1. Artificial Intelligence: Logic agents

2. AI in the News
April 6, 2005: GIDEON - Global Infectious Disease and Epidemiology Network. Media review by Vincent J. Felitti. JAMA, the Journal of the American Medical Association ( Vol. 293, No. 13, pages ; subscription req'd.). "GIDEON: The Global Infectious Disease and Epidemiology Network is a superbly designed expert system created to help physicians diagnose any infectious disease (337 recognized) in any country of the world (224 included). The program was created and has been progressively refined over more than a decade by a talented group of Americans, Canadians, and Israelis
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December 13, 2004: WebMed dispenses advice to students. By Robyn Shelton. Orlando Sentinel. "The site -- 24/7 WebMed -- takes students through questions, judges the severity of their symptoms and offers guidance for what to do next. ... 'It's decision-support systems, or artificial intelligence in a way,' said Dr. Scott Gettings, DSHI medical director. 'The system learns about you as you flow through and answer questions and determines how ill you are.' It makes no attempt to go further and diagnose the patient's illness -- but gauges the seriousness of the symptoms. 'This is not intended to take the place of human interaction, but to augment it,' said Dr. Michael Deichen, associate director of clinical services at the UCF Student Health Center. 'It really just helps the students know with what urgency they should be evaluated.'"
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3. “Thinking Rationally”
Computational models of human “thought” processes
Computational models of human behavior
Computational systems that “think” rationally
Computational systems that behave rationally
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4. Logical Agents
Reflex agents find their way from Arad to Bucharest by dumb luck
Chess program calculates legal moves of its king, but doesn’t know that a piece cannot be on 2 different squares at the same time
Logic (Knowledge-Based) agents combine general knowledge with current percepts to infer hidden aspects of current state prior to selecting actions
Crucial in partially observable environments
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5. Outline Knowledge-based agents Wumpus world Logic in general
Propositional and first-order logic
Inference, validity, equivalence and satifiability
Reasoning patterns
Resolution
Forward/backward chaining
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6. Knowledge Base
Knowledge Base : set of sentences represented in a knowledge representation language and represents assertions about the world.
Inference rule: when one ASKs questions of the KB, the answer should follow from what has been TELLed to the KB previously.
tell
ask
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7. Generic KB-Based Agent
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8. Abilities KB agent Agent must be able to:
Represent states and actions,
Incorporate new percepts
Update internal representation of the world
Deduce hidden properties of the world
Deduce appropriate actions
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9. A Typical Wumpus World
Wumpus
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10. Wumpus World PEAS Description
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11. Wumpus World Characterization
Observable?
Deterministic?
Episodic?
Static?
Discrete?
Single-agent?
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12. Wumpus World Characterization
Observable? No, only local perception
Deterministic? Yes, outcome exactly specified
Episodic? No, sequential at the level of actions
Static? Yes, Wumpus and pits do not move
Discrete? Yes
Single-agent? Yes, Wumpus is essentially a natural feature.
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13. Exploring the Wumpus World
[1,1] The KB initially contains the rules of the environment. The first percept is [none, none,none,none,none], move to safe cell e.g. 2,1
[2,1] breeze which indicates that there is a pit in [2,2] or [3,1], return to [1,1] to try next safe cell
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14. Exploring the Wumpus World
[1,2] Stench in cell which means that wumpus is in [1,3] or [2,2]
YET … not in [1,1]
YET … not in [2,2] or stench would have been detected in [2,1]
THUS … wumpus is in [1,3]
THUS [2,2] is safe because of lack of breeze in [1,2]
THUS pit in [1,3]
move to next safe cell [2,2]
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15. Exploring the Wumpus World
[2,2] move to [2,3]
[2,3] detect glitter , smell, breeze
THUS pick up gold
THUS pit in [3,3] or [2,4]
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16. What is a logic? A formal language
Syntax – what expressions are legal (well-formed)
Semantics – what legal expressions mean
in logic the truth of each sentence with respect to each possible world.
E.g the language of arithmetic
X+2 >= y is a sentence, x2+y is not a sentence
X+2 >= y is true in a world where x=7 and y =1
X+2 >= y is false in a world where x=0 and y =6
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17. Entailment One thing follows from another KB |= 
KB entails sentence  if and only if  is true in worlds where KB is true.
E.g. x+y=4 entails 4=x+y
Entailment is a relationship between sentences that is based on semantics.
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18. Models
Logicians typically think in terms of models, which are formally structured worlds with respect to which truth can be evaluated.
m is a model of a sentence  if  is true in m
M() is the set of all models of 
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19. Wumpus world model
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20. Wumpus world model
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21. Wumpus world model
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22. Wumpus world model
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23. Wumpus world model
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24. Wumpus world model
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25. Logical inference
The notion of entailment can be used for logic inference.
