1. Inference and Resolution for Problem Solving
Comp3710 Artificial Intelligence
Computing Science
Thompson Rivers University

2. Inference and Resolution
Course Outline
Part I – Introduction to Artificial Intelligence
Part II – Classical Artificial Intelligence, and Searching
Knowledge Representation
Searching
Search Methodologies
Advanced Search
Genetic Algorithms (relatively new study area)
Knowledge Represenation and Automated Reasoning
Propositinoal and Predicate Logic
Inference and Resolution for Problem Solving
Rules and Expert Systems
Part III – Machine Learning
Part IV – Advanced Topics
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Inference and Resolution

3. Inference and Resolution
Chapter Objectives
How to convert BNF to CNF
Use of the resolution rule for some problems
How to construct Horn clauses from given rules and facts
Use of forward chaining
Use of backward chaining
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4. Inference and Resolution
Chapter Outline
Introduction
Resolution in propositional logic
An application of resolution – graph coloring
Inference methods – forward and backward chaining
Horn clauses
Forward chaining
Backward chaining
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5. Inference and Resolution
Topics
1. Introduction
Problem solving using propositional calculus:
Based on rules, knowledge, observations, and facts,
Decide if a given query is valid.
Example
Rules and facts (or observation results)
If it is raining, Jenny does not go camping.
It is raining.
It is a hot day.
It is afternoon.
Question:
Does Jenny go fishing?
Can you convert them to a logical form, i.e., wff (or sentence)?
(Raining  ~Camping)  Raining  HotDay  Afternoon  ~Fishing ???
We need to see the above sentence is contradictory to answer “Does Jenny go fishing?”.
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6. 2. Resolution in Propositional Logic
CNF(Conjunctive Normal Form)
How to convert propositions to CNFs
Clauses instead of CNFs for convenience
The Resolution rule
Resolution refutation
Proof by refutation
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7. Conjunctive Normal Forms (CNF)2.
Conjunctive Normal Forms (CNF)
BNF (Backus-Naur Form) is not convenient to use.
A wff is in Conjunctive Normal Form (CNF) if it is a conjunction of disjunctions:
conjunction of disjunctions of literals
clauses
E.g., (A  B)  (B  C  D)
E.g., A  (B  C)  (¬A  ¬B  ¬C  D)
Every sentence of propositional logic is logically equivalent to a conjunction of disjunctions of literals.
[Q] Really? How to convert?
[Q] Why do we need to use CNF?
Similarly, a wff is in Disjunctive Normal Form (DNF) if it is a disjunction of conjunctions.
E.g., A  (B  C)  (¬A  ¬B  ¬C  D)
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8. Inference and Resolution
Converting to CNF
Any wff can be converted to CNF by using the following equivalences:
A ↔ B  (A → B)  (B → A) biconditional elimination
A → B  (¬A  B) implication elimination
¬(A  B)  (¬A  ¬B) de Morgan’s law
¬(A  B)  (¬A)  (¬B) de Morgan’s law
¬ ¬ A  A double-negation elimination
A  (B  C)  (A  B)  (A  C) distributivity
Importantly, this can be converted into an algorithm – this will be useful when we come to automating resolution.
However, the computational complexity of conversion to CNF: NP-hard
Examples – convert the following propositions to CNFs.
(¬A → B) → ¬C
A  (B  C)
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9. Inference and Resolution2.
Examples – convert the following BNF propositions to CNFs.
(~A  B)  ~C
~(~A  B)  ~C
(~~A  B)  ~C
(A  B  ~C)
A  (B  C)
(A  B)  (B  A)
(~A  B)  (~B  A)
A  (B  C)
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10. Inference and Resolution2.
Clauses
Having converted a wff to CNF, it is usual to write it as a set of clauses.
E.g.:
(A  B)  C
In CNF is:
(A  C)  (~B  C) [Q] How?
In clause form, we write:
{(A, C), (~B, C)}
[Q] How to use CNF?
Disjunction Conjunction
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11. Inference and Resolution2.
The Resolution Rule
The resolution rule is written as follows:
A  B, ¬B  C (A  B)  (¬B  C) ╞ (A  C)
A  C
A  B, ¬B (A  B)  ¬B ╞ A A
If the upper part is valid, then the lower part should be valid.
