Scalable Knowledge Representation and Reasoning Systems

1. Scalable Knowledge Representation and Reasoning Systems • Henry Kautz • AT&T 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 KR&R • Traditional approach: specializedlanguages / 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

6. 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!

7. 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

8. I. Compilation to Tractable Languages

9. 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

11. 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

12. 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)

16. 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

17. 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

19. 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

20. 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

22. II. Compilation to a Combinatorial Core

23. 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

24. 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)

25. 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)

26. 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)

27. 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! Practice

28. Planning as Satisfiability • SAT encodings are designed so that plans correspond to satisfying assignments • Use recent efficientsatisfiability procedures (systematic and stochastic) to solve • Evaluation performance on benchmark instances

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

30. SAT Encodings • Propositional CNF: no variables or quantifiers • Sets of clauses specified by axiom schemas • Usemodeling conventions(Kautz & Selman 1996) • Compile STRIPSoperators(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)

31. 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

32. 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)

33. 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.

34. 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

35. Scaling Up Logistics Planning

36. 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

37. 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

38. III. Improved Encodings: Compiling Control Knowledge

39. 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

40. 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

41. Axiomatic Control Knowledge • State Invariant: A truck is at only one location • at(truck,loc1,i) & loc1 ¹ loc2 ÉØ at(truck,loc2,i) • Optimality: Do not return a package to a location • at(pkg,loc,i) & Ø at(pkg,loc,i+1) & i<j ÉØ at(pkg,loc,j) • Simplifying Assumption: Once a truck is loaded, it should immediately move • Ø in(pkg,truck,i) & in(pkg,truck,i+1) & at(truck,loc,i+1) ÉØ at(truck,loc,i+2)

42. 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

43. Logistics - Control Knowledge

44. Scale Up with Compiled Control Knowledge • Significant scale-up using axiomatic control knowledge • Same knowledgeuseful for both systematic and local search engines • simple DP now scales from 1010 to 1016 states • order of magnitude speedup for Walksat • Control axioms summarizegeneral features of domain / good plans: nota 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)

45. IV. Improved SAT Solvers: Randomized Restarts

46. 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

47. Example: SATZ

48. Preview of Strategy • We’ll put variability / unpredictability to our advantage via randomization / averaging.

49. 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)