The Wumpus World in Artificial intelligence

Wumpus world:
The Wumpus world is a simple world example to illustrate the worth of a knowledge-based agent and to represent
knowledge representation. It was inspired by a video game Hunt the Wumpus by Gregory Yob in 1973.

The Wumpus world is a cave which has 4/4 rooms connected with passageways. So there are total 16 rooms which
are connected with each other. We have a knowledge-based agent who will go forward in this world. The cave has
a room with a beast which is called Wumpus, who eats anyone who enters the room. The Wumpus can be shot by
the agent, but the agent has a single arrow. In the Wumpus world, there are some Pits rooms which are
bottomless, and if agent falls in Pits, then he will be stuck there forever. The exciting thing with this cave is that in
one room there is a possibility of finding a heap of gold. So the agent goal is to find the gold and climb out the
cave without fallen into Pits or eaten by Wumpus. The agent will get a reward if he comes out with gold, and he will
get a penalty if eaten by Wumpus or falls in the pit.

Note: Here Wumpus is static and cannot move.

Following is a sample diagram for representing the Wumpus world. It is showing some rooms with Pits, one room
with Wumpus and one agent at (1, 1) square location of the world.

There are also some components which can help the agent to navigate the cave. These components are
given as follows:
 a. The rooms adjacent to the Wumpus room are smelly, so that it would have some stench.
b. The room adjacent to PITs has a breeze, so if the agent reaches near to PIT, then he will perceive the
breeze.
c. There will be glitter in the room if and only if the room has gold.
d. The Wumpus can be killed by the agent if the agent is facing to it, and Wumpus will emit a horrible
scream which can be heard anywhere in the cave.

PEAS description of Wumpus world:

To explain the Wumpus world we have given PEAS description as below:

Performance measure:

o +1000 reward points if the agent comes out of the cave with the gold.
o -1000 points penalty for being eaten by the Wumpus or falling into the pit.
o -1 for each action, and -10 for using an arrow.
o The game ends if either agent dies or came out of the cave.

Environment:

o A 4*4 grid of rooms.

o The agent initially in room square [1, 1], facing toward the right.
o Location of Wumpus and gold are chosen randomly except the first square [1,1].
o Each square of the cave can be a pit with probability 0.2 except the first square.

Actuators:

o Left turn,
o Right turn
o Move forward
o Grab
o Release
o Shoot.

Sensors:

o The agent will perceive the stench if he is in the room adjacent to the Wumpus. (Not diagonally).
o The agent will perceive breeze if he is in the room directly adjacent to the Pit.
o The agent will perceive the glitter in the room where the gold is present.
o The agent will perceive the bump if he walks into a wall.
o When the Wumpus is shot, it emits a horrible scream which can be perceived anywhere in the cave.
o These percepts can be represented as five element list, in which we will have different indicators for each
sensor.
 o Example if agent perceives stench, breeze, but no glitter, no bump, and no scream then it can be
represented as:
[Stench, Breeze, None, None, None].

The Wumpus world Properties:

o Partially observable: The Wumpus world is partially observable because the agent can only perceive the
close environment such as an adjacent room.
o Deterministic: It is deterministic, as the result and outcome of the world are already known.
o Sequential: The order is important, so it is sequential.
o Static: It is static as Wumpus and Pits are not moving.
o Discrete: The environment is discrete.
o One agent: The environment is a single agent as we have one agent only and Wumpus is not considered
as an agent.

Exploring the Wumpus world:

Now we will explore the Wumpus world and will determine how the agent will find its goal by applying logical
reasoning.

Agent's First step:

Initially, the agent is in the first room or on the square [1,1], and we already know that this room is safe for the
agent, so to represent on the below diagram (a) that room is safe we will add symbol OK. Symbol A is used to
represent agent, symbol B for the breeze, G for Glitter or gold, V for the visited room, P for pits, W for Wumpus.