This tells us that if we have two clauses that have a literal and its negation, we can combine them by removing that literal.
[Q] Can you prove it using the truth table?
E.g.: if we have {(A, C), (~A, D)}, we would apply resolution to get {(C, D)}
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12. Resolution Refutation2.
Resolution Refutation
Let us resolve: {(~A, B), (~B, C), A, ~C}
[Q] We begin by resolving the first clause with the second clause, thus eliminating B and ~B:
{(~A, B), (~B, C), A, ~C}
=> {(~A, C), A, ~C} B is eliminated.
=> {C, ~C} A is eliminated.
=> falsum
Why falsum? Because the initial assumption, ~C, is not valid. Therefore C is valid.
Now we can resolve both remaining literals, which gives falsum: ^
If we reach falsum, we have proved that our initial set of clauses were inconsistent.
This can be written as:
{(~A, B), (~B, C), A, ~C} ╞ ^
Note
{…, A, …, ~A, …}
╞ ^
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13. Inference and Resolution
Proof by Refutation
If we want to prove that a logical argument is valid, we negate its conclusion, convert it to clause form, and then try to derive falsum using resolution.
If we derive falsum, then our clauses were inconsistent, meaning the original argument was valid, since we negated its conclusion.
Proof by refutation is also called proof by contradiction.
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14. Inference and Resolution
[Q] Is the following argument valid?
(A ˄ ¬B) → C
A ˄ ¬B C
Start with the negation of the conclusion, and convert to clauses:
{(~A, B, C), A, ~B, ~C} [Q] How?
Let’s resolve:
{(~A, B, C), A, ~B, ~C}
=> {(B, C), ~B, ~C} How?
=> {C, ~C} How?
=> 
We have reached falsum.
[Q] Why?
Because of the assumption of ~C. Therefore C. Hence, our original argument is valid.
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15. Inference and Resolution
Another example:
A  B
B  C
C  D
D  E  F
 A  F Is this argument valid?
~(A  F) 
~(~A  F)  A  ~F
Clause form?
{(~A, B), (~B, C), (~C, D), (~D, E, F), A, ~F }
Resolution?
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16. Inference and Resolution2.
Topics
It is often sensible to represent a refutation proof in tree form:
In this case, the
proof has failed, as we
are left with E instead of falsum.
[Q] What does this mean?
A  F is not valid nor invalid.
How to automate the resolution?
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17. 3. Applications of Resolution
Resolution refutation can be used to determine if a solution exists for a particular combinatorial problem.
For example, for graph coloring, we represent the assignment of colors to the nodes and the constraints regarding edges as propositions, and attempt to prove that the complete set of clauses is satisfiable (i.e., to be deduced to non-false sum).
This does not tell us how to color the graph, simply that it is possible.
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18. Inference and Resolution
Topics
3-coloring problem: Available colors are r (red), g (green) and b (blue).
Ar means that node A has been coloured red.
Each node must have exactly one color:
Ar  Ag  Ab
¬Ar  ¬Ag ( ¬(Ar  Ag))
¬Ag  ¬Ab
¬Ab  ¬Ar
If (A, B) is an edge, then:
¬Ar  ¬Br ( ¬(Ar  Br))
¬Ab  ¬Bb
¬Ag  ¬Bg
(B, C), (A, C), (C, D) and (A, D) are also edges.
Now we construct the complete set of clauses for our graph, and try to see if they are satisfiable, using the resolution rule. It will take long time though.
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19. 4. Forward and backward chaining
Horn clauses – a simpler form
Forward chaining
Backward chaining
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20. Inference and Resolution
Horn Clauses
Horn Form (a more restricted version of propositional language) – conjunction of Horn clauses
Horn clause =
propositional symbol; or -- facts or observations
(conjunction of symbols)  symbol -- rules
E.g., C  (B  A)  (C  D  B)
Fact or observation: C
Rules: if B then A; if C and D then B
Modus Ponens (for Horn Form): complete for Horn KBs
α1[, … ,αn], α1 [ …  αn]  β β
Meaning: Facts and the rules with the facts  then  is a fact.
E.g., Rule: if red light, then stop; Fact: red light; then ???
Can be used with forward chaining or backward chaining. These algorithms are very natural and run in linear time.
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21. How to model Horn clauses?
