingencies : discrete contingency planning guaranteed results robust against all modeled modes of failure

QBF in the basic form takes approach (B)

Can be extended to weighted QBF in order to capture (A)

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Adversarial Uncertainty Modeled as QBF

• Two agents: self and adversary

• Both have their own set of actions, rules, etc.

• Self performs actions at time steps 1, 3, 5, …, T

• Adversary performs actions at time steps 2, 4, 6, …, T-1

There exists a self action at step 1 s.t.

for every adversary action at step 2

there exists a self action at step 3 s.t.

for every adversary action at step 4

…

there exists a self action at step T s.t.

( (initialState(time=1) and

self-respects-modeled-behavior(1,3,5,…,T) and goal(T))

OR (NOT adversary-respects-modeled-behavior(2,4,…,T-1)) )

The following QBF formulation is True if and only ifself can achieve the goal no matter what actions adversary takes

61

Tutorial Roadmap






Quantified Boolean logic Modeling problems as QBF Search space, worst-case complexity– Scaling in practice


5. Summary

62

QBF Search Space

Recall traditional SAT-type search space

Adversary action

Self action

Adversary action

Initial state

Self action

Goal GoalNo goal No goal

Self action

Self action

Initial state

Goal GoalNo goal

: self

: adversary

63

QBF Solution: A Policy or Strategy

Contingency plan

• A policy / strategy of actions for self

• A subtree of the QBF search tree (contrast with a linear sequence of actions in SAT-based planning)

Adversary action

Adversary action

Initial state

Self action

Self action

Self action

Self action

Initial state

GoalNo goal

Goal GoalGoal

No goal Goal

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Exponential Complexity Growth

Planning (single-agent): find the right sequence of actions

HARD: 10 actions, 10! = 3 x 106 possible plans

REALLY HARD: 10 x 92 x 84 x 78 x … x 2256 =

10224 possible contingency plans!

Contingency planning (multi-agent): actions may or may not produce the desired effect!

exponential

polynomial

…1 outof 10

2 outof 9

4 outof 8

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Exponential Complexity Growth:Chess Problems (benchmark domain for REAL)

Board size

No. of pieces

No. of moves

No. of vars.QBF model size (no. of constraints)

Boolean search space

RAW SPACE (legal moves)

4x4 3 3 600 ~ 4K 10180 107

4x4 5 4 1.3K ~ 10K 10390 1012

5x5 5 5 3K ~ 25K 10900 1014

6x6 8 6 10K ~ 100K 103,000 1019

7x7 10 7 24K ~ 200K 107,200 1025

8x8 16 11 90K ~ 500K 1027,000 1043

bxb p m O(pb3m) [highly board dependent] O(2pb3m) O(pmb3m)

(such problems solved in under 20 seconds by Qbf-Cornell)

Board Parameters

66

P

NP

PSPACE

EXP

NP-complete: SAT, scheduling, graph coloring, …

PSPACE-complete: QBF, adversarial planning, …

EXP-complete: games like Go, …

P-complete: circuit-SAT, …

Note: widely believed hierarchy; know P≠EXP for sure

In P: sorting, shortest path…

Computational Complexity Hierarchy

Easy

Hard

67

Tutorial Roadmap






Quantified Boolean logic Modeling problems as QBF Search space, worst-case complexity Scaling in practice


5. Summary

68

QBF Scaling Behavior in Practice

Although hard in the worst-case, QBF technology can scale well in practice.

• Real-world problems often have built-in structure and contain hidden tractable sub-problems that make QBF solvers much faster in practice.

(Note: This has helped make SAT-based single-agent reasoning scale to large problems with 1,000,000+ variables. We are pushing QBF toward this goal.)

• Qbf-Cornell and Qbf-CornellD can automatically exploit many hidden tractable sub-structures and regularities in the problem (scaling plots on the next few slides)

On many chess instances, we observe– either low-polynomial (or even near-linear) scaling– or exponential scaling but with a much lower exponent than raw search

69

Scaling: Number of PiecesQbf-CornellD compared with the raw space of legal chess moves

searched at 100,000,000 moves per second:

8K vars,150K constraints

17K vars,400K constraints

1

10

100

1000

10000

100000

1e+06

1e+07

1e+08

19 18 17 16 15 14 13 12 11 10 9 8

Run

time

in s

econ

ds (l

ogsc

ale)

Number of pieces (logscale)

Polynomial scaling with number of pieces (log-log plot)

Raw Search at 100M moves/secCornell QBF Solver

(1 day)

(1 year)

(1 minute)

70

Scaling: Number of Steps

The worst-case scaling is exponential in the number of steps. However, in practice,

the exponent is much smaller for the Qbf-CornellD than for raw search.