At Room [1,1] agent does not feel any breeze or any Stench which means the adjacent squares are also OK.
 Agent's second Step:

Now agent needs to move forward, so it will either move to [1, 2], or [2,1]. Let's suppose agent moves to the
room [2, 1], at this room agent perceives some breeze which means Pit is around this room. The pit can be in [3,
1], or [2,2], so we will add symbol P? to say that, is this Pit room?

Now agent will stop and think and will not make any harmful move. The agent will go back to the [1, 1] room. The
room [1,1], and [2,1] are visited by the agent, so we will use symbol V to represent the visited squares.

Agent's third step:

At the third step, now agent will move to the room [1,2] which is OK. In the room [1,2] agent perceives a stench
which means there must be a Wumpus nearby. But Wumpus cannot be in the room [1,1] as by rules of the game,
and also not in [2,2] (Agent had not detected any stench when he was at [2,1]). Therefore agent infers that
Wumpus is in the room [1,3], and in current state, there is no breeze which means in [2,2] there is no Pit and no
Wumpus. So it is safe, and we will mark it OK, and the agent moves further in [2,2].

Agent's fourth step:

At room [2,2], here no stench and no breezes present so let's suppose agent decides to move to [2,3]. At room
[2,3] agent perceives glitter, so it should grab the gold and climb out of the cave.

Propositional logic in Artificial intelligence

Propositional logic (PL) is the simplest form of logic where all the statements are made by propositions. A
proposition is a declarative statement which is either true or false. It is a technique of knowledge representation in
logical and mathematical form.

Example:

1. a) It is Sunday.  
2. b) The Sun rises from West (False proposition)  
3. c) 3+3= 7(False proposition)  
4. d) 5 is a prime number.   

Following are some basic facts about propositional logic:

o Propositional logic is also called Boolean logic as it works on 0 and 1.

o In propositional logic, we use symbolic variables to represent the logic, and we can use any symbol for a
representing a proposition, such A, B, C, P, Q, R, etc.
o Propositions can be either true or false, but it cannot be both.
o Propositional logic consists of an object, relations or function, and logical connectives.
o These connectives are also called logical operators.
o The propositions and connectives are the basic elements of the propositional logic.
o Connectives can be said as a logical operator which connects two sentences.
o A proposition formula which is always true is called tautology, and it is also called a valid sentence.
o A proposition formula which is always false is called Contradiction.
o A proposition formula which has both true and false values is called
o Statements which are questions, commands, or opinions are not propositions such as "Where is Rohini",
"How are you", "What is your name", are not propositions.

Syntax of propositional logic:

The syntax of propositional logic defines the allowable sentences for the knowledge representation. There are two
types of Propositions:

a. Atomic Propositions
b. Compound propositions

o Atomic Proposition: Atomic propositions are the simple propositions. It consists of a single proposition
symbol. These are the sentences which must be either true or false.

Example:

1. a) 2+2 is 4, it is an atomic proposition as it is a true fact.  
2. b) "The Sun is cold" is also a proposition as it is a false fact.   

o Compound proposition: Compound propositions are constructed by combining simpler or atomic

propositions, using parenthesis and logical connectives.

Example:

1. a) "It is raining today, and street is wet."  
2. b) "Ankit is a doctor, and his clinic is in Mumbai."   

