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Scalable Knowledge Representation and Reasoning

Systems

- Henry Kautz
- ATT Shannon Laboratories

Introduction

- In recent years, we've seen substantial progress

in scaling up knowledge representation and

reasoning systems - Shift from toy domains to real-world applications
- autonomous systems - NASA Remote Agent
- just in time manufacturing - I2, PeopleSoft
- deductive approaches to verification - Nitpick

(D. Jackson), bounded model checking (E. Clarke) - solutions to open problems in mathematics - group

theory (W. McCune, H. Zhang) - New emphasis on propositional reasoning and search

Approaches to Scaling Up KRR

- Traditional approach specialized languages /

specialized reasoning algorithms - difficult to share / evaluate results
- New direction
- compile combinatorial reasoning problems into a

common propositional form (SAT) - apply new, highly efficient general search

engines

SAT Encoding

Combinatorial Task

SAT Solver

Decoder

Methodology

- Compare with use of linear / integer programming

packages - emphasis on mathematical modeling
- after modeling, problem is handed to a state of

the art solver - Compare with reasoning under uncertainty
- convergence to Bayes nets and MDP's

Would specialized solver not be better?

- Perhaps theoretically, but often not in practice
- Rapid evolution of fast solvers
- 1990 100 variable hard SAT problems
- 1999 10,000 - 100,000 variables
- competitions encourage sharing of algorithms and

implementations - Germany 91 / China 96 / DIMACS-93/97/98
- Encodings can compensate for much of the loss due

to going to a uniform representation

Two Kinds of Knowledge Compilation

- Compilation to a tractable subset of logic
- shift inference costs offline
- guaranteed fast run-time response
- E.g. real-time diagnosis for NASA Deep Space One

- 35 msec response time! - fundamental limits to tractable compilation
- Compilation to a minimal combinatorial core
- can reduce SAT size by compiling together problem

spec control knowledge - inference for core still NP-hard
- new randomized SAT algorithms - low exponential

growth - E.g. optimal planning with 1018 states!

OUTLINE

- I. Compilation to tractable languages
- Horn approximations
- Fundamental limits
- II. Compilation to a combinatorial core
- SATPLAN
- III. Improved encodings
- Compiling control knowledge
- IV. Improved SAT solvers
- Randomized restarts

I. Compilation to Tractable Languages

Expressiveness vs. Complexity Tradeoff

- Consider problem of determining if a query

follows from a knowledge base - KB q ?
- Highly expressive KB languages make querying

intractable - ( ignition_on engine_off ) ?
- ( battery_dead V tank_empty )
- require general CNF - query answering is

NP-complete - Less expressive languages allow polynomial time

query answering - Horn clauses, binary clauses, DNF

Tractable Knowledge Compilation

- Goal guaranteed fast online query answering
- cost shifted to offline compilation
- Exact compilation often not possible
- Can approximate original theory
- yet retain soundness / completeness for queries
- (Kautz Selman 1993, 1996, 1999 Papadimitriou

1994)

expressive source language

tractable target language

Example Compilation into Horn

- Source clausal propositional theories
- Inference NP-complete
- example (a V b V c V d)
- equivalently (a b) ? (c V d)
- Target Horn theories
- Inference linear time
- at most one positive literal per clause
- example (a b) ? c
- strictly less expressive

Horn Bounds

- Idea compile CNF into a pair of Horn theories

that approximate it - Model truth assignment which satisfies a theory
- Can logically bound theory from above and below
- LB S UB
- lower bound fewer models logically stronger
- upper bound more models logically weaker
- BEST bounds LUB and GLB

Using Approximations for Query Answering

- S q ?
- If LUB q then S q
- (linear time)
- If GLB q then S q
- (linear time)
- Otherwise, use S directly
- (or return "don't know")
- Queries answered in linear time lead to

improvement in overall response time to a series

of queries

Computing Horn Approximations

- Theorem Computing LUB or GLB is NP-hard
- Amortize cost over total set of queries
- Query-algorithm still correct if weaker bounds

are used - anytime computation of bounds desirable

Computing the GLB

- Horn strengthening
- r ? (p V q) has two Horn-strengthenings
- r ? p
- r ? q
- Horn-strengthening of a theory conjunction of

one Horn-strengthening of each clause - Theorem Each LB of S is equivalent to some