Example:
Rules:
IF P THEN Q
IF L and M THEN P
IF B and L THEN M
IF A and P THEN L
IF A and B THEN L
Facts or observations: A B
Question:
Q ?
[Q] Horn clauses?
and-or graph
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22. Inference and Resolution
Example:
Rules:
If X croaks and X eats flies, then X is a frog.
If X chirps and X sings, then X is a canary.
If X is a frog, then X is green.
If X is a canary, then X is yellow.
Observations:
Fritz croaks.
Fritz eats flies.
Question:
What is the color of Fritz?
[Q] How to answer?
[Q] Can you convert the problem to Horn clauses?
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23. Inference and Resolution
Another more complex example – ZooKeeper:
If X has hair, then X is a mammal.
If X gives milk, then X is a mammal.
If X has feathers, then X is a bird.
If X flies and lays eggs, then X is a bird
If X is a mammal and X eats meat, then X is a carnivore.
If X is a mammal and X has pointed teeth and X has claws and X has forward pointing eyes, then X is a carnivore.
If X is a mammal and X has hoofs, then X is an ungulate.
If X is a mammal and X chews cud, then X is an ungulate.
If X is a carnivore and X has tawny color and X has dark spots, then X is a cheetah.
If X is a carnivore and X has tawny color and X has black strips, then X is a tiger.
If X is an ungulate and X has long legs and X has a long neck and X has tawny color and X has dark spots, then X is a giraffe.
If X is an ungulate and X has white color and X has black strips, then X is a zebra.
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24. Inference and Resolution4.
If X is a bird and X does not fly and X has long neck and X has long legs and X is black and white, then X is an ostrich.
If X is a bird and X does not fly and X swims and X is black and white, then X is a penguin.
If X is a bird and X is a good flier, then X is an albatross.
Facts/observations:
Stretch has hair.
Stretch chew cud.
Stretch has long legs.
Stretch has a long neck.
Stretch has tawny color.
Stretch has dark spots.
Question:
What animal is Stretch?
[Q] How to answer the above query? Can we use “proof by refutation”? Any problem with that?
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25. Inference and Resolution
Forward chaining
[Q] The query Q is valid?
[Q] Can we use “proof by refutation”? Any problem with that?
Knowledge base
Facts
Query = Q
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26. Inference and Resolution
Idea:
Start with facts, and fire any rule whose premises are satisfied in the KB; Keep adding its conclusion to the KB, until query is found.
Knowledge base OR
AND
Facts
Query = Q
AND-OR graph
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27. Inference and Resolution1 2 3 4 5
Query
Facts Rules Rule fired
triggered
A, B OR
AND
# of premises
Not inferred
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28. Inference and Resolution1 2 3 4 5
Facts Rules Rule fired
triggered
A, B
Reduced by 1
Set true
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29. Inference and Resolution1 2 3 4 5
Facts Rules Rule fired
triggered
A, B 5 OR
AND
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30. Inference and Resolution1 2 3 4 5
Facts Rules Rule fired
triggered
A, B 5 5 OR
AND
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31. Inference and Resolution1 2 3 4 5
Facts Rules Rule fired
triggered
A, B 5 5
A, B, L OR
AND
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32. Inference and Resolution1 2 3 4 5
Facts Rules Rule fired
triggered
A, B 5 5
A, B, L 3 3
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33. Inference and Resolution1 2 3 4 5
Facts Rules Rule fired
triggered
A, B 5 5
A, B, L 3 3
A, B, L, M
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34. Inference and Resolution1 2 3 4 5
Facts Rules Rule fired
triggered
A, B 5 5
A, B, L 3 3
A, B, L, M 2 2
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35. Inference and Resolution1 2 3 4 5
Facts Rules Rule fired
triggered
A, B 5 5
A, B, L 3 3
A, B, L, M 2 2
A, B, L, M, P
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36. Inference and Resolution1 2 3 4 5
Facts Rules Rule fired
triggered
A, B 5 5
A, B, L 3 3
A, B, L, M 2 2
A, B, L, M, P 4, 1
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37. Inference and Resolution1 2 3 4 5
Facts Rules Rule fired
triggered
A, B 5 5
A, B, L 3 3
A, B, L, M 2 2
A, B, L, M, P 4, 1 4
A, B, L, M, P 1
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38. Inference and Resolution1 2 3 4 5
Facts Rules Rule fired
triggered
A, B 5 5
A, B, L 3 3
A, B, L, M 2 2
A, B, L, M, P 4, 1 4
A, B, L, M, P 1 1
A, B, L, M, P, Q
STOP
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39. Inference and Resolution
Example – Fritz:
Rules
Croak  EatFly  Frog
Chirp  Sing  Canary
Frog  Green
Canary  Yellow
Facts
Croak
EatFly
Question
The color of Fritz.