Logical Connectives:
Logical connectives are used to connect two simpler propositions or representing a sentence logically. We can
create compound propositions with the help of logical connectives. There are mainly five connectives, which are
given as follows:
 1. Negation: A sentence such as ¬ P is called negation of P. A literal can be either Positive literal or negative
literal.
2. Conjunction: A sentence which has ∧ connective such as, P ∧ Q is called a conjunction.
Example: Rohan is intelligent and hardworking. It can be written as,
P= Rohan is intelligent,
Q= Rohan is hardworking. → P∧ Q.
3. Disjunction: A sentence which has ∨ connective, such as P ∨ Q. is called disjunction, where P and Q are
the propositions.
Example: "Ritika is a doctor or Engineer",
Here P= Ritika is Doctor. Q= Ritika is Doctor, so we can write it as P ∨ Q.
4. Implication: A sentence such as P → Q, is called an implication. Implications are also known as if-then
rules. It can be represented as
            If it is raining, then the street is wet.
        Let P= It is raining, and Q= Street is wet, so it is represented as P → Q
5. Biconditional: A sentence such as P⇔ Q is a Biconditional sentence, example If I am breathing,
then I am alive
            P= I am breathing, Q= I am alive, it can be represented as P ⇔ Q.

Following is the summarized table for Propositional Logic

Connectives:

Truth Table:
In propositional logic, we need to know the truth values of propositions in all possible scenarios. We can combine
all the possible combination with logical connectives, and the representation of these combinations in a tabular
format is called Truth table. Following are the truth table for all logical connectives:

Truth table with three propositions:

We can build a proposition composing three propositions P, Q, and R. This truth table is made-up of 8n Tuples as
we have taken three proposition symbols.

Precedence of connectives:
Just like arithmetic operators, there is a precedence order for propositional connectors or logical operators. This
order should be followed while evaluating a propositional problem. Following is the list of the precedence order for
operators:
 Precedence Operators

First Precedence Parenthesis

Second Precedence Negation

Third Precedence Conjunction(AND)

Fourth Precedence Disjunction(OR)

Fifth Precedence Implication

Six Precedence Biconditional

Note: For better understanding use parenthesis to make sure of the correct interpretations. Such as ¬R∨ Q, It
can be interpreted as (¬R) ∨ Q.

Logical equivalence:
Logical equivalence is one of the features of propositional logic. Two propositions are said to be logically equivalent
if and only if the columns in the truth table are identical to each other.

Let's take two propositions A and B, so for logical equivalence, we can write it as A⇔B. In below truth table we can
see that column for ¬A∨ B and A→B, are identical hence A is Equivalent to B

Properties of Operators:

o Commutativity:
o P∧ Q= Q ∧ P, or
o P ∨ Q = Q ∨ P.
o Associativity:
o (P ∧ Q) ∧ R= P ∧ (Q ∧ R),
o (P ∨ Q) ∨ R= P ∨ (Q ∨ R)
o Identity element:
o P ∧ True = P,
o P ∨ True= True.
o Distributive:
o P∧ (Q ∨ R) = (P ∧ Q) ∨ (P ∧ R).
o P ∨ (Q ∧ R) = (P ∨ Q) ∧ (P ∨ R).
o DE Morgan's Law:
o ¬ (P ∧ Q) = (¬P) ∨ (¬Q)
 o ¬ (P ∨ Q) = (¬ P) ∧ (¬Q).
o Double-negation elimination:
o ¬ (¬P) = P.

Limitations of Propositional logic:

o We cannot represent relations like ALL, some, or none with propositional logic. Example:

1. All the girls are intelligent.

2. Some apples are sweet.
b. Propositional logic has limited expressive power.
c. In propositional logic, we cannot describe statements in terms of their properties or logical relationships.

First-Order Logic in Artificial intelligence

In the topic of Propositional logic, we have seen that how to represent statements using propositional logic. But
unfortunately, in propositional logic, we can only represent the facts, which are either true or false. PL is not
sufficient to represent the complex sentences or natural language statements. The propositional logic has very
limited expressive power. Consider the following sentence, which we cannot represent using PL logic.

o "Some humans are intelligent", or

o "Sachin likes cricket."

To represent the above statements, PL logic is not sufficient, so we required some more powerful logic, such as
first-order logic.