Horn-strengthening of S. - Algorithm search space of Horn-strengthenings

for a local maxima (GLB)

Computing the LUB

- Basic strategy
- Compute all resolvents of original theory, and

collect all Horn resolvents - Problem
- Even a Horn theory can have exponentially many

Horn resolvents - Solution
- Resolve only pairs of clauses where exactly one

clause is Horn - Theorem Method is complete

Properties of Bounds

- GLB
- Anytime algorithm
- Not unique - any GLB may be used for query

answering - Size of GLB ? size of original theory
- LUB
- Anytime algorithm
- Is unique
- No space blow-up for Horn
- Can construct non-Horn theories with

exponentially larger LUB

Empirical Evaluation

- 1. Hard random theories, random queries
- 2. Plan-recognition domain
- e.g. query (obs1 obs2) ? (goal1 V goal2) ?
- Time to answer 1000 queries
- original with bounds
- rand100 340 45
- rand200 8600 51
- plan500 8950 620
- Cost of compilation amortized in less than 500

queries

Limits of Tractable Compilation

- Some theories have an exponentially-larger

clausal form LUB - QUESTION Can we always find a clever way to keep

the LUB small (new variables, non-clausal form,

structure sharing, ...)? - Theorem There do exist theories whose Horn LUB

is inherently large - any representation that enables polytime

inference is exponentially large - Proof based on non-uniform circuit complexity -

if false, polynomial hierarchy collapses to ?2

Other Tractable Target Languages

- Model-based representations
- (Kautz Selman 1992, Dechter Pear 1992,

Papadimitriou 1994, Roth Khardon 1996, Mannila

1999, Eiter 1999) - Prime Implicates
- (Reiter DeKleer 1987, del Val 1995, Marquis

1996, Williams 1998) - Compilation from nonmonotonic logics
- (Nerode 1995, Cadoli Donini 1996)
- Similar limits to compilability hold for all!

Truly Combinatorial Problems

- Tractable compilation not a universal solution

for building scalable KRR systems - often useful, but theoretical and empirical

limits - not applicable if you only care about a single

query no opportunity to amortize cost of

compilation - Sometimes must face NP-hard reasoning problems

head on - will describe how advances in modeling and SAT

solvers are pushing the envelope of the size

problems that can be handled in practice

II. Compilation to a Combinatorial Core

Example Planning

- Planning find a (partially) ordered set of

actions that transform a given initial state to a

specified goal state. - in most general case, can cover most forms of

problem solving - scheduling fixes set of actions, need to find

optimal total ordering - planning problems typically highly non-linear,

require combinatorial search

Some Applications of Planning

- Autonomous systems
- Deep Space One Remote Agent (Williams Nayak

1997) - Mission planning (Muscettola 1998)
- Natural language understanding
- TRAINS (Allen 1998) - mixed initiative dialog
- Software agents
- Softbots (Etzioni 1994)
- Goal-driven characters in games (Nareyek 1998)
- Help systems - plan recognition (Kautz 1989)
- Manufacturing
- Supply chain management (Crawford 1998)
- Software understanding / verification
- Bug-finding (goal undesired state) (Jackson

1998)

State-space Planning

- State complete truth assignment to a set of

variables (fluents) - Goal partial truth assignment (set of states)
- Operator a partial function State State
- specified by three sets of variables
- precondition, add list, delete list
- (STRIPS, Fikes Nilsson 1971)

Abdundance of Negative Complexity Results

- I. Domain-independent planning PSPACE-complete

or worse - (Chapman 1987 Bylander 1991 Backstrom 1993)
- II. Bounded-length planning NP-complete
- (Chenoweth 1991 Gupta and Nau 1992)
- III. Approximate planning NP-complete or worse
- (Selman 1994)

Practice

- Traditional domain-independent planners can

generate plans of only a few steps. - Most practical systems try to eliminate search
- Tractable compilation
- Custom, domain-specific algorithms
- Scaling remains problematic when state space is

large or not well understood!

