Don't solve this question otherwise by using AI $,$ I will give you $17$ dislike from my different account and your account will be revoked so don't do In the realm of computational intricacies, how does the Church$-$Turing thesis juxtapose the decidability problem with hypercomputational paradigms, and what ramifications does this have on the P vs NP conundrum?

Given a nondeterministic automaton operating over an alphabet of cardinality $,$ elucidate the implications of its transition function's surjectivity on the automaton's state space cardinality and the resultant language's topological properties.

In the context of algorithmic randomness, how does the Kolmogorov complexity of a string relate to its compressibility via universal coding schemes, and what does this infer about the string's entropy within the confines of Chaitin's incompleteness theorem?

Expound upon the interplay between G$$del$\text{'}$s incompleteness theorems and Tarski's undefinability theorem in the construction of self$-$referential formal systems, particularly within the framework of Peano arithmetic and first$-$order logic.

Within the spectrum of distributed consensus algorithms, how does the FLP impossibility result constrain the fault tolerance of asynchronous systems, particularly when considering Byzantine fault models and their probabilistic mitigations?

Examine the ramifications of Yao's principle in the context of communication complexity, especially when applied to the amortized analysis of randomized protocols for computing Boolean functions.

In the paradigm of quantum computing, how does the entanglement entropy of a bipartite quantum system influence the computational complexity class BQP$,$ and what implications does this have for quantum error correction and decoherence?

Delve into the intricacies of the Curry$-$Howard correspondence as it relates to the equivalence between proof systems in intuitionistic logic and type systems in lambda calculus, particularly focusing on the implications for functional programming languages.

Discuss the computational limits imposed by the arithmetical hierarchy on the expressibility of arithmetic predicates, especially in the context of $$ and $$ classes, and their relevance to the analytical hierarchy.

Analyze the implications of the PCP theorem on the approximability of combinatorial optimization problems, with a focus on how it redefines the hardness of approximation and the role of probabilistically checkable proofs in complexity theory.