Exponential speed-up overthe raw search time

Problems remain feasible much longer

1

100

10000

1e+06

1e+08

1e+10

1e+12

1e+14

3 5 7 9 11

Run

time

in s

econ

ds (l

ogsc

ale)

Number of steps

Scaling with number of steps (log-plot)


(1 day)

(1 year)

(1 minute)

5-18K vars,100-370K constraints

71

Scaling: Number of Steps

Many real-world problems contain tractable (but hidden) sub-problems.

Qbf-CornellD can often automatically learn and exploit these.

Reachability-based boards:

Near-linear scaling with number of steps!

1

10

100

1000

10000

3 5 7 9 11 13 15

Run

time

in s

econ

ds (l

ogsc

ale)

Number of steps

Scaling with no. of pieces, Hidden tractable sub-structure (log-plot)


y = x, for comparison

(1 hour)

(1 minute)

77K vars,2.2M constraints

72

Tutorial Roadmap




QBF reasoning (extends SAT) A new range of applications Two motivating examples



4. High-Performance QBF reasoning Key research advances– The technology behind QBF– A. New modeling techniques– B. Learning while reasoning– C. Structure discovery– Experimental Results

5. Summary

73

Key Technical Advances

1. Good scaling with increasing number of quantifier alternations

2. New problem modeling techniques• natural, generic, automated

• exploited by new solver techniques (Qbf-Cornell [’05], Qbf-CornellD [’06])

3. Effective constraint propagation across quantifiers• essential for SAT solver success; challenging for QBF

4. Addressing efficiency drop caused by auxiliary variables• the “illegal search space issue” specific to QBF

[more on these next]

74

Key Advance #1

• Most QBF benchmarks have only 2-3 quantifier levels– Might as well translate into SAT (it often works!)

– Early QBF solvers focused on such instances

– Benchmarks with many quantifier levels are often the hardest

• Practical issues in both modeling and solving become much more apparent with many quantifier levels

QBF solvers can now scale well with

10+ quantifier alternations

Achieved through better modeling and learning-while-reasoning techniques.

[details to follow]

75

Key Advance #2

QBF solvers are extremely sensitive to encoding!– Especially with many quantifier levels,

e.g., evader-pursuer chess instances [Madhusudan et al. 2003]

Instance (N, steps)

Model X [Madhusudan et al. 2003]

Model A [Cornell 2005]

Model B [Cornell 2005]

QuBEJ [Italy]

Semprop [Germany]

Quaffle [Princeton]

Best other solver

Qbf-Cornell

Best other solver

Qbf-Cornell

4 7 2030 >2030 >2030 7497 3 0.03 0.03

4 9 -- -- -- -- 28 0.06 0.04

8 7 -- -- -- -- 800 5 5

We now have generic QBF modeling techniques

that are simple and efficient for solvers[details to follow]

76

Key Advance #3

For QBF, traditional encodings hinder unit propagation– E.g. unsatisfiable “reachability” queries

– A SAT solver would have simply unit propagated

– Most QBF solvers need 1000’s of backtracks and relatively complex mechanisms like learning to achieve simple propagation

Best solverwith only unit propagation

Best solver(Qbf-Cornell)with learning

conf-r1 2.5 0.2

conf-r5 8603 5.4

conf-r6 >21600 7.1

q-unsat: too few steps for White

?

QBF solvers now achieve effective propagation across quantifiers

New solver (Qbf-CornellD) exploiting new modeling techniques [details to follow]

77

Example: Lack of Effective Propagation(in Traditional QBF Solvers)

Impossible! White has one toofew available moves

Question:Can White reach thepink square withoutbeing captured?

This instance should ideally be easy even with many additional (irrelevant) pieces!Unfortunately, all CNF-based QBF solvers scale exponentially

Good news: Qbf-CornellD based on dual encoding technology resolves this issue!

[ click image for video ]

78

Key Advance #4

QBF solvers suffer from the “illegal search space issue”[Cornell 2005]

– Auxiliary variables needed for conversion into Boolean form

– Can push solver into large irrelevant parts of search space

– Note: negligible impact on SAT solvers due to effective propagation

– Our first solution for QBF: Qbf-Cornell [2005]• Pass “flags” to the solver, which detect this event and trigger backtracking

Dual encoding based QBF solvers can

completely avoid the illegal search space issue

Achieved through better modeling and solving techniques (Qbf-CornellD).

