1. Knowledge and reasoning – second part
Knowledge representation
Logic and representation
Propositional (Boolean) logic
Normal forms
Inference in propositional logic
Wumpus world example
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2. Knowledge-Based Agent
Agent that uses prior or acquired knowledge to achieve its goals
Can make more efficient decisions
Can make informed decisions
Knowledge Base (KB): contains a set of representations of facts about the Agent’s environment
Each representation is called a sentence
Use some knowledge representation language, to TELL it what to know e.g., (temperature 72F)
ASK agent to query what to do
Agent can use inference to deduce new facts from TELLed facts
Knowledge Base
Inference engine
Domain independent algorithms
Domain specific content
TELL
ASK
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3. Generic knowledge-based agent
TELL KB what was perceived Uses a KRL to insert new sentences, representations of facts, into KB
ASK KB what to do. Uses logical reasoning to examine actions and select best.
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4. Wumpus world example
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5. Wumpus world characterization
Deterministic?
Accessible?
Static?
Discrete?
Episodic?
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6. Wumpus world characterization
Deterministic? Yes – outcome exactly specified.
Accessible? No – only local perception.
Static? Yes – Wumpus and pits do not move.
Discrete? Yes
Episodic? (Yes) – because static.
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7. Exploring a Wumpus world
A= Agent
B= Breeze
S= Smell
P= Pit
W= Wumpus
OK = Safe
V = Visited
G = Glitter
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8. Exploring a Wumpus world
A= Agent
B= Breeze
S= Smell
P= Pit
W= Wumpus
OK = Safe
V = Visited
G = Glitter
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9. Exploring a Wumpus world
A= Agent
B= Breeze
S= Smell
P= Pit
W= Wumpus
OK = Safe
V = Visited
G = Glitter
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10. Exploring a Wumpus world
A= Agent
B= Breeze
S= Smell
P= Pit
W= Wumpus
OK = Safe
V = Visited
G = Glitter
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11. Exploring a Wumpus world
A= Agent
B= Breeze
S= Smell
P= Pit
W= Wumpus
OK = Safe
V = Visited
G = Glitter
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12. Exploring a Wumpus world
A= Agent
B= Breeze
S= Smell
P= Pit
W= Wumpus
OK = Safe
V = Visited
G = Glitter
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13. Exploring a Wumpus world
A= Agent
B= Breeze
S= Smell
P= Pit
W= Wumpus
OK = Safe
V = Visited
G = Glitter
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14. Exploring a Wumpus world
A= Agent
B= Breeze
S= Smell
P= Pit
W= Wumpus
OK = Safe
V = Visited
G = Glitter
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15. Other tight spots
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16. Another example solution
No perception  1,2 and 2,1 OK
Move to 2,1
B in 2,1  2,2 or 3,1 P?
1,1 V  no P in 1,1
Move to 1,2 (only option)
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17. Example solution S and No S when in 2,1  1,3 or 1,2 has W
1,2 OK  1,3 W
No B in 1,2  2,2 OK & 3,1 P
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18. Logic in general
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19. Types of logic
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20. Physical Symbol System World
The Semantic Wall
Physical Symbol System World
+BLOCKA+
+BLOCKB+
+BLOCKC+
P1:(IS_ON +BLOCKA+ +BLOCKB+)
P2:((IS_RED +BLOCKA+)
Syntax: says what is allowed on the LHS
Semantics: says how what is on the LHS relates to what is on the RHS
Inference: says how you can manipulate (formally, i.e., with no reference to the RHS) the symbols. [remember PSSH]
Want to be able to trust the results: want whatever the inference procedure does to “respect” what’s true or what follows in the world. So this is where we’re headed; good to keep in mind as we go through all the definitions now to follow. There is a method in this madness...
algebra example (put on board): but don’t use “entails” instead convey the idea
> n m: is this true or false? don’t know
if n=3, m=5, and > has its usual meaning, then (>n m) is false
(> n m) and (> m p) entail (n p)
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21. Truth depends on Interpretation
Representation 1 World A B
ON(A,B) T
ON(A,B) F
ON(A,B) F A
ON(A,B) T B
block pictures
(on A B) and on(B C) entails (on A C)
(again don’t say “entails)
for this as well as for number example jus t did, need transitivity relationships
spend time on why it matters: if build a KB want to be able to verify its answers
introduce notion of “refer” “mean” idea of mapping into world; for this slide handwave about meaning of “on(a,b)”
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22. Entailment is different than inference
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23. Logic as a representation of the World
Facts
World
Fact
follows
Refers to
(Semantics)
Representation: Sentences
Sentence
entails
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24. Models
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25. Inference
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26. Basic symbols Expressions only evaluate to either “true” or “false.”
P “P is true”
¬P “P is false” negation
P V Q “either P is true or Q is true or both” disjunction
P ^ Q “both P and Q are true” conjunction
P => Q “if P is true, the Q is true” implication
P  Q “P and Q are either both true or both false” equivalence
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27. Propositional logic: syntax
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28. Propositional logic: semantics

