1. Knowledge Repn. & Reasoning Lec #11: Partitioning & Treewidth UIUC CS 498: Section EA Professor: Eyal Amir Fall Semester 2004

2. Last Time Resolution strategies –Deletion of clauses –Restricting resolution to some pairs –Ordering resolution between clauses Some strategies refutation complete, others only complete for Horn refutation

3. Today 1.We can partition reasoning while not hurting soundness and completeness 2.How to partition a KB with the best computational benefit Still maintaining soundness & completeness 3. Applications du jour: Planning

4. key   locked  can_open can_open  open  opened opened  fetch  broom key  opened open  opened  broom  fetch broom  dry  can_clean can_clean  cleaned broom  let_dry  dry time  let_dry drier  let_dry time  drier High-Level Structure in First- Order Logic

5. key   locked  can_open can_open  open  opened opened  fetch  broom key  opened open  opened  broom  fetch broom  dry  can_clean can_clean  cleaned broom  let_dry  dry time  let_dry drier  let_dry time  drier broom

6. Structured First-Order Reasoning Craig’s interpolation theorem (First-Order Logic): –If A B, then there is a formula C including only symbols from L(A)  L(B) such that A C and C B  clean    

7. Structured First-Order Reasoning Craig’s interpolation theorem (First-Order Logic): –If A B, then there is a formula C including only symbols from L(A)  L(B) such that A C and C B clean    

8. High-Level Structure in First- Order Logic key   locked  can_open can_open  open  opened opened  fetch  broom key  opened open  opened  broom  fetch broom  dry  can_clean can_clean  cleaned broom  let_dry  dry time  let_dry drier  let_dry time  drier broom

9. Structured First-Order Reasoning Craig’s interpolation theorem (First-Order Logic): –If A B, then there is a formula C including only symbols from L(A)  L(B) such that A C and C B clean   broom  

10. High-Level Structure in First- Order Logic key   locked  can_open can_open  open  opened opened  fetch  broom key  opened open  opened  broom  fetch broom  dry  can_clean can_clean  cleaned broom  let_dry  dry time  let_dry drier  let_dry time  drier broom

11. Start with a tree-decomposition partition graph Reasoning with partitions using MP MP Algorithm  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

12. Another Example Message Passing: Espresso machineEspresso machine SAT via partitioning

13. Benefits of Message-Passing Search space is restricted Allows parallel processing Sound and complete Can use different reasoners for each partition Small links imply short proofs Small partitions imply short proofs

14. High-Level Structure in First- Order Logic Has(key(x))   locked(x)  can_open(x) can_open(x)  open(x)  opened(x) opened(x)  fetch(y,x)  in(y,x)  has(y) Has(key(closet))  opened(closet) open(closet)  opened(closet) In(broom,closet) fetch(broom,closet) Has(broom)  dry(broom)  can_clean(x) can_clean(x)  cleaned(x) Has(y)  let_dry(y)  dry(y) Has(time)  let_dry(y) Has(drier)  let_dry(y) Has(time)  Has(drier) Has broom

15. Structured First-Order Reasoning Craig’s interpolation theorem (First-Order Logic): –If A B, then there is a formula C including only symbols from L(A)  L(B) such that A C and C B clean(room)   Has(broom)  

16. Structured First-Order Reasoning Craig’s interpolation theorem (First-Order Logic): –If A B, then there is a formula C including only symbols from L(A)  L(B) such that A C and C B clean(room)   Has(broom)  x Has(x)  Has(broom)  

17. Contents 1.We can partition reasoning while not hurting soundness and completeness 2.How to partition a KB with the best computational benefit Still maintaining soundness & completeness 3. Applications: Planning

18. key   locked  can_open can_open  open  opened opened  fetch  broom key  opened open  opened  broom  fetch broom  dry  can_clean can_clean  cleaned broom  let_dry  dry time  let_dry drier  let_dry time  drier Automatic Decomposition of a Theory

19. key   locked  can_open can_open  open  opened opened  fetch  broom key  opened open  opened  broom  fetch broom  dry  can_clean can_clean  cleaned broom  let_dry  dry time  let_dry drier  let_dry time  drier Automatic Decomposition of a Theory

