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--- title: "Theory of Computing" source: https://www.jemoka.com/posts/kbhtheory_of_computing/ --- Theory of Computing is a science which attempts to identify “what the most efficient way to solve a given computational task?” “efficient” …with respect to what resources? time space/memory randomness communication / interaction quantum-ness (https://theory.stanford.edu/~abouland/) “most” study of impossibilities: lower bounds For instance, we want a result like “SAT cannot be solved in Polynomial Time” (then in which case P != NP) history of the theory of computing computability theory computability theorists consider themselves to deal with the problem of: “what computational tasks can be solved at all?” We know that because of the halting problem, not everything is solvable. (Turing 1936) complexity theory Its a subclass of computability theory, but only with respect to problems that can be solve. grand challenges of complexity theory P vs. NP (belief P != NP): if the solution of a particular computational task can be verified quickly, does this mean the solution can be found quickly? NP vs. coNP (belief NP != coNP): are there theorems that are simple to state but require lengthy proofs? P vs. NC (belief P != NP): is every algorithm efficiently parallelizable? or are there inherently sequential tasks? P vs. L (belief P != L): can every P algorithm be recompiled to use “essentially no memory” P vs. PSPACE (belief P != PSPACE): does solvable without much memory implies solvable without much time P vs. BPP (belief P == BPP): can every efficient randomized algorithms be deterministic (or are there only fast random algorithms but no fast deterministic algorithms?)