29. Truth tables Truth value: whether a statement is true or false.
Truth table: complete list of truth values for a statement given all possible values of the individual atomic expressions.
Example:
P Q P V Q
T T T
T F T
F T T
F F F
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30. Truth tables for basic connectives
P Q ¬P ¬Q P V Q P ^ Q P=>Q PQ
T T F F T T T T
T F F T T F F F
F T T F T F T F
F F T T F F T T
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31. Propositional logic: basic manipulation rules
¬(¬A) = A Double negation
¬(A ^ B) = (¬A) V (¬B) Negated “and”
¬(A V B) = (¬A) ^ (¬B) Negated “or”
A ^ (B V C) = (A ^ B) V (A ^ C) Distributivity of ^ on V
A => B = (¬A) V B by definition
¬(A => B) = A ^ (¬B) using negated or
A  B = (A => B) ^ (B => A) by definition
¬(A  B) = (A ^ (¬B))V(B ^ (¬A)) using negated and & or …
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32. Propositional inference: enumeration method
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33. Enumeration: Solution
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34. Propositional inference: normal forms
“product of sums of
simple variables or
negated simple variables”
“sum of products of
simple variables or
negated simple variables”
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35. Deriving expressions from functions
Given a boolean function in truth table form, find a propositional logic expression for it that uses only V, ^ and ¬.
Idea: We can easily do it by disjoining the “T” rows of the truth table.
Example: XOR function
P Q RESULT
T T F
T F T P ^ (¬Q)
F T T (¬P) ^ Q
F F F
RESULT = (P ^ (¬Q)) V ((¬P) ^ Q)
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36. A more formal approach
To construct a logical expression in disjunctive normal form from a truth table:
Build a “minterm” for each row of the table, where:
- For each variable whose value is T in that row, include
the variable in the minterm
- For each variable whose value is F in that row, include
the negation of the variable in the minterm
- Link variables in minterm by conjunctions
The expression consists of the disjunction of all minterms.
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37. Example: adder with carry
Takes 3 variables in: x, y and ci (carry-in); yields 2 results: sum (s) and carry-out (co). To get you used to other notations, here we assume T = 1, F = 0, V = OR, ^ = AND, ¬ = NOT.
co is:
s is:
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38. Tautologies
Logical expressions that are always true. Can be simplified out.
Examples: T
T V A
A V (¬A)
¬(A ^ (¬A))
A  A
((P V Q)  P) V (¬P ^ Q)
(P  Q) => (P => Q)
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39. Validity and satisfiability
Theorem
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40. Proof methods
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41. Inference Rules
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42. Inference Rules
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43. li  …  li-1  li+1  …  lk  m1  …  mj-1  mj+1 ...  mn
Resolution
Conjunctive Normal Form (CNF)
conjunction of disjunctions of literals clauses
E.g., (A  B)  (B  C  D)
Resolution inference rule (for CNF):
li …  lk, m1  …  mn
li  …  li-1  li+1  …  lk  m1  …  mj-1  mj+1 ...  mn
where li and mj are complementary literals.
E.g., P1,3  P2,2, P2,2
P1,3
Resolution is sound and complete for propositional logic

44. Conversion to CNF B1,1  (P1,2  P2,1)
1) Eliminate , replacing α  β with (α  β)(β  α).
(B1,1  (P1,2  P2,1))  ((P1,2  P2,1)  B1,1)
2) Eliminate , replacing α  β with α β.
(B1,1  P1,2  P2,1)  ((P1,2  P2,1)  B1,1)
Move  inwards using de Morgan's rules and double-negation:
(B1,1  P1,2  P2,1)  ((P1,2  P2,1)  B1,1)
4) Apply distributivity law ( over ) and flatten:
(B1,1  P1,2  P2,1)  (P1,2  B1,1)  (P2,1  B1,1)

