$1$ Propositional Logic

$15+15=30$

Consider the following propositional logic sentences:

$\backslash$phi $1=$ A $->(($A $->$ B$)($B $->$ C$))$

$\backslash$phi $2=($A $$ B$)(($B $->$ C$)->(($C $$ A$)$ A$))$

$3=($A $$ B$)(($A $$C$)($B $$ D$))$

Relative to these sentences,

$($a$)$ Show clearly whether they are either valid, satisfiable $($but not valid$),$ or unsatisfiable;

in each case, justify your answer.

$($b$)$ Convert them into Conjunctive Normal Form $($CNF$)$ and indicate if they are the Horn Clause. $2$ Backward$/$Forward Reasoning

$10+10+10=30$

Convert the expressions below into propositional logic sentences:

If a person is likely to vomit and looks pale and is thirsty, then is sick.

Always, a person does not have a high temperature or has slept well or is thirsty.

If a person does not look pale, then feels well or has slept well.

If a person has a fever, then their temperature is high.

It is known that when a person looks pale, then they should drink water.

If a person has a high temperature and does not feel well, then is likely to vomit.

It is always the case that a person does not feel well or does not have a fever.

If a person has slept well, then is not exhausted.

With the propositional logic sentences obtained from the above expressions in English, show

whether or not it can be proved that a person is sick if one assumes both that the person

has a fever and that the person is exhausted. Do this reasoning using:

$($a$)$ Forward chaining; and show the graph resulting from the application of the rules.

Provide a table showing which variables are in your AGENDA, are INFERRED, and

their COUNT throughout the application of the algorithm.

$($b$)$ Backward chaining; and provide a sequence of rules to prove the goal.

S : is Sick

V: likely to Vomit

P: looks Pale

T: is Thirsty

H: has High temperature

NS: has Not Slept well

NF: does Not Feel well

F: has Fever

W: drink Water

E: is Exhausted mgu$($most general unifier$)-$the mgu g of $\{Ei\}$ has the property that if $s$ is any unifier of $\{Ei\},$

yielding Ei