Knowledge Representation using First-Order Logic

Domain

• Domain is a section of the knowledge representation.

- The Kinship domain

- Mathematical sets

- Assertions and queries in first order logic

- The Wumpus World

Kinship domain

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.

Examples

The kinship domain:• Brothers are siblings

x,y Brother(x,y) => Sibling(x,y)

• Male and female are disjoint categoriesx, Male(x) ¬Female(x)

• Parent and child are inverse relationsp,c Parent(p,c) Child(c,p)

Mathematical sets

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)

Assertions and Queries in first-order logic

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.

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.

, ( )x Person x

Wumpus world

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.

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

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.

• 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.

Two kinds of synchronic rules

1.Diagnostic rules

2.Casual rules

Deducing hidden properties

• Squares are breezy near a pit:

– Diagnostic rule---infer cause from effects Breezy(s) r Adjacent(r,s) Pit(r)

– Causal rule---infer effect from cause

r Pit(r) [s Adjacent(r,s) Breezy(s)]

Knowledge engineering in FOL

Steps1. 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

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

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

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.

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

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

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))

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))

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:

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))

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.

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

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.