45. Resolution algorithm
Proof by contradiction, i.e., show KBα unsatisfiable

46. Resolution example
KB = (B1,1  (P1,2 P2,1))  B1, α = P1,2

47. Forward and backward chaining
Horn Form (restricted)
KB = conjunction of Horn clauses
Horn clause =
proposition symbol; or
(conjunction of symbols)  symbol
E.g., C  (B  A)  (C  D  B)
Modus Ponens (for Horn Form): complete for Horn KBs
α1, … ,αn α1  …  αn  β β
Can be used with forward chaining or backward chaining.
These algorithms are very natural and run in linear time

48. Idea: fire any rule whose premises are satisfied in the KB,
Forward chaining
Idea: fire any rule whose premises are satisfied in the KB,
add its conclusion to the KB, until query is found

49. Forward chaining algorithm
Forward chaining is sound and complete for Horn KB

50. Forward chaining example

51. Forward chaining example

52. Forward chaining example

53. Forward chaining example

54. Forward chaining example

55. Forward chaining example

56. Forward chaining example

57. Forward chaining example

58. FC derives every atomic sentence that is entailed by KB
Proof of completeness
FC derives every atomic sentence that is entailed by KB
FC reaches a fixed point where no new atomic sentences are derived
Consider the final state as a model m, assigning true/false to symbols
Every clause in the original KB is true in m
a1  …  ak  b
Hence m is a model of KB
If KB╞ q, q is true in every model of KB, including m

59. to prove q by BC, Backward chaining
check if q is known already, or
prove by BC all premises of some rule concluding q
Avoid loops: check if new subgoal is already on the goal stack
Avoid repeated work: check if new subgoal
has already been proved true, or
has already failed

60. Backward chaining example

61. Backward chaining example

62. Backward chaining example

63. Backward chaining example

64. Backward chaining example

65. Backward chaining example

66. Backward chaining example

67. Backward chaining example

68. Backward chaining example

69. Backward chaining example

70. Forward vs. backward chaining
FC is data-driven, automatic, unconscious processing,
e.g., object recognition, routine decisions
May do lots of work that is irrelevant to the goal
BC is goal-driven, appropriate for problem-solving,
e.g., Where are my keys? How do I get into a PhD program?
Complexity of BC can be much less than linear in size of KB

71. Efficient propositional inference
Two families of efficient algorithms for propositional inference:
Complete backtracking search algorithms
DPLL algorithm (Davis, Putnam, Logemann, Loveland)
Incomplete local search algorithms
WalkSAT algorithm

72. The DPLL algorithm
Determine if an propositional logic sentence (in CNF) is satisfiable.
Improvements over truth table enumeration:
Early termination
A clause is true if any literal is true.
A sentence is false if any clause is false.
Pure symbol heuristic
Pure symbol: always appears with the same "sign" in all clauses.
e.g., In the three clauses (A  B), (B  C), (C  A), A and B are pure, C is impure.
Make a pure symbol literal true.
Unit clause heuristic
Unit clause: only one literal in the clause
The only literal in a unit clause must be true.

73. The DPLL algorithm

74. The WalkSAT algorithm
Incomplete, local search algorithm
Evaluation function: The min-conflict heuristic of minimizing the number of unsatisfied clauses
Balance between greediness and randomness

75. The WalkSAT algorithm

76. Inference-based agents in the wumpus world
A wumpus-world agent using propositional logic:
P1,1
W1,1
Bx,y  (Px,y+1  Px,y-1  Px+1,y  Px-1,y)
Sx,y  (Wx,y+1  Wx,y-1  Wx+1,y  Wx-1,y)
W1,1  W1,2  …  W4,4
W1,1  W1,2
W1,1  W1,3 …
 64 distinct proposition symbols, 155 sentences

78. Facts: Percepts inject (TELL) facts into the KB
Wumpus world: example
Facts: Percepts inject (TELL) facts into the KB
[stench at 1,1 and 2,1]  S1,1 ; S2,1
Rules: if square has no stench then neither the square or adjacent square contain the wumpus
R1: !S1,1 !W1,1  !W1,2  !W2,1
R2: !S2,1 !W1,1 !W2,1  !W2,2  !W3,1 …
Inference:
KB contains !S1,1 then using Modus Ponens we infer !W1,1  !W1,2  !W2,1
Using And-Elimination we get: !W1,1 !W1,2 !W2,1
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79. Limitations of Propositional Logic
1. It is too weak, i.e., has very limited expressiveness:
Each rule has to be represented for each situation: e.g., “don’t go forward if the wumpus is in front of you” takes 64 rules
2. It cannot keep track of changes:
If one needs to track changes, e.g., where the agent has been before then we need a timed-version of each rule. To track 100 steps we’ll then need 6400 rules for the previous example.
Its hard to write and maintain such a huge rule-base
Inference becomes intractable
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80. Summary
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