Model checking (see wumpus example): enumerate all possible models and check whether  is true.
If an algorithm only derives entailed sentences it is called sound or truth preserving.
Otherwise it just makes things up.
i is sound if whenever KB |-i  it is also true that KB|= 
Completeness : the algorithm can derive any sentence that is entailed.
i is complete if whenever KB |=  it is also true that KB|-i 
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26. Schematic perspective
If KB is true in the real world, then any sentence  derived
From KB by a sound inference procedure is also true in the
real world.
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27. Inference
KB |-i a
Soundness: Inference procedure i is sound if whenever KB |-i a, it is also true that KB |= a
Completeness: Inference procedure i is complete if whenever KB |= a, it is also true that KB |-i a
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28. Validity and Inference
((P V H) ^ ØH) => P P H P V H (P H) Ø V ^ H
((P V H) ^
ØH) => P T T T F T T F T T T F T T F T F F F F T
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29. Rules of Inference a |- b a b Valid Rules of Inference: Modus Ponens
And-Elimination
And-Introduction
Or-Introduction
Double Negation
Unit Resolution
Resolution
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30. Examples in Wumpus World
Modus Ponens: a => b, a |- b (WumpusAhead ^ WumpusAlive) => Shoot, (WumpusAhead ^ WumpusAlive) |- Shoot
And-Elimination: a ^ b |- a (WumpusAhead ^ WumpusAlive) |- WumpusAlive
Resolution: a V b, Øb V g |- a V g (WumpusDead V WumpusAhead), (ØWumpusAhead V Shoot) ` (WumpusDead V Shoot)
a => b a b
a ^ b a
a V b Øb V g a V g
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31. Proof Using Rules of Inference
Prove A => B, (A ^ B) => C, Therefore A => C
A => B |- ØA V B
A ^ B => C |- Ø(A ^ B) V C |- ØA V ØB V C
So ØA V B resolves with ØA V ØB V C deriving
ØA V C
This is equivalent to A => C
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32. Rules of Inference (continued)
And-Introduction a1, a2, …, an a1 ^ a2 ^ … ^ an
Or-Introduction ai a1 V a2 V …ai … V an
Double Negation ØØa a
Unit Resolution (special case of resolution) a V b Alternatively: Øa => b Øb Øb a a
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33. Wumpus World KB Proposition Symbols for each i,j:
Let Pi,j be true if there is a pit in square i,j
Let Bi,j be true if there is a breeze in square i,j
Sentences in KB
“There is no pit in square 1,1” R1: ØP1,1
“A square is breezy iff pit in a neighboring square” R2: B1,1  (P1,2 V P2,1) R3: B1,2  (P1,1 V P1,3 V P2,2)
“Square 1,1 has no breeze”, “Square 1,2 has a breeze” R4: ØB1,1 R5: B1,2
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34. Inference in Wumpus World
Apply biconditional elimination to R2: R6: (B1,1 => (P1,2 V P2,1)) ^ ((P1,2 V P2,1) => B1,1)
Apply AE to R6: R7: ((P1,2 V P2,1) => B1,1)
Contrapositive of R7: R8: (ØB1,1 => Ø(P1,2 V P2,1))
Modus Ponens with R8 and R4 (ØB1,1): R9: Ø(P1,2 V P2,1)
de Morgan: R10: ØP1,2 ^ ØP2,1
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35. Searching for Proofs
Finding proofs is exactly like finding solutions to search problems.
Can search forward (forward chaining) to derive goal or search backward (backward chaining) from the goal.
Searching for proofs is not more efficient than enumerating models, but in many practical cases, it’s more efficient because we can ignore irrelevant propositions
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36. Full Resolution Rule Revisited
Start with Unit Resolution Inference Rule:
Full Resolution Rule is a generalization of this rule:
For clauses of length two:
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37. Resolution Applied to Wumpus World
At some point we determine the absence of a pit in square 2,2: R13: ØP2,2
Biconditional elimination applied to R3 followed by modus ponens with R5: R15: P1,1 V P1,3 V P2,2
Resolve R15 and R13: R16: P1,1 V P1,3
Resolve R16 and R1: R17: P1,3
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38. Resolution: Complete Inference Procedure
Any complete search algorithm, applying only the resolution rule, can derive any conclusion entailed by any knowledge base in propositional logic.
Refutation completeness: Resolution can always be used to either confirm or refute a sentence, but it cannot be used to enumerate true sentences.