[details to follow]

79

OriginalSearch Space

2N

Search SpaceSAT Encoding

2N+M

Space Searchedby SAT Solvers

2N/C ; Nlog(N); Poly(N)

Original2N

Intuition for Illegal Search Space:Search Space for SAT Approaches

In practice, formany real-worldapplications, polytime scaling.

80

OriginalSearch Space

2N

Search SpaceQBF Encoding

2N+M’

Space Searchedby Qbf-Cornell

with Streamlining

Search Space of QBFSearch Space

Standard QBF Encoding2N+M’’

Original2N

81

Tutorial Roadmap







4. High-Performance QBF reasoning Key research advances The technology behind QBF– A. New modeling techniques– B. Learning while reasoning– C. Structure discovery– Experimental Results

5. Summary

82

Progress in QBF Reasoning

QBF technology has grown rapidly and can now scale to 100,000 variables, compared to 200 variables 5 years ago. (Advances largely due to research supported by Darpa’s REAL program.)

What were the key techniques that made these advances possible?

83

#1. New QBF Modeling Techniques

#2. Learning while Reasoning

#3. Structure Exploitation:

Streamlining

gain factor > 104

(w.r.t. best QBF solver, 2004)

gain factor> 102

gain factor> 103

Overall performance gainover state of the art in 2004 = 109x

Goal: solve 80% chess boards in < 5 secs [12 pieces, 5 moves, 10K vars, 40K rules]

Result: achieved 88.5% success![10-12 pieces, 7 moves, 12K vars, 100K rules]

Reveals:• Power of QBF for developing and analyzing

strategies in multi-agent settings• As tool for augmented cognition, e.g. in

military adversarial analysis

High-Performance Multi-Agent Reasoning

Technology behindQbf-Cornell [’05] and Qbf-CornellD [’06]

84

Qbf-Cornell on a Chess Instance

16 pieces11 steps

Raw searchspace size~ 1043

Solved in~ 20 seconds

QBF solution gives a strategy with which White king can provably reach the goal square

– no matter what Black tries to do!

[ click image for video of a sample run ]

85

Generality of the QBF Approach

General reasoning very different from specialized programs.

1. Generality: Improved general reasoning faster solutions for all domains– If you can model a domain (logistics, security protocols, network configuration, etc.),

you can solve it with general reasoning

2. Flexibility: End-user can ask different (sub)-queries– “Can two black pieces get within two squares from white king in 4 moves?”

– “Can my troops capture that bridge in one day without using heavy artillery?

– Provides strategic analysis capabilities

QBF Reasoning Technology offers even more:

3. Optimality: Guarantees optimality in any multi-agent scenario– Can analyze worst-case

e.g. “Can my troops capture that bridge no matter what the enemy does?”

4. Complex internal reasoning: – Exploits interplay between representations (e.g. move-based & location-based)

– Can go beyond a human programmer

86

Tutorial Roadmap







4. High-Performance QBF reasoning Key research advances– The technology behind QBF A. New modeling techniques– B. Learning while reasoning– C. Structure discovery– Experimental Results

5. Summary

87

A. New QBF Modeling Techniques

a) Logarithmic encoding• Succinctly represents collective behavior of objects

e.g. “in chess, only one move can be taken at each time step”• Technique extended to QBF and our setting

b) Both moves and locations used as building blocks• Allows interplay between move-based and location-based reasoning / representation• Most heuristic chess programs use only move-based search

c) Trigger-based pruning of search space• Automatically generated Indicator variables

flag illegal actions of the universal agent

e.g. in chess, illegal moves of the black player when white plays first• QBF-Cornell implements flag-based pruning

d) Dual-encoding, combining conjunctive and disjunctive rules (CNF and DNF)• Exploit a dichotomy between the existential and universal agents• Qbf-CornellD exploits this dual format to further boost QBF-Cornell

RESULT: Over six orders of magnitude improvement

88

Encoding: The Traditional Approach

Problemof interest

e.g. circuit minimization

CNF-basedQBF encoding QBF Solver

Solution!Any discrete

adversarial task

89

Encoding: Our Game-Based Approach

AdversarialTask

e.g. circuit minimization

Game G:

players E & U,states, actions,

rules, goal

“Planning as Satisfiability”framework

[Selman-Kautz ’96]

Create CNF encodingseparately for E and U:

initial state axioms,action implies precondition,

fact implies achieving action,frame axioms,goal condition

Dual (split)CNF-DNF encoding

QBF SolverQbf-CornellD

[2006]Negate

CNF part for U(creates DNF)

Solution!

