Problem P₁:

"Does a given finite state machine accept a given string?"

Analysis of P₁:

Finite State Machines (FSMs):

Can be deterministic (DFA) or nondeterministic (NFA).

Both DFAs and NFAs have a finite number of states and transitions.

Decidability:

Decidable Problem: A problem for which an algorithm can provide a yes/no answer for every input instance in finite time.

Algorithm for P₁:

For DFAs:

Simulate the DFA on the input string by transitioning through states according to the input symbols.

If the DFA reaches an accepting state after processing the entire string, it accepts the string.

The simulation takes time proportional to the length of the input string.

For NFAs:

Since NFAs can have multiple possible transitions for a given input symbol, we can:

Use a depth-first or breadth-first search to explore all possible paths.

Alternatively, convert the NFA to an equivalent DFA (subset construction), though this may increase the number of states exponentially.

As both the NFA and the string are finite, the simulation will terminate.

Conclusion for P₁:

Decidable.

There exists an algorithm that can determine whether the FSM accepts the given string in finite time.

Problem P₂:

"Does a given context-free grammar generate an infinite number of strings?"

Analysis of P₂:

Context-Free Grammars (CFGs):

Consist of a finite set of production rules that generate strings over some alphabet.

May produce finite or infinite languages.

Decidability:

Decidable Problem: Existence of an algorithm to determine whether the language generated by a CFG is infinite.

Algorithm for P₂:

Step 1: Identify Generating Non-Terminals

Useful Symbols:

Non-terminals that can derive terminal strings.

Compute the set of symbols that derive terminal strings using a fix-point algorithm.

Step 2: Build the Dependency Graph

Dependency Graph:

Nodes represent non-terminals.

An edge from $ A $ to $ B $ exists if $ A $ can derive a string containing $ B $.

Step 3: Detect Cycles Involving Useful Symbols

Cycles:

A cycle involving useful symbols indicates the possibility of generating strings of unbounded length.

Use graph traversal algorithms (e.g., DFS) to detect cycles.

Step 4: Determine Unbounded Derivations

If the start symbol can reach itself through a cycle involving useful symbols, the language is infinite.

Otherwise, if all derivations are acyclic or cycles do not lead to increased string length, the language is finite.

Conclusion for P₂:

Decidable.

The algorithm terminates after finite steps and determines whether the CFG generates infinitely many strings.

Final Conclusion:

P₁ is decidable: We can simulate the FSM on the input string.

P₂ is decidable: We can analyze the CFG to determine if it generates an infinite language.

Therefore, both P₁ and P₂ are decidable.

Answer: Option A