First-Order logic:
o First-order logic is another way of knowledge representation in artificial intelligence. It is an extension to
propositional logic.
o FOL is sufficiently expressive to represent the natural language statements in a concise way.
o First-order logic is also known as Predicate logic or First-order predicate logic. First-order logic is a
powerful language that develops information about the objects in a more easy way and can also express
the relationship between those objects.
o First-order logic (like natural language) does not only assume that the world contains facts like
propositional logic but also assumes the following things in the world:
o Objects: A, B, people, numbers, colors, wars, theories, squares, pits, wumpus, ......
o Relations: It can be unary relation such as: red, round, is adjacent, or n-any relation such
as: the sister of, brother of, has color, comes between
o Function: Father of, best friend, third inning of, end of, ......
o As a natural language, first-order logic also has two main parts:

o Syntax
o Semantics
Syntax of First-Order logic:
The syntax of FOL determines which collection of symbols is a logical expression in first-order logic. The basic
syntactic elements of first-order logic are symbols. We write statements in short-hand notation in FOL.

Basic Elements of First-order logic:

Following are the basic elements of FOL syntax:

Constant 1, 2, A, John, Mumbai, cat,....

Variables x, y, z, a, b,....

Predicates Brother, Father, >,....

Function sqrt, LeftLegOf, ....

Connectives ∧, ∨, ¬, ⇒, ⇔

Equality ==

Quantifier ∀, ∃

Atomic sentences:

o Atomic sentences are the most basic sentences of first-order logic. These sentences are formed from a
predicate symbol followed by a parenthesis with a sequence of terms.
o We can represent atomic sentences as Predicate (term1, term2, ......, term n).

Example: Ravi and Ajay are brothers: => Brothers(Ravi, Ajay).

                Chinky is a cat: => cat (Chinky).

Complex Sentences:

o Complex sentences are made by combining atomic sentences using connectives.

First-order logic statements can be divided into two parts:

o Subject: Subject is the main part of the statement.

o Predicate: A predicate can be defined as a relation, which binds two atoms together in a statement.

Consider the statement: "x is an integer.", it consists of two parts, the first part x is the subject of the
statement and second part "is an integer," is known as a predicate.
Quantifiers in First-order logic:
o A quantifier is a language element which generates quantification, and quantification specifies the quantity
of specimen in the universe of discourse.
o These are the symbols that permit to determine or identify the range and scope of the variable in the
logical expression. There are two types of quantifier:

1. Universal Quantifier, (for all, everyone, everything)

2. Existential quantifier, (for some, at least one).

Universal Quantifier:
Universal quantifier is a symbol of logical representation, which specifies that the statement within its range is true
for everything or every instance of a particular thing.

The Universal quantifier is represented by a symbol ∀, which resembles an inverted A.

Note: In universal quantifier we use implication "→".

If x is a variable, then ∀x is read as:

o For all x
o For each x
o For every x.

Example:
All man drink coffee.

Let a variable x which refers to a cat so all x can be represented in UOD as below:

∀x man(x) → drink (x, coffee).

It will be read as: There are all x where x is a man who drink coffee.

Existential Quantifier:
Existential quantifiers are the type of quantifiers, which express that the statement within its scope is true for at
least one instance of something.

It is denoted by the logical operator ∃, which resembles as inverted E. When it is used with a predicate variable
then it is called as an existential quantifier.
Note: In Existential quantifier we always use AND or Conjunction symbol (∧).

If x is a variable, then existential quantifier will be ∃x or ∃(x). And it will be read as:

o There exists a 'x.'

o For some 'x.'
o For at least one 'x.'

Example:
Some boys are intelligent.

∃x: boys(x) ∧ intelligent(x)

It will be read as: There are some x where x is a boy who is intelligent.

Points to remember:
o The main connective for universal quantifier ∀ is implication →.
o The main connective for existential quantifier ∃ is and ∧.

Properties of Quantifiers:
o In universal quantifier, ∀x∀y is similar to ∀y∀x.
o In Existential quantifier, ∃x∃y is similar to ∃y∃x.
o ∃x∀y is not similar to ∀y∃x.

