Partition-Based Logical Reasoning Bill MacCartney (KSL), Sheila A. McIlraith (KSL), Eyal Amir (FRG/Berkeley), Tomas Uribe (SRI) Richard Fikes and John McCarthy Knowledge Systems Lab | Formal Reasoning Group Stanford University with thanks to Mark Stickel and Vinay Chaudhri of SRI

11/14/02Bill MacCartney, Stanford KSL Motivation With large KBs, general-purpose reasoners suffer from combinatorial explosion.  Can we focus reasoning by decomposing the KB into a network of minimally-connected partitions? Special-purpose reasoners can be highly efficient in specific domains, but how to integrate them?  Given a network of (possibly heterogeneous) knowledge systems, how can we achieve efficient global reasoning? Can we exploit implicit structure of knowledge to make reasoning more focused & efficient?

11/14/02Bill MacCartney, Stanford KSL Overview Algorithms and theoretical results  Automatic partitioning of large KBs  Reasoning with partitions using message passing (MP) Experimental testing  Empirical validation of the effectiveness of partitioning  Even better when combined with good local strategies Surprising, productive results  Partitioning can induce near-optimal symbol orderings  MP can integrate special-purpose reasoners  Many new research questions

11/14/02Bill MacCartney, Stanford KSL Automatic partitioning Begin with a KB in PL or FOL Efficient reasoning depends on keeping partition sizes and link sizes small Construct symbol graph  Edges join symbols which appear together in an axiom Apply tree decomposition algorithm  “Alg 5”: a variant of min-fill  “Alg 6”: a divide-and-conquer tree-width algorithm Partition axioms correspondingly  Each partition has its own vocabulary  “Link languages” are defined by shared vocabulary

11/14/02Bill MacCartney, Stanford KSL Reasoning with partitions: an example A simple propositional theory Theory {Q R S T U V W X Y Z} Partition 1 {Q R S T}Partition 2 {T U V W}Partition 3 {W X Y Z} {T}{T} {W}{W} Partition 1 {Q R S T}Partition 2 {T U V W}Partition 3 {W X Y Z} {T}{T} {W}{W} (1)  Q   R  T (2)  S  T (3)  S   R (4)S  R (5)  T   U   V  W (6)T   W (7)U   W (8)V   W (9)  W   X  Z (10)X  Y (11)  W   Y  Z (15)  Z (12)Q(13)U (14)V (16)  R  T (17) S  T (18) T (19)  U   V  W (20)  V  W (21)W (22)  W  Y  Z (23)  W  Z (24)Z (25)  Using partitioning, this query took just 10 resolution steps. Using set-of-support, the same query can take 28 steps. Query: Q  U  V  Z ?

11/14/02Bill MacCartney, Stanford KSL Start with a tree-structured partition graph Reasoning with partitions using MP MP Algorithm [Amir & McIlraith 2000]  Pass messages in L i toward goal Identify goal partition Direct edges toward goal (fixing outbound link language L i for each partition) Concurrently, in each partition:  Generate consequences in L i

11/14/02Bill MacCartney, Stanford KSL Reasoning is performed locally in each partition Relevant results propagate toward goal partition Globally sound & complete … provided each local reasoner is sound & complete for L i -consequence finding Performance is worst-case exponential within partitions, but linear in tree structure Characteristics of MP Minimizes between-partition deduction Supports parallel processing Different reasoners in different partitions Focuses within-partition deduction

11/14/02Bill MacCartney, Stanford KSL Experimental Testing Do “real world” KBs exhibit inherent structure?  Can we generate partitionings in which both partition sizes and link language sizes are small?  Can partition-based reasoning outperform other strategies? Experimental testbed  Theorem prover: SNARK –Thanks to Mark Stickel and SRI  KB: Cyc –A subset on spatial relationships, ~750 axioms, ~150 symbols –We’re working on adding SUMO, Geo-Logica, RCC-8  Queries –Cyc queries provided by Vinay Chaudhri

11/14/02Bill MacCartney, Stanford KSL Results: automatic partitioning Partition graph is largely independent of query  But edges may need to be redirected We’re experimenting with multiple algorithms Alg 5Alg 6 Number of partitions12440 Max symbols/partition1619 Max symbols/link1417 Max axioms/partition8095 Max partitions/axiom2528 Axioms in multiple partitions152

11/14/02Bill MacCartney, Stanford KSL Testing MP “Vanilla” MP vs. common restriction strategies  Use MP with no local strategy  Compare to no strategy, ordered resolution, set-of-support “Smart” MP vs. set-of-support  In SNARK testbed, we use MP + set-of-support to approximate MP with smart local strategy  Within-partition restriction strategies should do better Partition-derived symbol ordering  Use partitioning to induce symbol ordering  Compare partition-derived ordering with set-of-support  What if we combine them?

