There are two sections as explained below. Provide answers to the following questions specified in the corresponding section.

Section $1$: Inference Logic Exercise: Applying Inference Rules $(70$ marks$)$

For each question, you are provided with a set of premises and a conclusion. Your task is to apply the appropriate inference rules to prove the conclusion based on the given premises. Clearly state the inference rule$($s$)$ used in each step of your proof. Ensure that your proof follows a logical sequence and that each step is justified by the application of a valid inference rule.

Q$1.$ Premises: $PQ,P$

Conclusion: Q

Q$2.$ Premises: $P^{{?}^{?}}Q$

Conclusion: P

Q$3.$ Premises: $PvvQ,$notP

Conclusion: Q

Q$4.$ Premises: $P^{{?}^{?}}(QR),$notR

Conclusion: notP

Q$5.$ Premises: $(P^{{?}^{?}}Q)R,P^{{?}^{?}}$notR

Conclusion: notQ

Q$6.$ Premises: $(PvvQ{)}^{{?}^{?}}$notR,$PS,QS$

Conclusion: S

Q$7.$ Premises: $P(Q^{{?}^{?}}R),$not$(Q^{{?}^{?}}R)$

Conclusion: notP

Q$8.$ Premises: $P(QvvR),$notR

Conclusion: $PQ$

Q$9.$ Premises: $(PQ{)}^{{?}^{?}}(QR)$

Conclusion: $PR$

Q$10.$ Premises: $(P^{{?}^{?}}Q)R,$notR

Conclusion: notPvvnotQ

Note: Submit your solutions with clear explanations for each step, indicating the inference rule$($s$)$ applied. Ensure clarity, coherence, and correctness in your proofs. Use proper notation and formatting to present your proofs.

Description:

Classic planning in artificial intelligence involves generating a sequence of actions to achieve a desired goal state from an initial state, adhering to specified constraints and preconditions. In this assignment, you will be presented with a classic planning problem scenario, and you will use inference rules to deduce the sequence of actions required to accomplish the task.

Scenario:

Imagine a scenario where a robot needs to navigate through a $4$ x $4$ maze to reach a goal location, one step at a time avoiding any obstacle $($i$.$e$.,$ squares which are highlighted in the dark color$).$ The maze is represented as a grid with walls blocking $($obstacles$)$ certain paths. The robot can move in four directions: up$,$ down, left, and right. Additionally, it has sensors to detect walls and determine its current position.

Initial State

Given:

The robot is located within the maze environment.

The maze is represented as a grid with walls forming barriers between cells.

Inference Symbols:

R : The robot is present within the maze environment.

$P(x,y)$ : The robot is positioned at cell $(x,y)$ within the maze.

W$($x$,$y$)$: There is a wall present at cell $(x,y)$ within the maze.

$M(x,y)$ : Cell $(x,y)$ is part of the maze environment.

Using inference logic symbols, we can represent the initial state of the maze environment as follows:

By considering the mentioned facts the Initial State will be:

AAx,yinZ:$(M(x,y)((R^{{?}^{?}}P(x,y))vvW(x,y)))$