Scalable Knowledge Representation and Reasoning Systems

1
Scalable Knowledge Representation and Reasoning
Systems

• Henry Kautz
• ATT Shannon Laboratories

2
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

3
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
4
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

5
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

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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!

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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

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I. Compilation to Tractable Languages
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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

10
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
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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

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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

13
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

14
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

15
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)

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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

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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

18
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

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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

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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!

21
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

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II. Compilation to a Combinatorial Core
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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

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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)

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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)

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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)

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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!

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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

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SATPLAN
instantiated propositional clauses
instantiate
axiom schemas
problem description
length
SAT engine(s)
interpret
satisfying model
plan
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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)

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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

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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)

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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.

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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

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Scaling Up Logistics Planning
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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

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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

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III. Improved Encodings Compiling Control
Knowledge
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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

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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

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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)

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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
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Logistics - Control Knowledge
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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)

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IV. Improved SAT Solvers Randomized Restarts
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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

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Example SATZ
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Preview of Strategy

• Well put variability / unpredictability to our
  advantage via randomization / averaging.

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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,...

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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

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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)

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Rapid Restart on LOG.D
Note Log Scale Exponential speedup!
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Increased Predictability
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• 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,

59
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

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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

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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)

62
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

63
Logistics Increased Predictability
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Example SATZ