Green
Yellow
Frog
Canary
Croak
EatFly
Chirp
Sing
Facts Rules triggered Rule fired
Croak, EatFly
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40. Inference and Resolution
Example – Fritz:
Rules
Croak  EatFly  Frog
Chirp  Sing  Canary
Frog  Green
Canary  Yellow
Facts
Croak
EatFly
Question
The color of Fritz.
Green
Yellow
Frog
Canary
Croak
EatFly
Chirp
Sing
Facts Rules triggered Rule fired
Croak, EatFly
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41. Inference and Resolution
Example – Fritz:
Rules
Croak  EatFly  Frog
Chirp  Sing  Canary
Frog  Green
Canary  Yellow
Facts
Croak
EatFly
Question
The color of Fritz.
Green
Yellow
Frog
Canary
Croak
EatFly
Chirp
Sing
Facts Rules triggered Rule fired
Croak, EatFly
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42. Inference and Resolution
Example – Fritz:
Rules
Croak  EatFly  Frog
Chirp  Sing  Canary
Frog  Green
Canary  Yellow
Facts
Croak
EatFly
Question
The color of Fritz.
Green
Yellow
Frog
Canary
Croak
EatFly
Chirp
Sing
Facts Rules triggered Rule fired
Croak, EatFly 1 1
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43. Inference and Resolution4.
Example – Fritz:
Rules:
Croak  EatFly  Frog
Chirp  Sing  Canary
Frog  Green
Canary  Yellow
Facts:
Croak
EatFly
Question:
The color of Fritz?
Green
Yellow
Frog
Canary
Croak
EatFly
Chirp
Sing
Facts Rules triggered Rule fired
Croak, EatFly 1 1
…, Frog …
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44. Inference and Resolution
Backward chaining
[Q] Problem of forward chaining (FC)?
FC starts with all the asserted symbols, i.e., facts -> longer execution time
(We do not have to use all the facts to conclude the given query.)
Idea of backward chaining (BC): work backwards from the query q:
to prove q by BC,
check if q is known already, or
prove by BC – only the premises of some rules concluding q
Something to consider
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
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45. Inference and Resolution
Unknown
Facts Goals Matching rules
A, B Q Q:? 1 2 3 4 5
TRUE
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46. Inference and Resolution
Unknown
(subgoal)
Facts Goals Matching rules
A, B Q 1
A, B P P:? 1 2 3 4 5
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47. Inference and Resolution
Facts Goals Matching rules
A, B Q 1
A, B P 2
A, B [L, M] L: ? 1 2 3 4 5
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48. Inference and Resolution
Already in the stack of subgoals
Facts Goals Matching rules
A, B Q 1
A, B P 2
A, B [L, M] L: 4, 5 1 2 3 4 5
Not fired
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49. Inference and Resolution
Facts Goals Matching rules
A, B Q 1
A, B P 2
A, B [L, M] L: 4, 5 1 2 3 4 5
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50. Inference and Resolution
Facts Goals Matching rules
A, B Q 1
A, B P 2
A, B [L, M] L: 4, 5
A, B, L M M:? 1 2 3 4 5
Became TRUE
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51. Inference and Resolution
Facts Goals Matching rules
A, B Q 1
A, B P 2
A, B [L, M] L: 4, 5
A, B, L M 3 1 2 3 4 5
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52. Inference and Resolution
Facts Goals Matching rules
A, B Q 1
A, B P 2
A, B [L, M] L: 4, 5
A, B, L M 3
A, B, L, M 
STOP 1 2 3 4 5
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53. Inference and Resolution1 2 3 4 5
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54. Inference and Resolution1 2 3 4 5
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55. Inference and Resolution
Example – Fritz:
Rules
Croak  EatFly  Frog
Chirp  Sing  Canary
Frog  Green
Canary  Yellow
Facts
Croak
EatFly
Question
The color of Fritz?
