# turing machines Published articles for turing machines. This is one page of public article previews, not the complete archive. Follow Next page to continue. Summaries are not the original full articles. ## Methods of Proof -- Diagonalization DevFeed: [Methods of Proof -- Diagonalization]() Original publisher: [Read original article]() Published: 2015-06-08T09:00:00Z Content type: tutorial Language: en Sources: [Jeremy Kun]() Topics: [Mathematics](), [Math and Logic](), [Collections]() Tags: [bijections](), [cardinality](), [diagonalization](), [halting-problem](), [mathematics](), [methods-of-proof](), [turing-machines](), [uncountability]() ### AI overview This tutorial introduces diagonalization as an advanced method of mathematical proof. It explains the table-and-diagonal construction and presents the theorem that no bijection exists between the natural numbers and the real numbers. ### Source excerpt A while back we featured a post about why learning mathematics can be hard for programmers, and I claimed a major issue was not understanding the basic methods of proof (the lingua franca between intuition and rigorous mathematics). I boiled these down to the "basic four," direct implication, contrapositive, contradiction, and induction. But in mathematics there is an ever growing supply of proof methods. There are books written about the "probabilistic method," and I recently went to a lecture where the "linear algebra method" was displayed. ## Information Distance -- A Primer DevFeed: [Information Distance -- A Primer]() Original publisher: [Read original article]() Published: 2012-12-04T08:30:08Z Content type: tutorial Language: en Sources: [Jeremy Kun]() Topics: [Compression](), [Machine learning]() Tags: [compression](), [computing](), [information-theory](), [kolmogorov-complexity](), [machine-learning](), [metric](), [turing-machines](), [universality]() ### AI overview This primer introduces information distance through Kolmogorov complexity. It explains how the shortest program relating two strings can reflect their shared information and behave similarly to a metric, with applications to compression and machine learning. ### Source excerpt This post assumes familiarity with our primer on Kolmogorov complexity. We recommend the uninformed reader begin there. We will do our best to keep consistent notation across both posts. Kolmogorov Complexity as a Metric Over the past fifty years mathematicians have been piling up more and more theorems about Kolmogorov complexity, and for good reason. One of the main interpretations of the Kolmogorov complexity function $ K$ is that for a given string $ x$, $ K(x)$ is the best theoretical compression of $ x$ under any compression scheme. ## Other Complexity Classes DevFeed: [Other Complexity Classes]() Original publisher: [Read original article]() Published: 2012-02-29T17:12:35Z Content type: article Language: en Sources: [Jeremy Kun]() Topics: [Algorithms, Complexity](), [Computing](), [Math and Logic]() Tags: [complexity](), [computational-complexity](), [halting-problem](), [kolmogorov-complexity](), [nondeterminism](), [open-questions](), [oracle](), [primer](), [pspace](), [turing-machine](), [turing-machines]() ### AI overview This article introduces complexity classes beyond P and NP, including PSPACE and NPSPACE, and discusses oracle-based classes. It explains that PSPACE equals NPSPACE while many relationships among complexity classes remain open. ### Source excerpt Not Just Time, But Space Too! So far on this blog we've introduced models for computation, focused on Turing machines and given a short overview of the two most fundamental classes of problems: P and NP. While the most significant open question in the theory of computation is still whether P = NP, it turns out that there are hundreds (almost 500, in fact!) other "classes" of problems whose relationships are more or less unknown. ## P vs. NP, A Primer (And a Proof Written in Racket) DevFeed: [P vs. NP, A Primer (And a Proof Written in Racket)]() Original publisher: [Read original article]() Published: 2012-02-23T19:57:35Z Content type: tutorial Language: en Sources: [Jeremy Kun]() Topics: [Computing](), [Algorithms, Complexity](), [Racket](), [Programming]() Tags: [algorithm](), [algorithms](), [analysis](), [big-o-notation](), [class](), [computational-complexity](), [computing](), [decidability](), [efficiency](), [fast](), [halting-problem](), [input](), [language](), [machine](), [np-completeness](), [p-versus-np](), [primer](), [racket](), [solve](), [theory](), [turing-machine](), [turing-machines]() ### AI overview A primer on the distinction between decidability and computational efficiency, introducing the complexity class P and its definition in terms of polynomial-time decision by a Turing machine. It also discusses efficient algorithms and gives a card-sorting example. ### Source excerpt Decidability Versus Efficiency In the early days of computing theory, the important questions were primarily about decidability. What sorts of problems are beyond the power of a Turing machine to solve? As we saw in our last primer on Turing machines, the halting problem is such an example: it can never be solved a finite amount of time by a Turing machine. However, more recently (in the past half-century) the focus of computing theory has shifted away from possibility in favor of determining feasibility. ## 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.