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Knowledge Representation using First-Order Logic
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Domain• Domain is a section of the knowledge representation. - The Kinship domain - Mathematical sets - Assertions and queries in first order logic - The Wumpus World
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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)
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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.
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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.
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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.
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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
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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.
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• 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.
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Two kinds of synchronic rules 1.Diagnostic rules 2.Casual rules
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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)]
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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
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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
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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
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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.
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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
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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
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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))
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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))
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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:
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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.
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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
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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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