To determine which processes to abort in order to make the system deadlock-free, we need to analyze the resource allocation graph and identify deadlock-causing cycles.

• Processes: \(P = \{P_1, P_2, P_3, P_4\}\)
• Resources: \(R = \{R_1, R_2, R_3, R_4\}\)
• Assignment Edges:
  • \(R_1 \rightarrow P_1\)
  • \(R_2 \rightarrow P_2\)
  • \(R_3 \rightarrow P_3\)
  • \(R_4 \rightarrow P_4\)
• Claim Edges:
  • \(P_1 \rightarrow R_2\)
  • \(P_2 \rightarrow R_3\)
  • \(P_3 \rightarrow R_1\)
  • \(P_2 \rightarrow R_4\)
  • \(P_4 \rightarrow R_2\)

A system is said to be in a deadlock state if there exists a cycle in the resource allocation graph. The cycle here can be detailed as:

• Cycle 1: \(P_1 \rightarrow R_2 \rightarrow P_2 \rightarrow R_3 \rightarrow P_3 \rightarrow R_1 \rightarrow P_1\)
• Cycle 2: \(P_2 \rightarrow R_4 \rightarrow P_4 \rightarrow R_2 \rightarrow P_2\)

To break these cycles, we need to consider the effects of aborting different processes:

• Option 1: Aborting \(P_1\)
  • Breaks Cycle 1 by removing \(P_1\) from the graph, but Cycle 2 persists.
• Option 2: Aborting \(P_3\)
  • Breaks part of Cycle 1, but Cycle 2 persists, and a portion of Cycle 1 can still exist via \(P_1\rightarrow R_2 \rightarrow P_2\rightarrow R_4\rightarrow P_4\).
• Option 3: Aborting \(P_2\)
  • Breaks both Cycle 1 and Cycle 2 effectively.
• Option 4: Aborting \(P_1\) and \(P_4\)
  • Breaks both Cycle 1 and Cycle 2 by removing all dependencies associated with the deadlock cycles.

Hence, the correct deadlock-breaking choices are:

• Aborting \(P_2\) makes the system deadlock-free.
• Aborting \(P_1\) and \(P_4\) makes the system deadlock-free.