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39. Conjunctive Normal Form
Conjunctive Normal Form is a disjunction of literals.
Example: (A V B V ØC) ^ (B V D) ^ (Ø A) ^ (BVC)
literals
clause
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40. CNF Example Example: (A V B)  (C => D) Eliminate  Eliminate =>
((A V B) => (C => D)) ^ ((C => D) => (A V B)
Eliminate =>
(Ø (A V B) V (ØC V D)) ^ (Ø(ØC V D) V (A V B) )
Drive in negations ((ØA ^ ØB) V (ØC V D)) ^ ((C ^ ØD) V (A V B))
Distribute (ØA V ØC V D) ^ (ØB V ØC V D) ^ (C V A V B) ^ (ØD V A V B)
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41. Resolution Algorithm
To show KB |= a, we show (KB ^ Øa) is unsatisfiable.
This is a proof by contradiction.
First convert (KB ^ Øa) into CNF.
Then apply resolution rule to resulting clauses.
The process continues until:
there are no new clauses that can be added (KB does not entail a)
two clauses resolve to yield empty clause (KB entails a)
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42. Simple Inference in Wumpus World
KB = R2 ^ R4 = (B1,1  (P1,2 V P2,1)) ^ ØB1,1
Prove ØP1,2 by adding the negation P1,2
Convert KB ^ P1,2 to CNF
PL-RESOLUTION algorithm
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43. Horn Clauses Real World KB’s are often a conjunction of Horn clauses
proposition symbol; or
(conjunction of symbols) => symbol
Examples: C (B => A) (C ^ D => B)
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44. Forward Chaining Fire any rule whose premises are satisfied in the KB.
Add its conclusion to the KB until query is found.
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45. Forward Chaining Example
P => Q
L ^ M => P
B ^ L => M
A ^ P => L
A ^ B => L A B
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46. Forward Chaining Example
P => Q
L ^ M => P
B ^ L => M
A ^ P => L
A ^ B => L A B
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47. Forward Chaining Example
P => Q
L ^ M => P
B ^ L => M
A ^ P => L
A ^ B => L A B
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48. Forward Chaining Example
P => Q
L ^ M => P
B ^ L => M
A ^ P => L
A ^ B => L A B
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49. Forward Chaining Example
P => Q
L ^ M => P
B ^ L => M
A ^ P => L
A ^ B => L A B
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50. Forward Chaining Example
P => Q
L ^ M => P
B ^ L => M
A ^ P => L
A ^ B => L A B
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51. Forward Chaining Example
P => Q
L ^ M => P
B ^ L => M
A ^ P => L
A ^ B => L A B
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52. Forward Chaining Example
P => Q
L ^ M => P
B ^ L => M
A ^ P => L
A ^ B => L A B
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53. Backward Chaining
Motivation: Need goal-directed reasoning in order to keep from getting overwhelmed with irrelevant consequences
Main idea:
Work backwards from query q
To prove q:
Check if q is known already
Prove by backward chaining all premises of some rule concluding q
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54. Backward Chaining Example
P => Q
L ^ M => P
B ^ L => M
A ^ P => L
A ^ B => L A B
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55. Backward Chaining Example
P => Q
L ^ M => P
B ^ L => M
A ^ P => L
A ^ B => L A B
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56. Backward Chaining Example
P => Q
L ^ M => P
B ^ L => M
A ^ P => L
A ^ B => L A B
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57. Backward Chaining Example
P => Q
L ^ M => P
B ^ L => M
A ^ P => L
A ^ B => L A B
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58. Backward Chaining Example
P => Q
L ^ M => P
B ^ L => M
A ^ P => L
A ^ B => L A B
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59. Backward Chaining Example
P => Q
L ^ M => P
B ^ L => M
A ^ P => L
A ^ B => L A B
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60. Backward Chaining Example
P => Q
L ^ M => P
B ^ L => M
A ^ P => L
A ^ B => L A B
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61. Backward Chaining Example
P => Q
L ^ M => P
B ^ L => M
A ^ P => L
A ^ B => L A B
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62. Backward Chaining Example
P => Q
L ^ M => P
B ^ L => M
A ^ P => L
A ^ B => L A B
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63. Backward Chaining Example
P => Q
L ^ M => P
B ^ L => M
A ^ P => L
A ^ B => L A B
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64. Backward Chaining Example
P => Q
L ^ M => P
B ^ L => M
A ^ P => L
A ^ B => L A B
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65. Forward Chaining vs. Backward Chaining
FC is data-driven—it may do lots of work irrelevant to the goal
BC is goal-driven—appropriate for problem-solving
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