Flag-basedCNF encoding

QBF SolverQbf-Cornell

[2005]

Solution!

96

Tutorial Roadmap







4. High-Performance QBF reasoning Key research advances– The technology behind QBF A. New modeling techniques B. Learning while reasoning– C. Structure discovery– Experimental Results

5. Summary

97

B. Learning while Reasoning

• A key component of the QBF solver is the ability to learn new constraints during the search.

• The learned information is captured by new clauses that are added to the formula. These clauses can dramatically prune the remaining search.

Com

pu

tati

onal

Cos

t

50 var

40 var

20 var

0

1000

3000

2000

4000

Ratio of clauses to variables

Computational cost

(secs)

Hardness complexity curves

Add RulesDelete RulesRecall:

98

Encoding: Before Learning

Before learning, constraints in the encoding specify the rules of the game(i.e. what it means for a move to be legal) and the query as Boolean logic. E.g.

Rule : “if White Rook moves from a2 to a5 at step 7, then a4 must be empty at 7.”Encoding : (move_wRook_a2_a5_7 empty_a4_7)

Rule : “Only one piece can move at each step t.”

Encoding : Several constraints of the form: (move_pieceA_cell1_cell2_t NOT move_pieceB_cell3_cell4_t)

Query : “Can White King reach cell e2 in at most 6 steps?”

Encoding : (location_wKing_e2_1 OR location_wKing_e2_2 OR … OR location_wKing_e2_6)

1

2

3

4

5

6

a b c d e f

True/False variables

99

Encoding: After Learning

After learning, new constraints specify 1000’s of more complex observations logically deduced from the board.

These observations typically combine spatial and temporal knowledge, often in terms of “if-then” facts. Some of these are used subsequently in the search, and the rest are gradually discarded. E.g.

Knowledge : “if White King is at c5 and White Pawn is at d3 initially, then White King

cannot be at d3 after making 2 moves (its own piece is blocking d3).”

Encoding : ((location_wKing_c5_1 AND location_wPawn_d3_1) NOT location_wKing_d3_5) Note: White moves on odd steps, Black on even steps.

Knowledge : “if White moves the bottom Rook at step 1, then wKing cannot be in row 2 at step 7.”

Encoding : For each cell X in {a2,b2,c2,d2,e2,f2}, a constraint: (NOT location_wRook_f2_2) (NOT location_wKing_X_7) 1

2

3

4

5

6

a b c d e f

100

Learning while Reasoning

E.g. can w-king reach top-left cell in 5 moves?

Without Learningexplores > 20K

possibilities

With Learningexplores < 5possibilities

QBF reasoning engine has no hard-coded knowledge of chessor distance functions! – still learns automatically, e.g., that a square is not reachable! – continuously adds learned/inferred constraints from failure, simplifying future search

NOTE: Such knowledge must be hand-programmed into conventional solvers

101

[ click images for video ]

Query: Can the white king reach the top-left cell in 5 moves?

Without Learning(> 20K possibilities explored)

With Learning(< 5 possibilities explored)

RECALL: No hard-coded knowledge. Learns automatically about reachability

Example: Learning “Reachability”

102

Learning while Reasoning: A Visualization

RESULT: Over two orders of magnitude improvement

Solver learns, e.g., to focus the searchto the most relevant areas of the boardwithout explicitly being told anythingabout it!

103

Tutorial Roadmap







4. High-Performance QBF reasoning Key research advances– The technology behind QBF A. New modeling techniques B. Learning while reasoning C. Structure discovery– Experimental Results

5. Summary

104

C. Structure Exploitation: Streamlining

Search space

solutions

Exploit domain and problem structure

to narrow down search to

small regions very likely to have a solution!

E.g in chess,

Try region R1: white never moves a piece away from the goal : no solution found

Try region R2: white doesn’t use the two of its far away pawns : solution found!

R1

R2

Key invariant: if an agent’s actions are restricted and it still succeeds, it will succeed in the original setting as well

The art lies in choosing “good” restrictions – not too strong, not too weak!

