# computability theory Published articles for computability theory. This is one page of public article previews, not the complete archive. Follow Next page to continue. Summaries are not the original full articles. ## Kolmogorov Complexity--A Primer DevFeed: [Kolmogorov Complexity--A Primer]() Original publisher: [Read original article]() Published: 2012-04-21T18:23:41Z Content type: article Language: en Sources: [Jeremy Kun]() Topics: [Algorithms, Complexity](), [Math and Logic]() Tags: [complexity-theory](), [computability-theory](), [computational-complexity](), [kolmogorov-complexity](), [mathematics](), [primer]() ### AI overview This primer introduces Kolmogorov complexity as a way to formalize how random or complex a string is. It defines the concept using the length of the shortest program that outputs a string, then begins developing a more rigorous formulation using binary programs and Turing-machine concepts. ### Source excerpt The Complexity of Things Previously on this blog (quite a while ago), we've investigated some simple ideas of using randomness in artistic design (psychedelic art, and earlier randomized css designs). Here we intend to give a more thorough and rigorous introduction to the study of the complexity of strings. This naturally falls into the realm of computability theory and complexity theory, and so we refer the novice reader to our other primers on the subject (Determinism and Finite Automata, Turing Machines, and Complexity Classes; but Turing machines will be the most critical to this discussion). ## Turing Machines--A Primer DevFeed: [Turing Machines--A Primer]() Original publisher: [Read original article]() Published: 2011-07-04T16:35:44Z Content type: tutorial Language: en Sources: [Jeremy Kun]() Topics: [Automaton](), [function]() Tags: [acceptance](), [computability-theory](), [computers](), [concepts](), [decidability](), [halting-problem](), [input](), [machine](), [mathematics](), [primer](), [process](), [pushdown-automata](), [state](), [symbols](), [turing-machines]() ### AI overview This tutorial introduces Turing machines as a model of computation. It defines their states, tape symbols, input symbols, initial state, accepting and rejecting states, and transition function, then explains how the tape, read-write head, state transitions, and halting behavior work. ### Source excerpt We assume the reader is familiar with the concepts of determinism and finite automata, or has read the corresponding primer on this blog. The Mother of All Computers Last time we saw some models for computation, and saw in turn how limited they were. Now, we open Pandrora's hard drive: Definition: A Turing machine is a tuple $ (S, \Gamma, \Sigma, s_0, F, \tau)$, where $ S$ is a set of states, $ \Gamma$ is a set of tape symbols, including a special blank symbol $ b$, $ \Sigma \subset \Gamma$ is a set of input symbols, not including $ b$, $ s_0$ is the initial state, $ A \subset S$ is a set of accepting states, $ R \subset S$ is a set of rejecting states, $ \tau: S - (A \cup R) \times \Gamma \to S \times \Gamma \times \left \{ L, R \right \}$ is a partial function called the transition function, where $ L, R$ correspond to "left shift" and "right shift," respectively. ## Determinism and Finite Automata--A Primer DevFeed: [Determinism and Finite Automata--A Primer]() Original publisher: [Read original article]() Published: 2011-07-02T23:41:30Z Content type: tutorial Language: en Sources: [Jeremy Kun]() Topics: [Automaton](), [function]() Tags: [computability-theory](), [computing](), [context-free-languages](), [deterministic-finite-automata](), [finite-state-machines](), [function](), [input](), [language](), [mathematics](), [nondeterministic-finite-automata](), [primer](), [pushdown-automata](), [regular-languages](), [string](), [turing-machines]() ### AI overview This primer introduces computation as a function that maps input to output, then develops a more rigorous model based on finite alphabets and strings. It explains how a computational model recognizes a language by accepting or rejecting strings. ### Source excerpt The first step in studying the sorts of possible computations (and more interestingly, those things which cannot be computed) is to define exactly what we mean by a "computation." At a high level, this is easy: a computation is simply a function. Given some input, produce the appropriate output. Unfortunately this is much too general. For instance, we could define almost anything we want in terms of functions. Let $ f$ be the function which accepts as input the date of California Super Lotto drawings, and returns the set of winning numbers for that date. ## Turing Machines and Conway's Dreams DevFeed: [Turing Machines and Conway's Dreams]() Original publisher: [Read original article]() Published: 2011-06-30T20:50:04Z Content type: tutorial Language: en Sources: [Jeremy Kun]() Topics: [Automaton]() Tags: [cellular](), [cellular-automata](), [computability-theory](), [computational-complexity](), [conway](), [infinite](), [life](), [mathematics](), [patterns](), [period](), [programming](), [recursion](), [theory](), [turing-machine](), [turing-machines]() ### AI overview The article examines Conway's Game of Life patterns that do not stabilize, including Gosper's glider gun and puffers that produce continuing activity. It connects these patterns to computability, explaining that infinite looping is necessary for Turing-complete computation. ### Source excerpt Additional Patterns Last time we left the reader with the assertion that Conway's game of life does not always stabilize. Specifically, there exist patterns which result in unbounded cell population growth. Although John Conway's original conjecture was that all patterns eventually stabilize (and offered $50 to anyone who could provide a proof or counterexample), he was proven wrong. Here we have the appropriately named glider gun, whose main body oscillates, expelling a glider once per period. ## Introduction to Cellular Automata DevFeed: [Introduction to Cellular Automata]() Original publisher: [Read original article]() Published: 2011-06-29T21:08:45Z Content type: tutorial Language: en Sources: [Jeremy Kun]() Topics: [Cellular automaton](), [Automaton](), [Finite-state machine]() Tags: [cellular-automata](), [computability-theory](), [conus](), [conway](), [life](), [mathematica](), [mathematics](), [patterns](), [programming](), [rules](), [simulation](), [turing-machine]() ### AI overview An introductory tutorial on cellular automata, explaining cells, states, transition rules, simultaneous updates, and a one-dimensional binary example. ### Source excerpt Cellular Automata There is a long history of mathematical models for computation. One very important one is the Turing Machine, which is the foundation of our implementations of actual computers today. On the other end of the spectrum, one of the simpler models of computation (often simply called a system) is a cellular automaton. Surprisingly enough, there are deep connections between the two. But before we get ahead of ourselves, let's see what these automata can do.