Some Examples of FOL using quantifier:

1. All birds fly.

In this question the predicate is "fly(bird)."
And since there are all birds who fly so it will be represented as follows.
              ∀x bird(x) →fly(x).

2. Every man respects his parent.

In this question, the predicate is "respect(x, y)," where x=man, and y= parent.
Since there is every man so will use ∀, and it will be represented as follows:
              ∀x man(x) → respects (x, parent).

3. Some boys play cricket.

In this question, the predicate is "play(x, y)," where x= boys, and y= game. Since there are some boys so we will
use ∃, and it will be represented as:
              ∃x boys(x) → play(x, cricket).
4. Not all students like both Mathematics and Science.
In this question, the predicate is "like(x, y)," where x= student, and y= subject.
Since there are not all students, so we will use ∀ with negation, so following representation for this:
              ¬∀ (x) [ student(x) → like(x, Mathematics) ∧ like(x, Science)].

5. Only one student failed in Mathematics.

In this question, the predicate is "failed(x, y)," where x= student, and y= subject.
Since there is only one student who failed in Mathematics, so we will use following representation for this:
              ∃(x) [ student(x) → failed (x, Mathematics) ∧∀ (y) [¬(x==y) ∧ student(y) → ¬failed (x,
Mathematics)].

Free and Bound Variables:

The quantifiers interact with variables which appear in a suitable way. There are two types of variables in First-
order logic which are given below:

Free Variable: A variable is said to be a free variable in a formula if it occurs outside the scope of the quantifier.

          Example: ∀x ∃(y)[P (x, y, z)], where z is a free variable.

Bound Variable: A variable is said to be a bound variable in a formula if it occurs within the scope of the
quantifier.

          Example: ∀x [A (x) B( y)], here x and y are the bound variables.

Knowledge Engineering in First-order logic

What is knowledge-engineering?
The process of constructing a knowledge-base in first-order logic is called as knowledge- engineering.
In knowledge-engineering, someone who investigates a particular domain, learns important concept of that
domain, and generates a formal representation of the objects, is known as knowledge engineer.

In this topic, we will understand the Knowledge engineering process in an electronic circuit domain, which is
already familiar. This approach is mainly suitable for creating special-purpose knowledge base.

The knowledge-engineering process:

Following are some main steps of the knowledge-engineering process. Using these steps, we will develop a
knowledge base which will allow us to reason about digital circuit (One-bit full adder) which is given below

1. Identify the task:

The first step of the process is to identify the task, and for the digital circuit, there are various reasoning tasks.

At the first level or highest level, we will examine the functionality of the circuit:

o Does the circuit add properly?

 o What will be the output of gate A2, if all the inputs are high?

At the second level, we will examine the circuit structure details such as:

o Which gate is connected to the first input terminal?

o Does the circuit have feedback loops?

2. Assemble the relevant knowledge:

In the second step, we will assemble the relevant knowledge which is required for digital circuits. So for digital
circuits, we have the following required knowledge:

o Logic circuits are made up of wires and gates.

o Signal flows through wires to the input terminal of the gate, and each gate produces the corresponding
output which flows further.
o In this logic circuit, there are four types of gates used: AND, OR, XOR, and NOT.
o All these gates have one output terminal and two input terminals (except NOT gate, it has one input
terminal).

3. Decide on vocabulary:
The next step of the process is to select functions, predicate, and constants to represent the circuits, terminals,
signals, and gates. Firstly we will distinguish the gates from each other and from other objects. Each gate is
represented as an object which is named by a constant, such as, Gate(X1). The functionality of each gate is
determined by its type, which is taken as constants such as AND, OR, XOR, or NOT. Circuits will be identified by a
predicate: Circuit (C1).

For the terminal, we will use predicate: Terminal(x).

For gate input, we will use the function In(1, X1) for denoting the first input terminal of the gate, and for output
terminal we will use Out (1, X1).

