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UIUC CS 497: Section EA Lecture #5

UIUC CS 497: Section EA Lecture #5. Reasoning in Artificial Intelligence Professor: Eyal Amir Spring Semester 2004. Last Time. DL entailment using subsumption and Tableau (for satisfiability) Applications: Games and natural language generation Medical informatics. Today.

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UIUC CS 497: Section EA Lecture #5

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  1. UIUC CS 497: Section EALecture #5 Reasoning in Artificial Intelligence Professor: Eyal Amir Spring Semester 2004

  2. Last Time • DL entailment using subsumption and Tableau (for satisfiability) • Applications: • Games and natural language generation • Medical informatics

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

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

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

  11. Reasoning with partitions using MP MP Algorithm • Start with a tree-decomposition partition graph • Identify goal partition • Direct edges toward goal • (fixing outbound link language Li for each partition) • Concurrently, in each partition: • Generate consequences in Li • Pass messages in Li toward goal

  12. Another Example • Message Passing: Espresso 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) Ù open(x)  opened(x) opened(x) Ù fetch(y,x) Ù in(y,x)  has(y) Has(key(closet))  opened(closet) open(closet)  opened(closet) In(broom,closet) fetch(broom,closet) Has(broom) Ù dry(broom)  can_clean(x) can_clean(x)  cleaned(x) Has(y) Ù 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 • We can partition reasoning while not hurting soundness and completeness • How to partition a KB with the best computational benefit • Still maintaining soundness & completeness • Applications: Planning

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

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

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

  21. Automatic Decomposition of a Theory key locked can_open open can_clean cleaned opened fetch dry time broom let_dry drier

  22. Automatic Decomposition of a Theory key locked can_open open can_clean cleaned opened fetch dry time broom let_dry drier

  23. Automatic Decomposition of a Theory broom key locked can_open open can_clean cleaned opened fetch dry time broom let_dry drier

  24. Automatic Decomposition of a Theory broom key locked can_open open can_clean cleaned broom opened fetch dry time broom let_dry drier

  25. Automatic Decomposition of a Theory key  locked  can_open can_open Ù open  opened opened Ù fetch broom key  opened open  opened  broom fetch broomÙ dry  can_clean can_clean  cleaned broomÙ 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-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 • Given a graph G; List L - empty • Add to L a node v with minimum number of neighbors • Make a clique from v’s neighbors • Remove v from G • If G is empty, return L • Go to 2

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

  32. Summary: Characteristics of MP • Reasoning is performed locally in each partition • Specialized reasoning procedures in every partition • Globally sound & complete… provided each local reasoner is sound & complete for Li-consequence finding • Performance is worst-caseexponential within partitions, but linear in tree structure Minimizesbetween-partitiondeduction Focuseswithin-partitiondeduction Supports parallel processing Different reasoners in different partitions

  33. Contents • We can partition reasoning while not hurting soundness and completeness • How to partition a KB with the best computational benefit • Still maintaining soundness & completeness • 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. 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×22w+2k) • Factoring can be done in polynomial time • Goal can be distributed over partitions by adding at most 2 features per partition

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

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