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- 1. Knowledge Representation using First-Order Logic
- 2. Domain• Domain is a section of the knowledge representation. - The Kinship domain - Mathematical sets - Assertions and queries in first order logic - The Wumpus World
- 3. Kinship domain
- 4. Kinship domain(family relationship)It consists of• Object – People• Unary Predicates - Male and Female• Binary Predicates - Parent,Brother,Sister• Functions – Father, Mother• Relations – Brotherhood, sisterhood.
- 5. ExamplesThe kinship domain:• Brothers are siblings ∀x,y Brother(x,y) => Sibling(x,y)• Male and female are disjoint categories ∀x, Male(x) ⇔ ¬Female(x)• Parent and child are inverse relations ∀p,c Parent(p,c) ⇔ Child(c,p)
- 6. Mathematical sets
- 7. Mathematical set representation• Constant – Empty set (s = {})• Predicate – Member and subset (s1 ⊆ s2)• Functions – Intersection( ∩ ) and union (∪ )• Example: Two sets are equal if and only if each is a subset of the other. ∀s1,s2 (s1=s2) ⇔(subset(s1,s2) ∧ subset(s2,s1))Other eg: ∀x,s1,s2 x ∈ (s1 ∩ s2) ⇔ (x ∈ s1 ∧ x ∈ s2) ∀x,s1,s2 x ∈ (s1 ∪ s2) ⇔ (x ∈ s1 ∨ x ∈ s2)
- 8. Assertions and Queries in first-order logic
- 9. Assertions• Sentences are added to a knowledge base using TELL are called assertions.• We want to TELL things to the KB, e.g. TELL(KB, King(John)) TELL(KB, ∀ x king(x) => Person(x)) John is a king and that king is a person.
- 10. Queries• Questions are asked to the knowledge base using ASK called as queries or goals.• We also want to ASK things to the KB, ASK(KB, ∃x , Person (x ) ) returns true by substituting john to a x.
- 11. Wumpus world
- 12. Agent Architectures• Reflex agents: Classify their percept and act accordingly.• Model based agents: Construct an internal representation of the world and use it to act.• Goal based agent : Form goals and try to achieve them.
- 13. FOL Version of Wumpus World• Typical percept sentence: Percept([Stench,Breeze,Glitter,None,None],3)• In this sentence: Percept - predicate Stench, Breeze and glitter – Constants 3 – Integer to represent time• Actions: Turn Right), Turn Left), Forward, Shoot, Grab, Release, Climb
- 14. Cont..,• To determine best action, construct query: ∀ a BestAction(a,5)• ASK solves this query and returns {a/Grab} – Agent program then calls TELL to record the action which was taken to update the KB.
- 15. • Percept sequences 1. Synchronic sentences (same time). - sentences dealing with time. 2. Diachronic sentences (across time). - agent needs to know how to combine information about its previous location to current location.
- 16. Two kinds of synchronic rules 1.Diagnostic rules 2.Casual rules
- 17. Deducing hidden properties• Squares are breezy near a pit: – Diagnostic rule---infer cause from effect ∀s Breezy(s) ⇔ ∃ r Adjacent(r,s) ∧ Pit(r) – Causal rule---infer effect from cause ∀r Pit(r) ⇒ [∀s Adjacent(r,s) ⇒ Breezy(s)]
- 18. Knowledge engineering in FOL
- 19. Steps1. Identify the task3. Assemble the relevant knowledge5. Decide on a vocabulary of predicates, functions, and constants7. Encode general knowledge about the domain9. Encode a description of the specific problem instance11. Pose queries to the inference procedure and get answers13. Debug the knowledge base
- 20. The electronic circuits domainOne-bit full adderPossible queries: - does the circuit function properly? - what gates are connected to the first input terminal? - what would happen if one of the gates is broken? and so on
- 21. The electronic circuits domain1. Identify the task – Does the circuit actually add properly?2. Assemble the relevant knowledge – Composed of wires and gates; Types of gates (AND, OR, XOR, NOT) – Two input terminals and one output terminal
- 22. 3. Decide on a vocabulary• Alternatives: Type(X1) = XOR (function) Type(X1, XOR) (binary predicate) XOR(X1) (unary predicate) It can be represented by either binary predicate or individual type.
- 23. 4. Encode general knowledge of the domain 1.If two terminals are connected, then they havethe same signal. ∀t1,t2 Connected(t1, t2) ⇒ Signal(t1) = Signal(t2)2.The signal at every terminal is either 1 or 0 (but not both) ∀t Signal(t) = 1 ∨ Signal(t) = 0 1≠0
- 24. 3. Connected is a commutative predicate. ∀t1,t2 Connected(t1, t2) ⇒ Connected(t2, t1)4. An OR gate’s output is 1 if and only if any of its input is 1. ∀g Type(g) = OR ⇒ Signal(Out(1,g)) = 1 ⇔ ∃n Signal(In(n,g)) = 1
- 25. 5. An AND gate’s output is 0 if and only if any of its input is 0. ∀g Type(g) = AND ⇒ Signal(Out(1,g)) = 0 ⇔ ∃n Signal(In(n,g)) = 06. An XOR gate’s output is 1 if and only if any of its inputs are different: ∀g Type(g) = XOR ⇒Signal(Out(1,g)) = 1 ⇔ Signal(In(1,g)) ≠ Signal(In(2,g))
- 26. 7. An XOR gate’s output is 1 if and only if any of its inputs are different: ∀g Type(g) = NOT ⇒ Signal(Out(1,g)) ≠ Signal(In(1,g))
- 27. 5. Encode the specific problem instance• First we categorize the gates: Type(X1) = XOR Type(X2) = XOR Type(A1) = AND Type(A2) = AND Type(O1) = OR – Then show the connections between them:
- 28. Connected(Out(1,X1),In(1,X2)) Connected(In(1,C1),In(1,X1))Connected(Out(1,X1),In(2,A2)) Connected(In(1,C1),In(1,A1))Connected(Out(1,A2),In(1,O1)) Connected(In(2,C1),In(2,X1))Connected(Out(1,A1),In(2,O1)) Connected(In(2,C1),In(2,A1))Connected(Out(1,X2),Out(1,C1)) Connected(In(3,C1),In(2,X2))Connected(Out(1,O1),Out(2,C1)) Connected(In(3,C1),In(1,A2))
- 29. 6. Pose queries to the inference procedure and get answers For the given query the inferenceprocedure operate on the problemspecific facts and derive the answers.
- 30. What are the possible sets of values of all the terminals for the adder circuit?∃i1,i2,i3,o1,o2 Signal(In(1,C1)) = i1 ∧ Signal(In(2,C1)) = i2 ∧ Signal(In(3,C1)) = i3 ∧Signal(Out(1,C1)) = o1 ∧ Signal(Out(2,C1)) = o2
- 31. 7. Debug the knowledge base• For the given query, if the result is not a user expected one then KB is updated with relevant axioms.• The KB is checked with different constraints.eg:prove any output for the circuit i.e.,0 or 1.

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