CS498EA Reasoning in AI Lecture

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CS498-EAReasoning in AILecture 3

• Professor Eyal Amir
• Fall Semester 2008

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Today

• What can we say about propositional reasoning
  methods?
• Correctness (Soundness, Completenes)
• Time efficiency theory, practice
• Partitioned reasoning
• Graphs of partitions, tree structures

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Propositional Resolution

• Resolution algorithm (saturation)
• While there are unresolved C1,C2
• Select C1, C2 in KB
• If C1, C2 are resolvable, resolve them into a new
  clause C3
• Add C3 to KB
• If C3 (empty clause),
• we got a contradiction.
• STOP

C1 p1 ? C1 C2 ?p1 ? C2 --------------------
C3 C1 ? C2
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Properties of Resolution

• Theorem Resolution is sound
• Resolving clauses in KB generates valid
  consequences of KB
• Theorem Resolution is refutation complete
• Resolution of KB with ?Q yields the empty clause
  iff KB Q

-
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Properties of Resolution

• Resolution does not always generate Q
• KB a,b, ?a,b, b,c
• Q b ? ?c b,?c
• Theorem Resolution always generates a clause
  that subsumes Q iff KB Q
• Example Resolving KB generates b

-
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Simple Enhancements

• Remove subsumed clauses
• p subsumes p , q
• p , q subsumes p , q, r
• ?p does not subsume p , q
• Contract same literals
• p , p , q becomes p , q
• Unit resolution resolve unit clauses first

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What can we say about Resolution?
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Related to Prop. Resolution

• Clause selection for resolution
• Consequence finding
• Prime implicates/implicants

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Partitioning

• 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

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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

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High-Level Structure inLogic
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
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High-Level Structure inLogic
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
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Structured Reasoning

• Craigs 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

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Structured Reasoning

• Craigs 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

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High-Level Structure 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
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Structured First-Order Reasoning

• Craigs 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
17
High-Level Structure 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
18
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

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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

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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

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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
22
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
23
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
24
Automatic Decomposition of a Theory
key
locked
can_open
open
cleaned
can_clean
opened
fetch
dry
time
broom
let_dry
drier
25
Automatic Decomposition of a Theory
key
locked
can_open
open
cleaned
can_clean
opened
fetch
dry
time
broom
let_dry
drier
26
Automatic Decomposition of a Theory
broom
key
locked
can_open
open
cleaned
can_clean
opened
fetch
dry
time
broom
let_dry
drier
27
Automatic Decomposition of a Theory
broom
key
locked
can_open
open
cleaned
can_clean
broom
opened
fetch
dry
time
broom
let_dry
drier
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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
broom
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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

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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

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Why Tree Decomposition?

• Example BREAK-CYCLES

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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

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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 vs neighbors
• Remove v from G
• If G is empty, return L
• Go to 2

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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 vs neighbors
• Remove v from G
• If G is empty, return L
• Go to 2

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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
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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

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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

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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

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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

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Factored Planning Analysis

• Planner is sound and complete
• Running time for finding plans of width w with m
  partitions of treewidth k is O(mw22w2k)
• Factoring can be done in polynomial time
• Goal can be distributed over partitions by adding
  at most 2 features per partition

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Next Time

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