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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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Kinship domain

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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.

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Examples The 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)

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Mathematical sets

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Mathematical set representation Constant – Empty set (s = {}) Predicate – Member and subset (s 1 s 2 ) Functions – Intersection( ) and union ( ) Example: Two sets are equal if and only if each is a subset of the other. s 1,s 2 (s 1 =s 2 ) (subset(s 1,s 2 ) subset(s 2,s 1 )) Other eg: x,s 1,s 2 x (s 1 s 2 ) (x s 1 x s 2 )

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Assertions and Queries in first-order logic

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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, ) returns true by substituting john to a x.

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Wumpus world

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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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Knowledge engineering in FOL

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Steps 1.Identify the task 2.Assemble the relevant knowledge 3.Decide on a vocabulary of predicates, functions, and constants 4.Encode general knowledge about the domain 5.Encode a description of the specific problem instance 6.Pose queries to the inference procedure and get answers 7.Debug the knowledge base

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The electronic circuits domain One-bit full adder Possible 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 domain 1.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(X 1 ) = XOR (function) Type(X 1, XOR) (binary predicate) XOR(X 1 ) (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 have the same signal. t 1,t 2 Connected(t 1, t 2 ) Signal(t 1 ) = Signal(t 2 ) 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. t 1,t 2 Connected(t 1, t 2 ) Connected(t 2, t 1 ) 4. An OR gates 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 gates 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)) = 0 6. An XOR gates 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 gates 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(X 1 ) = XOR Type(X 2 ) = XOR Type(A 1 ) = AND Type(A 2 ) = AND Type(O 1 ) = OR –Then show the connections between them:

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Connected(Out(1,X 1 ),In(1,X 2 )) Connected(In(1,C 1 ),In(1,X 1 )) Connected(Out(1,X 1 ),In(2,A 2 )) Connected(In(1,C 1 ),In(1,A 1 )) Connected(Out(1,A 2 ),In(1,O 1 )) Connected(In(2,C 1 ),In(2,X 1 )) Connected(Out(1,A 1 ),In(2,O 1 )) Connected(In(2,C 1 ),In(2,A 1 )) Connected(Out(1,X 2 ),Out(1,C 1 )) Connected(In(3,C 1 ),In(2,X 2 )) Connected(Out(1,O 1 ),Out(2,C 1 )) Connected(In(3,C 1 ),In(1,A 2 ))

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6. Pose queries to the inference procedure and get answers For the given query the inference procedure operate on the problem specific facts and derive the answers.

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What are the possible sets of values of all the terminals for the adder circuit? i 1,i 2,i 3,o 1,o 2 Signal(In(1,C1)) = i 1 Signal(In(2,C 1 )) = i 2 Signal(In(3,C 1 )) = i 3 Signal(Out(1,C 1 )) = o 1 Signal(Out(2,C 1 )) = o 2

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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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