Planning as Satisfiability

- SAT encodings are designed so that plans

correspond to satisfying assignments - Use recent efficient satisfiability procedures

(systematic and stochastic) to solve - Evaluation performance on benchmark instances

SATPLAN

instantiated propositional clauses

instantiate

axiom schemas

problem description

length

SAT engine(s)

interpret

satisfying model

plan

SAT Encodings

- Propositional CNF no variables or quantifiers
- Sets of clauses specified by axiom schemas
- Use modeling conventions (Kautz Selman 1996)
- Compile STRIPS operators (Kautz Selman 1999)
- Discrete time, modeled by integers
- upper bound on number of time steps
- predicates indexed by time at which fluent holds

/ action begins - each action takes 1 time step
- many actions may occur at the same step
- fly(Plane, City1, City2, i) É at(Plane, City2, i

1)

Solution to a Planning Problem

- A solution is specified by any model (satisfying

truth assignment) of the conjunction of the

axioms describing the initial state, goal state,

and operators - Easy to convert back to a STRIPS-style plan

Satisfiability Testing Procedures

- Systematic, complete procedures
- Davis-Putnam (DP)
- backtrack search unit propagation (1961)
- little progress until 1993 - then explosion of

improved algorithms implementations - satz (1997) - best branching heuristic
- See SATLIB 1998 / Hoos Stutzle
- csat, modoc, rel_sat, sato, ...
- Stochastic, incomplete procedures
- Walksat (Kautz, Selman Cohen 1993)
- greedy local search noise to escape local

minima - outperforms systematic algorithms on random

formulas, graph coloring, (DIMACS 1993, 1997)

Walksat Procedure

- Start with random initial assignment.
- Pick a random unsatisfied clause.
- Select and flip a variable from that clause
- With probability p, pick a random variable.
- With probability 1-p, pick greedily
- a variable that minimizes the number of

unsatisfied clauses - Repeat until time limit reached.

Planning Benchmark Test Set

- Extension of Graphplan benchmark set
- Graphplan (Blum Furst 1995) - best

domain-independent state-space planning algorithm - logistics - complex, highly-parallel

transportation domain, ranging up to - 14 time slots, unlimited parallelism
- 2,165 possible actions per time slot
- optimal solutions containing 150 distinct actions
- Problems of this size (1018 configurations) not

previously handled by any state-space planning

system

Scaling Up Logistics Planning

What SATPLAN Shows

- General propositional theorem provers can compete

with state of the art specialized planning

systems - New, highly tuned variations of DP surprising

powerful - result of sharing ideas and code in large SAT/CSP

research community - specialized engines can catch up, but by then new

general techniques - Radically new stochastic approaches to SAT can

provide very low exponential scaling - 2 orders magnitude speedup on hard benchmark

problems - Reflects general shift from first-order

non-standard logics to propositional logic as

basis of scalable KRR systems

Further Paths to Scale-Up

- Efficient representations and new SAT engines

extend the range of domain-independent planning - Ways for further improvement
- Better SAT encodings
- Better general search algorithms

III. Improved Encodings Compiling Control

Knowledge

Kinds of Control Knowledge

- About domain itself
- a truck is only in one location
- airplanes are always at some airport
- About good plans
- do not remove a package from its destination

location - do not unload a package and immediate load it

again - About how to search
- plan air routes before land routes
- work on hardest goals first

Expressing Knowledge

- Such information is traditionally incorporated in

the planning algorithm itself - or in a special programming language
- Instead use additional declarative axioms
- (Bacchus 1995 Kautz 1998 Chen, Kautz, Selman

1999) - Problem instance operator axioms initial and

goal axioms control axioms - Control knowledge constraints on search and

solution spaces - Independent of any search engine strategy

Axiomatic Control Knowledge

- State Invariant A truck is at only one location
- at(truck,loc1,i) loc1 ¹ loc2 É Ø

at(truck,loc2,i) - Optimality Do not return a package to a location
- at(pkg,loc,i) Ø at(pkg,loc,i1) iltj É Ø

at(pkg,loc,j) - Simplifying Assumption Once a truck is loaded,

it should immediately move - Ø in(pkg,truck,i) in(pkg,truck,i1)

at(truck,loc,i1) É Ø at(truck,loc,i2)

Adding Control Kx to SATPLAN

Problem Specification Axioms

Control Knowledge Axioms

Instantiated Clauses

As control knowledge increases, Core shrinks!

