---
title: "Theory of Computing"
type: concept
related: [P Vs Bpp, Theory Of Computing]
source: https://www.jemoka.com/posts/kbhtheory_of_computing/
confidence: high
status: active
---

Theory of Computing is a science which attempts to identify &ldquo;what the most efficient way to solve a given computational task?&rdquo;
&ldquo;efficient&rdquo; &hellip;with respect to what resources?
time space/memory randomness communication / interaction quantum-ness (https://theory.stanford.edu/~abouland/) &ldquo;most&rdquo; study of impossibilities: lower bounds For instance, we want a result like &ldquo;SAT cannot be solved in Polynomial Time&rdquo; (then in which case P != NP)
history of the theory of computing computability theory computability theorists consider themselves to deal with the problem of: &ldquo;what computational tasks can be solved at all?&rdquo; 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 &ldquo;essentially no memory&rdquo; 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?)