(white player wins)

RESULT: Over three orders of magnitude improvement

105

Structure Exploitation: Structure Discovery

Planning problem encoding843 vars, 7301 constraints, approx. min backdoor 16

(“backdoor” set = reasoning shortcut)

Constraint graph of areasoning problem

One node per variable:edge between two variablesif they share a constraint.

(Gomes et al. ’03, ’04)

106


After setting 5 backdoor vars(out of 800)

After setting just 12 backdoor vars – problem almost solved!

MAP-6-7.cnf infeasible planning instances. Strong backdoor of size 3.392 vars, 2,578 clauses.


After setting 2 (out of 392) backdoor vars -- complexity almost gone!


109

Dynamic view: Running SAT solver(no backdoor detection)


110

SAT solver detects backdoor set


111

Tractable Sub-Structure Can Lead toNear-Linear Scaling in QBF as Well!

Many real-world problems contain tractable (but hidden) sub-problems.

Qbf-CornellD can often automatically learn and exploit these.

Reachability-based boards:

Near-linear scaling with number of steps!

1

10

100

1000

10000

3 5 7 9 11 13 15

Run

time

in s

econ

ds (l

ogsc

ale)

Number of steps

Scaling with no. of pieces, Hidden tractable sub-structure (log-plot)


y = x, for comparison

(1 hour)

(1 minute)

77K vars,2.2M constraints

112

Tutorial Roadmap







4. High-Performance QBF reasoning Key research advances The technology behind QBF A. New modeling techniques B. Learning while reasoning C. Structure discovery Experimental Results

5. Summary

113

Experimental Results

xChess instance Pure CNF EncodingDual

Encoding

name T/FSemprop

[Germany]sKizzo

[France]Quaffle

[Princeton]Qbf-Cornell

[2005]Qbf-CornellD

[2006]

conf-r1 F 12 4.0 15 1.3 0.01

conf-r2 F 25 5.86 33 2.5 0.02

conf-r3 F 55 9.3 62 4.1 0.03

conf-r4 F 85 26 124 6.4 0.04

conf-r5 F 985 84 676 34 0.08

conf-r6 F 2042 73 713 49 0.10

conf01 F 1225 492 -- 539 6.4

conf02 F 93 30 6.0 1.0 0.0

conf03 T -- 1532 -- 83 1.4

conf04 T -- -- 2352 100 3.5

conf05 F 3290 448 510 196 0.1

conf06 F -- memout -- 633 30.6

conf07 F 261 42 78 3.5 0.0

conf08 T -- 1509 -- 1088 31.2

5-15quantifier

levels(reachability)

7-9quantifier

levels

114

Experimental Results, contd.

xChess instance Pure CNF EncodingDual

Encoding

name T/FSemprop

[Germany]sKizzo

[France]Quaffle

[Princeton]Qbf-Cornell

[2005]Qbf-CornellD

[2006]

conf1a T 627 83 -- 161 1.8

conf1b F 682 176 2939 124 1.3

conf1c T 659 804 -- 156 2.1

conf1d F 706 1930 1473 148 2.2

conf2a T -- -- -- 438 65.9

conf2b F -- -- -- 275 56.9

conf3a T -- memout -- 653 5.2

conf3b F -- -- 2128 327 2.2

conf4 F -- -- -- 274 32.0

conf5 F 1018 427 142 11 0.1

7-9quantifier

levels

Qbf-Cornell and Qbf-CornellD dramatically outperform all leading QBF solvers on these challenging instances

115

Tutorial Roadmap







High-Performance QBF reasoning Key research advances The technology behind QBF A. New modeling techniques B. Learning while reasoning C. Structure discovery Experimental results

5. Summary

116

Where Does QBF Reasoning Stand?

We have come a long way since the first QBF solvers 5 years ago!

(thanks to the Darpa REAL program)

• From 200 variable problems to 100,000 variable problems(Qbf-Cornell and Qbf-CornellD)

• From 2-3 quantifier alternations to 10+ quantifiers

• New techniques for modeling and solving• A better understanding of issues like

propagation across quantifiers and illegal search space• Many more benchmarks and test suites

117

Summary

QBF Reasoning: a promising new automated reasoning technology!