Green
Yellow
Frog
Canary
Croak
EatFly
Chirp
Sing
Facts Goals Matching rules
Croak, EatFly Yellow
Croak, EatFly Canary
Croak, EatFly [Chirp, Sing] 
Failure
Croak, EatFly Green
… … …
=> Chirp –> false
=> Canary –> false
=> Yellow –> false
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56. Inference and Resolution
How to use BC to solve ZooKeeper?
Animal candidates – Cheetah, Tiger, Giraffe, Zebra, Ostrich, Penguin, Albatross
For each of the above animals,
Run BC to see if the animal can be deduced.
Can you implement BC for ZooKeeper?
One good thing of Horn clauses is the conclusion part of if-then rules is a propositional symbol.
The condition part of if-then rules is a conjunction of propositional symbols, and hence AND operation.
There can be multiple rules having the same conclusion, i.e., propositional symbol. Anyone of the rules can be used, and hence OR operation.
As you can see in the example of Yellow in the previous slide, recursion can be used.
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57. Inference and Resolution
Recursion with the next representation of rules and facts.
If X is a mammal AND X eats meat, then X is a carnivore.
If X is a mammal AND X has pointed teeth AND X has claws AND X has forward pointing eyes, then X is a carnivore. …
Stretch has hair.
Stretch chew cud.
var rules = [];
rules['Carnivore'] = [
['Mammal', 'EatMeat'],
['Mammal', 'PointedTooth', 'Claw', 'ForwardPointingEye']
]; // (Mammal  EatMeat) 
// (Mammal  PointedTooth  Claw  ForwardPointingEye)
...
var facts = [];
facts[0] = 'Hair';
facts[1] = 'ChewCud'; OR
DNF (Disjunctive Normal Form)
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58. Inference and Resolution
// DNF for rules
// rules: associative array in which property values are 2D arrays.
rules['Mammal'] = [['Hair'], ['Milk']];
rules['Ungulate'] = [['Mammal', 'Hoof'], ['Mammal', 'ChewCud']];
rules['Giraffe'] = [['Ungulate', 'LongLeg', 'LongNeck', 'TawnyColor', 'DarkSpot']];
rules['Zebra'] = [['Ungulate', 'WhiteColor', 'BlackStrip']];
// Linear array
// This array starts with the facts obtained by observing an animal.
facts = ['ChewCud', 'DarkSpot', 'Hair', 'LongLeg', LongNeck', 'TawnyColor'];
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59. Inference and Resolution
Facts:
Stretch has hair.
Stretch chew cud.
Stretch has long legs.
Stretch has a long neck.
Stretch has tawny color.
Stretch has dark spots.
Let’s find if that animal is giraffe, using backward chaining – recursion.
* BC – the recursive function for backward chaining
* Facts? – is the propositional symbol fact?
* Rules? – is there any rule used for the propositional symbol?
BC(…)
Fact?
Rule?
Rule0
Rule1
Return; Action
BC(‘Giraffe’) No
Yes
[‘Ungulate’, ‘LongLeg’, ‘LongNeck’, ‘TawnyColor’, ‘DarkSpot’]
BC(‘Ungulate’) && BC(‘LongLeg’) && BC(‘LongNeck’) && BC(‘TawnyColor’) && BC(‘DarkSpot’)
BC(‘Ungulate’)
[‘Mammal’, ‘Hoof’]
[‘Mammal’, ‘ChewCud’]
BC(‘Mammal’) && BC(‘Hoof’) || BC(‘Mammal’) && BC(‘ChewCud’)
BC(‘Mammal’)
[‘Hair’]
[‘Milk’]
BC(‘Hair’) || BC(‘Milk’)
BC(‘Hair’)
TRUE
...
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60. Forward vs. backward chaining4.
Forward vs. backward chaining
Topics
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?
E.g., How do I get into a PhD program?
E.g., My car won’t start. What is the problem?
Complexity of BC can be much less than linear in size of KB
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61. Topics
Project ideas
Resolution refutation for the ZooKeeper problem, with explanation
Forward chaining for the ZooKeeper problem, with explanation
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