The function Arity(c, i, j) is used to denote that circuit c has i input, j output.

The connectivity between gates can be represented by predicate Connect(Out(1, X1), In(1, X1)).

We use a unary predicate On (t), which is true if the signal at a terminal is on.

4. Encode general knowledge about the domain:

To encode the general knowledge about the logic circuit, we need some following rules:

o If two terminals are connected then they have the same input signal, it can be represented as:

1. ∀  t1, t2 Terminal (t1) ∧ Terminal (t2) ∧ Connect (t1, t2) → Signal (t1) = Signal (2).   

o Signal at every terminal will have either value 0 or 1, it will be represented as:

1. ∀  t Terminal (t) →Signal (t) = 1 ∨Signal (t) = 0.  

o Connect predicates are commutative:

1. ∀  t1, t2 Connect(t1, t2)  →  Connect (t2, t1).       

o Representation of types of gates:

1. ∀  g Gate(g) ∧ r = Type(g) → r = OR ∨r = AND ∨r = XOR ∨r = NOT.   

o Output of AND gate will be zero if and only if any of its input is zero.

1. ∀  g Gate(g) ∧ Type(g) = AND →Signal (Out(1, g))= 0 ⇔  ∃n Signal (In(n, g))= 0.   

o Output of OR gate is 1 if and only if any of its input is 1:

1. ∀  g Gate(g) ∧ Type(g) = OR → Signal (Out(1, g))= 1 ⇔  ∃n Signal (In(n, g))= 1   

o Output of XOR gate is 1 if and only if its inputs are different:

1. ∀  g Gate(g) ∧ Type(g) = XOR → Signal (Out(1, g)) = 1 ⇔  Signal (In(1, g)) ≠ Signal (In(2, g)).  

o Output of NOT gate is invert of its input:

1. ∀  g Gate(g) ∧ Type(g) = NOT →   Signal (In(1, g)) ≠ Signal (Out(1, g)).  

o All the gates in the above circuit have two inputs and one output (except NOT gate).

1. ∀  g Gate(g) ∧ Type(g) = NOT →   Arity(g, 1, 1)   
2. ∀  g Gate(g) ∧ r =Type(g)  ∧ (r= AND ∨r= OR ∨r= XOR) →  Arity (g, 2, 1).   

o All gates are logic circuits:

1. ∀  g Gate(g) → Circuit (g).   

5. Encode a description of the problem instance:

Now we encode problem of circuit C1, firstly we categorize the circuit and its gate components. This step is easy if
ontology about the problem is already thought. This step involves the writing simple atomics sentences of
instances of concepts, which is known as ontology.

For the given circuit C1, we can encode the problem instance in atomic sentences as below:

Since in the circuit there are two XOR, two AND, and one OR gate so atomic sentences for these gates will be:

1. For XOR gate: Type(x1)= XOR, Type(X2) = XOR  
2. For AND gate: Type(A1) = AND, Type(A2)= AND  
3. For OR gate: Type (O1) = OR.    

And then represent the connections between all the gates.

Note: Ontology defines a particular theory of the nature of existence.

6. Pose queries to the inference procedure and get answers:

 In this step, we will find all the possible set of values of all the terminal for the adder circuit. The first query will be:

What should be the combination of input which would generate the first output of circuit C1, as 0 and a second
output to be 1?

1. ∃ i1, i2, i3 Signal (In(1, C1))=i1  ∧  Signal (In(2, C1))=i2  ∧ Signal (In(3, C1))= i3  
2.  ∧ Signal (Out(1, C1)) =0 ∧ Signal (Out(2, C1))=1  

7. Debug the knowledge base:

Now we will debug the knowledge base, and this is the last step of the complete process. In this step, we will try to
debug the issues of knowledge base.

In the knowledge base, we may have omitted assertions like 1 ≠ 0.