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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 (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 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 Identify the task Assemble the relevant knowledge**

Decide on a vocabulary of predicates, functions, and constants Encode general knowledge about the domain Encode a description of the specific problem instance Pose queries to the inference procedure and get answers 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**

Identify the task Does the circuit actually add properly? 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 have the 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)) = 0 6. 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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**Signal(Out(1,g)) ≠ Signal(In(1,g))**

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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**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))

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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?**

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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