1. Artificial Intelligence
Lecture 9 – Knowledge-Based Agents and Logic
Dr. Muhammad Adnan Hashmi
26 December 2018

2. Outline Knowledge-based agents Wumpus world
Logic in general - models and entailment
Propositional (Boolean) logic
Equivalence, validity, satisfiability
Inference rules and theorem proving
Forward chaining
Backward chaining
Resolution.
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3. Knowledge Base Knowledge base: Set of sentences in a formal language
Declarative approach for building an agent
TELL it what it needs to know
Then it can ASK itself what to do - answers should follow from the KB
Agents can be viewed at the knowledge level
What they know, regardless of implementation
Or at the implementation level
Data structures in KB and algorithms that manipulate them.
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4. A Knowledge-based Agent
The agent must be able to:
Represent states, actions, etc.
Incorporate new percepts
Update internal representations of the world
Deduce hidden properties of the world
Deduce appropriate actions.
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5. Wumpus World
Performance measure: Gold +1000, Death -1000, -1 per step, -10 for using the arrow
Environment:
Squares adjacent to wumpus are smelly
Squares adjacent to pit are breezy
Glitter iff gold is in the same square
Shooting kills wumpus if you are facing it
Shooting uses up the only arrow
Grabbing picks up gold if in same square
Releasing drops the gold in same square
Actuators: Left turn, Right turn, Forward, Grab, Release, Shoot
Sensors: Breeze, Glitter, Smell.
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6. Wumpus World Characterization
Observable? No - only local perception
Deterministic? Yes -outcomes exactly specified
Episodic? No - sequential at the level of actions
Static? Yes -Wumpus and Pits do not move
Discrete? Yes
Single-agent? Yes
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7. Exploring the Wumpus World
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8. Exploring the Wumpus World
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9. Exploring the Wumpus World
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10. Exploring the Wumpus World
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11. Exploring the Wumpus World
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12. Exploring the Wumpus World
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13. Exploring the Wumpus World
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14. Exploring the Wumpus World
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15. Logic In General
Logics are formal languages for representing information, such that conclusions can be drawn
Syntax defines the sentences in the language
Semantics define the “meaning" of sentences, i.e., define truth of a sentence in a world
E.g., the language of arithmetic
x + 2 ≥ y is a sentence; x2 + y > is not a sentence
x + 2 ≥ y is true iff the number x + 2 is no less than the number y
x + 2 ≥ y is true in a world where x=7; y =1
x + 2 ≥ y is false in a world where x=0; y =6
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16. Entailment Entailment means that one thing follows from another
Knowledge base KB entails sentence B iff B is true in all worlds where KB is true
E.g., the KB containing “the Giants won" and “the Reds won” entails “Either the Giants won or the Reds won“
E.g., x + y =4 entails 4=x + y
Entailment is a relationship between sentences (i.e., syntax) that is based on semantics.
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17. Models
Logicians typically think in terms of models, which are formally structured worlds with respect to which truth can be evaluated
We say m is a model of a sentence B if B is true in m
M(B) is the set of all models of B
Then KB |= B if and only if M(KB) M(B)
E.g. KB = Giants won and Reds won
B = Giants won
M(B) could be also true for worlds that are different than the worlds of KB
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18. Models
M(B)
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19. Entailment in the Wumpus World
Situation after detecting nothing in [1,1], moving right, breeze in [2,1]
Consider possible models for ?s, assuming only pits
3 Boolean choices, i.e., 8 possible models
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20. Wumpus Models
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21. Wumpus Models
KB = Wumpus World Rules + Observations
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22. Wumpus Models KB = Wumpus World Rules + Observations
Alpha_1 = “(1,2) is safe, KB |= Alpha_1
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23. Wumpus Models
KB = Wumpus World Rules + Observations
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24. Wumpus Models KB = Wumpus World Rules + Observations
Alpha_2 = “(2,2) is safe, KB |= Alpha_2 is false
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25. Inference
First Order Logic (FOL): Allows complete and sound inference procedures
Will answer any question whose answer follows from what is known by the KB.
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26. Propositional Logic Order of Precedence If S1 then S2 If and Only If
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If and Only If 26

27. Propositional Logic: Semantics
If S1 is true, then I am claiming that S2 is true
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28. Truth Table for Connectives
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29. Wumpus World Sentences
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30. Wumpus World Sentences
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31. Wumpus World Truth Table
128 possible rows, only 3 make the KB true
P1,2 is false for all 3; hence, no pit in P1,2
Evaluate the entailed sentences in these rows
P2,2 is both false and true; hence, no conclusion can be drawn about the pit being in P2,2
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32. Inference by Enumeration
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33. Logical Equivalence
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34. Validity and Satisfiability
If KB is true, alpha is always true. Hence, I can say that alpha follows from KB
Suppose that KB=true. Then, this will become unsatisfiable only when alpha is true. Hence, I can say that alpha follows from KB
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35. Proof (A Sequence of Application of Inference Rules)
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36. Two Famous Inference Rules
Modus Ponens
And Rule
Given that P implies Q, and I know that P is true, then I can infer Q
Given that P AND Q is true, I can infer that P is true, and I can also infer that Q is true.
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37. Resolution
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38. Conversion to CNF
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39. Resolution Algorithm
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40. Example
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41. Resolution Example Add P1,2 P2,1 already added Del P2,1 Del P1,2
This returns an empty clause. So, alpha is true.
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42. Forward and Backward Chaining
If this is true, and we know that the set of alphas is true, then we can infer beta.
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43. A and B are facts (leaves)
Forward Chaining
What you want to check whether it can be entailed (or not)
A and B are facts (leaves)
of the AND-OR graph)
AND-OR Graph – Multiple links with Arcs indicate Conjunction, and without Arcs indicate Disjunction
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44. Forward Chaining Algorithm
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45. Forward Chaining Example
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46. Forward Chaining Example
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47. Forward Chaining Example
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48. Forward Chaining Example
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49. Forward Chaining Example
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50. Forward Chaining Example
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51. Forward Chaining Example
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52. Forward Chaining Example
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53. Soundness and Completeness
It is sound: every inference is an application of Modus Ponens
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54. Backward Chaining
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55. Backward Chaining Example
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56. Backward Chaining Example
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57. Backward Chaining Example
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58. Backward Chaining Example
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59. Backward Chaining Example
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60. Backward Chaining Example
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61. Backward Chaining Example
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62. Backward Chaining Example
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63. Backward Chaining Example
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64. Backward Chaining Example
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65. Backward Chaining Example
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66. FC vs. BC
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67. Questions
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