SAT Simplifier

SAT Core

SAT Engine

Logistics - Control Knowledge

Scale Up with Compiled Control Knowledge

- Significant scale-up using axiomatic control

knowledge - Same knowledge useful for both systematic and

local search engines - simple DP now scales from 1010 to 1016 states
- order of magnitude speedup for Walksat
- Control axioms summarize general features of

domain / good plans not a detailed program! - Obtained benefits using only admissible control

axioms no loss in solution quality (Cheng,

Kautz, Selman 1999) - Many kinds of control knowledge can be created

automatically - Machine learning (Minton 1988, Etzioni 1993,

Weld 1994, Kambhampati 1996) - Type inference (Fox Long 1998, Rintanen 1998)
- Reachability analysis (Kautz Selman 1999)

IV. Improved SAT Solvers Randomized Restarts

Background

- Combinatorial search methods often exhibit
- a remarkable variability in performance. It is
- common to observe significant differences
- between
- different heuristics
- same heuristic on different instances
- different runs of same heuristic with different

random seeds

Example SATZ

Preview of Strategy

- Well put variability / unpredictability to our

advantage via randomization / averaging.

Cost Distributions

- Consider distribution of running times of

backtrack search on a large set of equivalent

problem instances - renumber variables
- change random seed used to break ties
- Observation (Gomes 1997) distributions often

have heavy tails - infinite variance
- mean increases without limit
- probability of long runs decays by power law

(Pareto-Levy), rather than exponentially (Normal)

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Heavy-Tailed Distributions

- infinite variance infinite mean
- Introduced by Pareto in the 1920s
- probabilistic curiosity
- Mandelbrot established the use of heavy-tailed

distributions to model real-world fractal

phenomena. - Examples stock-market, earth-quakes, weather,...

How to Check for Heavy Tails?

- Log-Log plot of tail of distribution
- should be approximately linear.
- Slope gives value of
- infinite mean and

infinite variance - infinite variance

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Heavy 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

Randomized 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 - Provably Eliminates heavy tails
- In practice rapid restarts with low cutoff can

dramatically improve performance - (Gomes 1996, Gomes, Kautz, and Selman 1997,

1998)

Rapid Restart on LOG.D

Note Log Scale Exponential speedup!

Increased Predictability

- Overall insight
- Randomized tie-breaking with
- rapid restarts can boost
- systematic search
- Related analysis Luby Zuckerman 1993 Alt

Karp 1996. - Other applications sports scheduling, circuit

synthesis, quasigroup competion,

Conclusions

- Discussed approaches to scalable KRR systems

based on propositional reasoning and search - Shift to 10,000 variables and 106 clauses has
- opened up new applications
- Methodology
- Model as SAT
- Compile away as much complexity as possible
- Use off-the-shelf SAT Solver for remaining core
- Analogous to LP approaches

Conclusions, cont.

- Example AI planning / SATPLAN system
- Order of magnitude improvement (last

3yrs) - 10 step to 200 step optimal plans
- Huge economic impact possible with 2 more!
- up to 20,000 steps ...
- Discussed themes in Encodings Solvers
- Local search
- Control knowledge
- Heavy-tails / Randomized restarts

Tractable Knowledge Compilation Summary

- Many techniques have been developed for compiling

general KR languages to computationally tractable

languages - Horn approximations (Kautz Selman 1993, Cadoli

1994, Papadimitriou 1994) - Model-based representations (Kautz Selman 1992,

Dechter Pearl 1992, Roth Khardon 1996,

Mannila 1999, Eiter 1999) - Prime Implicates (Reiter DeKleer 1987, del Val

1995, Marquis 1996, Williams 1998)

Limits to Compilability

- While practical for some domains, there are

fundamental theoretical limitations to the

approach - some KBs cannot be compiled into a tractable

form unless polynomial hierarchy collapses

(Kautz) - Sometimes must face NP-hard reasoning problems

head on - will describe how advances in modeling and SAT

solvers are pushing the envelope of the size

problems that can be handled in practice

Logistics Increased Predictability

Example SATZ