11/14/02Bill MacCartney, Stanford KSL “Vanilla” MP vs. common strategies

11/14/02Bill MacCartney, Stanford KSL “Smart” MP vs. set-of-support

11/14/02Bill MacCartney, Stanford KSL Partition-derived ordering (PDO)

11/14/02Bill MacCartney, Stanford KSL MP and PDO vs. SoS

11/14/02Bill MacCartney, Stanford KSL Ongoing research Testing on more KBs Partition-derived symbol orderings  Can we beat hand-crafted symbol orderings? Within-partition restriction strategies  Focus reasoning on L i -consequence finding Completeness results  When is partitioning + set-of-support complete? Distributed implementations  Demonstrate integration of heterogeneous reasoners

11/14/02Bill MacCartney, Stanford KSL Conclusions Partitioning can speed up reasoning  Makes large KBs tractable by exploiting implicit structure  Reasoning becomes significantly more focused and efficient  Smarter local strategies should do even better Partition-derived ordering is surprisingly effective  Especially when combined with set-of-support  Automatic alternative to hand-crafted orderings Partitioning supports heterogeneous local reasoners  Efficient special-purpose reasoners can be cleanly integrated  MP ensures global soundness & completeness

11/14/02Bill MacCartney, Stanford KSL Web Papers Amir, E. and McIlraith, S., “Partition-Based Logical Reasoning for First-Order and Propositional Theories,” Artificial Intelligence journal, accepted for publication. McIlraith, S. and Amir, E., “Theorem Proving with Structured Theories,” 17th International Joint Conference on Artificial Intelligence (IJCAI-01), Amir, E., “Efficient Approximation for Triangulation of Minimum Treewidth,” 17th Conference on Uncertainty in Artificial Intelligence (UAI ’01), Amir, E. and McIlraith, S., “Solving Satisfiability using Decomposition and the Most Constrained Subproblem.” Proceedings of SAT 2001, Amir, E. and McIlraith, S., “Partition-Based Logical Reasoning,” 7 th International Conference on Principles of Knowledge Representation and Reasoning (KR ’2000), References

11/14/02Bill MacCartney, Stanford KSL The End

11/14/02Bill MacCartney, Stanford KSL Example (1)ok-pump  on-pump  water (2)man-fill  water (3)man-fill   on-pump (4)  man-fill  on-pump (5)water  ok-boiler  on-boiler  steam (6)  water   steam (7)  on-boiler   steam (8)  ok-boiler   steam (9)steam  coffee  hot-drink (10)steam  tea  hot-drink (11)coffee  tea The espresso machine theory

11/14/02Bill MacCartney, Stanford KSL (1)  ok-pump   on-pump  water (2)  man-fill  water (3)  man-fill   on-pump (4)man-fill  on-pump (5)  water   ok-boiler   on-boiler  steam (6)water   steam (7)on-boiler   steam (8)ok-boiler   steam (9)  steam   coffee  hot-drink (10)  steam   tea  hot-drink (11)coffee  tea Example: partitioning hot-drinktea coffee steam on-boiler ok-boiler water on-pump ok-pumpman-fill Step 1: construct symbol graph

11/14/02Bill MacCartney, Stanford KSL Example: partitioning hot-drinktea coffee steam on-boiler ok-boiler water on-pump ok-pumpman-fill steam water hot-drinktea coffee steam on-boiler ok-boiler water on-pump ok-pumpman-fill water steam Step 2: graph decomposition steam water

11/14/02Bill MacCartney, Stanford KSL Example: partitioning steam water hot-drinktea coffee steamon-boiler ok-boiler water on-pump ok-pumpman-fill water steam (1)  ok-pump   on-pump  water (2)  man-fill  water (3)  man-fill   on-pump (4)man-fill  on-pump (5)  water   ok-boiler   on-boiler  steam (6)water   steam (7)on-boiler   steam (8)ok-boiler   steam (9)  steam   coffee  hot-drink (10)  steam   tea  hot-drink (11)coffee  tea steam water Step 3: generate partition graph

11/14/02Bill MacCartney, Stanford KSL Example: add query to partition graph (1)  ok-pump   on-pump  water (2)  man-fill  water (3)  man-fill   on-pump (4)man-fill  on-pump (5)  water   ok-boiler   on-boiler  steam (6)water   steam (7)on-boiler   steam (8)ok-boiler   steam (9)  steam   coffee  hot-drink (10)  steam   tea  hot-drink (11)coffee  tea steam water Query: If the pump is OK and the boiler is OK and the boiler is on, do we get a hot drink? (12)ok-pump (13)ok-boiler (14)on-boiler (15)  hot-drink

11/14/02Bill MacCartney, Stanford KSL (1)  ok-pump   on-pump  water (2)  man-fill  water (3)  man-fill   on-pump (4)man-fill  on-pump (5)  water   ok-boiler   on-boiler  steam (6)water   steam (7)on-boiler   steam (8)ok-boiler   steam (9)  steam   coffee  hot-drink (10)  steam   tea  hot-drink (11)coffee  tea steam water (12)ok-pump (13)ok-boiler (14)on-boiler (15)  hot-drink Example of MP (16)  on-pump  water (17) man-fill  water (18) water water steam (19)  ok-boiler   on-boiler  steam (20) steam (21)  steam  tea  hot-drink (22)  steam  hot-drink (23) hot-drink Using set-of-support, SNARK took 28 steps to prove this. Using partitioning, SNARK took just 11 steps. (24) 

11/14/02Bill MacCartney, Stanford KSL Automatic partitioning

11/14/02Bill MacCartney, Stanford KSL Queries hd-q1If the pump is OK and the boiler is OK and the boiler is on, do we get a hot drink? cyc-p5If A and B are inside C, can C be inside A? cyc-p7If A and B are part of C and C is at D, where is A? cyc-p1Suppose that A is touching B and B is inside C and C is at D. Is A at D? cyc-v5A has parts B, C, and D. B has parts E, and F. Is F near A? cyc-p3If C is between A and B, and both A and B are inside D, and D is at E, is C at E? cyc-p4If C is between A and B, and both A and B are at D, is C also at D?