20. key   locked  can_open can_open  open  opened opened  fetch  broom key  opened open  opened  broom  fetch broom  dry  can_clean can_clean  cleaned broom  let_dry  dry time  let_dry drier  let_dry time  drier Automatic Decomposition of a Theory

21. keylocked can_openopen openedfetch broom dry cleaned let_dry time drier can_clean

22. Automatic Decomposition of a Theory keylocked can_openopen openedfetch broom dry cleaned let_dry time drier can_clean

23. Automatic Decomposition of a Theory keylocked can_openopen openedfetch broom dry cleaned let_dry time drier can_clean broom

24. Automatic Decomposition of a Theory broom keylocked can_openopen openedfetch broom dry cleaned let_dry time drier can_clean broom

25. Automatic Decomposition of a Theory key   locked  can_open can_open  open  opened opened  fetch  broom key  opened open  opened  broom  fetch broom  dry  can_clean can_clean  cleaned broom  let_dry  dry time  let_dry drier  let_dry time  drier broom

26. Automatic Partitioning Begin with a KB in PL or FOL Construct symbol graph –Edges join symbols which appear together in an axiom Find a tree decomposition of low width –Roughly, generalizes balanced vertex cut Partition axioms correspondingly –Each partition has its own vocabulary –Edge labels defined by shared vocabulary

27. Automatic Partitioning Find a tree decomposition of minimum width: –A tree in which each node corresponds to a set of vertices from the original graph –The tree satisfies the running intersection property: if v appears in two nodes in the tree, then v appears in all the nodes on the path connecting them –The width of the tree is the size of its largest node

28. Why Tree Decomposition? Example: BREAK-CYCLESBREAK-CYCLES

29. Automatic Partitioning Treewidth: [Robertson & Seymour ’86], … Approximation Algorithms: –General theories: [A. & McIlraith ’00] –O(Log(OPT))-approximation for general graphs: [A. ’01] –Constant factor approximation for planar graphs: [Seymour & Thomas ’94], [A., Krauthgamer & Rao ’03]

30. Automatic Partitioning: Heuristics Heuristic: min-degree 1.Given a graph G; List L - empty 2.Add to L a node v with minimum number of neighbors 3.Make a clique from v’s neighbors 4.Remove v from G 5.If G is empty, return L 6.Go to 2

31. Automatic Partitioning: Heuristics Heuristic: min-fill 1.Given a graph G; List L - empty 2.Add to L a node v with minimum number of edges missing between neighbors 3.Make a clique from v’s neighbors 4.Remove v from G 5.If G is empty, return L 6.Go to 2

32. Reasoning is performed locally in each partition Specialized reasoning procedures in every 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 Summary: Characteristics of MP Minimizes between-partition deduction Supports parallel processing Different reasoners in different partitions Focuses within-partition deduction

33. Contents 1.We can partition reasoning while not hurting soundness and completeness 2.How to partition a KB with the best computational benefit Still maintaining soundness & completeness 3. Applications: Planning

34. Application: Planning General-purpose planning problem: –Given: Domain features (fluents) Action descriptions: effects, preconditions Initial state Goal condition –Find: Sequence of actions that is guaranteed to achieve the goal starting from the initial state

35. Application: Planning with partitions PartPlan Algorithm Start with a tree- structured partition graph Identify goal partition Direct edges toward goal In each partition –Generate all plans possible with depth d and width k –Pass messages toward goal

36. Planning with partitions PartPlan Algorithm Start with a tree-structured partition graph Identify goal partition Direct edges toward goal In each partition –Generate all plans possible with depth d and width k: “if you give me a block, I can return it to you painted”, “if you give me a block, let me do a few things, and then give me another block, then I can return the two painted and glued together.” –Pass messages toward goal: All preconditions/effects for which there are feasible action sequences

37. Factored Planning: Analysis Planner is sound and complete Running time for finding plans of width w with m partitions of treewidth k is O(m  w  2 2w+2k ) Factoring can be done in polynomial time Goal can be distributed over partitions by adding at most 2 features per partition

38. Next Time Probabilistic Graphical Models: –Directed models: Bayesian Networks –Undirected models: Markov Fields Requires prior knowledge of: –Treewidth and graph algorithms –Probability theory