On the road to a whole new range of applications:

• Strategic decision making

• Performance guarantees in complex multi-agent scenarios

• Secure communication and data networks in hostile environments

• Robust logistics planning in adversarial settings

• Large scale contingency planning

• Provably robust and secure software and hardware

118

Formal Models. Problem structure, BackdoorsH. Chen (Cornell)John Hopcroft (Cornell)Jon Kleinberg (Cornell)R. Williams (CMU)Joerg Hoffman (Max-Planck Inst.)

Creating the Next Generation ReasoningTechnology at the Intelligent Information Systems Institute (IISI), Cornell University

Information TheoryS. Wicker (Cornell)

Branching ProcessesK. Athreya (Cornell)

Robustness vs.FragilityJohn Doyle (Caltech)Walter Willinger (AT&T Labs)

Power laws vs. Small-world S. Strogatz (Cornell)T. Walsh (U. New South Wales)

Director: Dr. Carla Gomes

Learning Dynamic Restart StrategiesE. Horvitz (Micrsoft Research)H. Kautz and Y. Ruan (U. Washington)Nudelman and Shoham (Stanford)

Random CSP ModelsC. Fernandez, M. Valls (U. Lleida)C. Bessiere (LIRMM-CNRS)C. Moore (U. New Mexico)

Connections and Collaborations

Approximations and RandomizationLucian Leahu (Cornell)David Shmoys (Cornell)

Computer Science

Mathematics

Operations Research

Physics Cognitive Science

Economics

Cross-fertilizationof ideas for the study

and design ofIntelligent SystemsPhase transition

Engineering

Research part of Cornell’s Intelligent Information Systems Institute.Director: Carla Gomes.

Additional Material

121

Ease of Use / Portability

Q: Can someone else easily use our techniques for problems other than chess?A: Yes, it is straightforward once they understand the concept of a QBF encoding.

– Solver: they can use the Qbf-Cornell and Qbf-CornellD off-the-shelf as a black box.

– Encoder: A general encoder can be written for each application domain. Our chess encoder will serve as a useful starting point.

– Our technique is described in research publications [AAAI-05, SAT-06] and has already been explored and extended by others in various contexts [e.g. Zhang AAAI-06, Tang-Malik SAT-06, Bessiere-Verger CP-06]. (Our related work received best paper awards at the AAAI-06 and CP-06 conferences.)

– We have an initial proposal for a generic automated QBF encoder for a higher-level adversarial planning language. This language will be similar to commonly used single-agent planning specification languages like STRIPS.

Q: Are we creating a general solution?A: Yes, the approach provides a fully general platform for adversarial reasoning.

– Solver: generic; no chess-based or other specific knowledge embedded.– Encoder technology: the core is applicable to any discrete adversarial reasoning task.– Chess encoder: can be adapted manually for other tasks.

122

Encoding: Generation

• The encoding process is automatic and very efficient.– The “encoding” is a re-statement of the problem in the language of the solver.– We use a fully automated chess-to-QBF encoder.– Each of the 600 go/no-go boards required < 0.3 sec to encode.– The hardest instances we have tried required < 2 sec to encode.– For chess and most future applications of interest, encoder time will be polynomial in

the problem parameters. The challenge is always the solver time.

The encoding process takes a fraction of the time compared to the solver.

123

Encoding: Effect of Number of Steps

The number of steps is simply a parameter to the encoder.

If a different number of steps is desired, the new encoding can be generated within seconds by re-running the encoder.

Additional information:– Decreasing the number of steps (and possibly also changing the query itself) :

Requires changing only the few constraints that encode the new query (< 1% of the encoding). The rest of the encoding (> 99%) remains

untouched.

– Increasing the number of steps in the query: Requires additional variables and constraints for the extra steps,

which are simply appended to the original encoding.

– Transfer learning technology can be exploited to re-use knowledge gained from the original encoding when the query is changed; we are exploring this.

124

From Adversarial Tasks To Games

Example #2:

The Chromatic Number Problem: Given a graph G and a positive number k, does G have chromatic number k?– Chromatic number: minimum number of colors needed to color G so

that every two adjacent vertices get different colors

– A game with 2 turns• Moves : First, E produces a coloring S of G; second, U

produces a coloring T of G

• Rules : S must be a legal k-coloring of G; T must be a legal

(k-1)-coloring of G

• Goal : E wins if S is valid and T is not

– “E wins” iff graph G has chromatic number k

Documents

Quantified Boolean Formula (QBF) Reasoning Bart Selman, Carla Gomes, Ashish Sabharwal Cornell University Feb 13, 2007 Tutorial for Dr. Charles Holland