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How does the halting problem illustrate the limitations of algorithmic computability, and what implications does it have for the development of decision procedures in formal languages?

In the domain of complexity theory, how does the class NP

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complete relate to the class P

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Explain the significance of the lambda calculus in the foundations of functional programming languages, and discuss its equivalence to Turing machines in terms of computational expressiveness.

Discuss the concept of a universal Turing machine and its role in demonstrating the universality of computation, including its impact on modern computing architectures.

How does the concept of information entropy relate to the efficiency of data transmission in communication networks, and what are the practical applications of this relationship in network design?

In distributed systems, what are the key challenges of achieving consensus in the presence of Byzantine faults, and how do algorithms like Paxos and Raft address these challenges?

Analyze the impact of quantum superposition and entanglement on the computational power of quantum computers, particularly in relation to classical algorithms.

Discuss the role of automata theory in the design and analysis of formal languages, with a focus on the differences between deterministic and nondeterministic finite automata.

Explain the significance of the von Neumann architecture in the development of modern computers, and discuss how it compares to alternative computing models such as parallel and distributed computing.

How do probabilistic algorithms differ from deterministic algorithms in terms of performance and accuracy, and what are some key applications where probabilistic algorithms provide significant advantages?