# Turing Machines and Turing machines AB Ch 1

Turing, Machines, and, Turing machines. [AB, Ch 1, 3]

The Digital Era A Visualization of the various routes through a portion of the Internet.

The Jacquard loom (1801) The Jacquard loom, on display at the Museum of Science and Industry in Manchester, England, was one of the first programmable devices.

Bomba (1938) Cryptologic bomb. Diagram from Marian Rejewski's papers. 1: Rotors (for clarity, only one 3 -rotor set is shown). 2: Electric motor. 3: Switches.

The Columbia supercomputer

The Weather Eye of Hurricane Isabel from the International Space Station, September 15, 2003.

Cup of tea

What is a computer? 1. A computer is a programmable machine that receives input, stores and manipulates data, and provides output in a useful format. 2. A computer is any physical process that acts on input and changes its configuration.

Alan Turing (1912 -1954)

Alan Turing 1918 -1930 – School 1930 - Turing's hopes and ambitions at school were raised by the close friendship he developed with a slightly older fellow student, Christopher Morcom, who was Turing's first love interest. Morcom died suddenly only a few weeks into their last term at Sherborne, from complications of bovine tuberculosis, contracted after drinking infected cow's milk as a boy. [13] Turing's religious faith was shattered and he became an atheist. He adopted the conviction that all phenomena, including the workings of the human brain, must be materialistic. [14]

Alan Turing 1931 – 1934 Kings College 1936 - In his momentous paper "On Computable Numbers, with an Application to the Entscheidungsproblem "[15] (submitted on 28 May 1936), Turing reformulated Kurt Gödel's 1931 results on the limits of proof and computation, replacing Gödel's universal arithmetic-based formal language with what are now called Turing machines, formal and simple devices. He proved that some such machine would be capable of performing any conceivable mathematical computation if it were representable as an algorithm. … the halting problem for Turing machines is undecidable

Alan Turing 1936 – 1938 Princeton. Working with Church. his dissertation introduced the notion of relative computing, where Turing machines are augmented with so-called oracles, allowing a study of problems that cannot be solved by a Turing machine.

Alan Turing 1938 – 1945 Cryptanalysis. UK, USA. • There should be no question in anyone's mind that Turing's work was the biggest factor in Hut 8's success. In the early days he was the only cryptographer who thought the problem worth tackling and not only was he primarily responsible for the main theoretical work within the Hut but he also shared with Welchman and Keen the chief credit for the invention of the Bombe. It is always difficult to say that anyone is absolutely indispensable but if anyone was indispensable to Hut 8 it was Turing. The pioneer's work always tends to be forgotten when experience and routine later make everything seem easy and many of us in Hut 8 felt that the magnitude of Turing's contribution was never fully realized by the outside world. —Alexander, Sir C. Hugh O'D. Cryptographic History of Work on the German Naval Enigma. The National Archives, Kew, Reference HW 25/1.

Alan Turing In 1945, Turing was awarded the OBE for his wartime services

Alan Turing 1945 -1947, National Physical Laboratory, the first detailed design of a stored-program computer. 1948 -. . proposed an experiment now known as the Turing test, an attempt to define a standard for a machine to be called "intelligent". The idea was that a computer could be said to "think" if it could fool an interrogator into thinking that the conversation was with a human.

Alan Turing In January 1952 Turing picked up 19 -year-old Arnold Murray outside a cinema in Manchester. After a lunch date, Turing invited Murray to spend the weekend with him at his house, an invitation which Murray accepted although he did not show up. The pair met again in Manchester the following Monday, when Murray agreed to accompany Turing to the latter's house. A few weeks later Murray visited Turing's house again, and apparently spent the night there. [36] After Murray helped an accomplice to break into his house, Turing reported the crime to the police. During the investigation, Turing acknowledged a sexual relationship with Murray. Homosexual acts were illegal in the United Kingdom at that time, [6] and so both were charged with gross indecency under Section 11 of the Criminal Law Amendment Act 1885, the same crime that Oscar Wilde had been convicted of more than fifty years earlier. [37] Turing was given a choice between imprisonment or probation conditional on his agreement to undergo hormonal treatment designed to reduce libido. He accepted chemical castration via oestrogen hormone injections, [38] one of the side effects of which was that he grew breasts. [37]

Alan Turing On 8 June 1954, Turing's cleaner found him dead; he had died the previous day. A post-mortem examination established that the cause of death was cyanide poisoning. When his body was discovered an apple lay half-eaten beside his bed, and although the apple was not tested for cyanide, [41] it is speculated that this was the means by which a fatal dose was delivered. An inquest determined that he had committed suicide, and he was cremated at Woking crematorium on 12 June 1954.

2009 - Apology Thousands of people have come together to demand justice for Alan Turing and recognition of the appalling way he was treated. While Turing was dealt with under the law of the time and we can't put the clock back, his treatment was of course utterly unfair and I am pleased to have the chance to say how deeply sorry I and we all are for what happened to him. . . So on behalf of the British government, and all those who live freely thanks to Alan's work I am very proud to say: we're sorry, you deserved so much better.

TM: Formally Q finite set of states • q 0 start state • qacc Q accept state • qreject qaccept qrej Q reject state •

b Turing Machine: Schematic Moves: a a b a a left b right _ _ Infinite Tape, Not random access 20

TM: Formally Q finite set of states • q 0 start state • qacc Q accept state • qrej Q reject state • input alphabet : a finite set • tape alphabet • qreject qaccept Excluding “_” and _ : Q Q {L, R} - the transition function • 21

q 0 Computations Initial Configuration ” • For input “ Head On leftmost cell • a a b _ _ State starting • Input – from left 23

q 0 Computation Step q 6 (q 0, a)=(q 6, b, R) aa aa b b __ __ 24

L={a nbncn | n 0} My first TM Examples: Q • = {q 0, q 1, q 2, q 3, q 4, qaccept , qreject } • = {a, b, c} q 0 ccc f L: aaabbb o r e b m e M : aaabbcccc L f o r e b m Non-Me • = {a, b, c, _, X, Y, Z} • specified next. . . • - the start state. qacc • Q - the accept state. qrej • Q - the reject state. 25

The Transitions Function a a, R Y Y, R a X, R q 0 q 1 b Y, R Y Y, R q 3 qac _ _, R q 2 c Z, L X X, R _ _, R b b, R Z Z, R b b, L a a, L Y Y, L Z Z, L q 4 Y Y, R Z Z, R Complexity ©D. Moshkovitz 26

The thing to remember Turing machines (algorithms) are objects : • with constant description size, and yet, • They work on inputs of arbitrary size. A uniform model of computation.

Multi-Tape TMs Q finite set of states • input alphabet : a finite set • tape alphabet • q 0 start state • qacc ” Excluding “ _ and _ : Q k Q ( {L, R})k - the transition function • pes- is ta f o r e b m u n e h -t k some constant Q accept state • qrej Q reject state • qreject qaccept 28

Multi-Tape Turing Machines At start: input on first tape a _ _ b _ _ _ - a b - - - 29

The universal TM Thm: There exists a TM U, such that U(x, k)=Mk(x) If Mk halts on x within T steps, then U(x, k) halts within O(T log T) steps. The hidden constant depends only on Mk’s alphabet size, number of tapes and number of states, and does not depend on the input length.

Languages Def: A language is some subset S {0, 1}* Example: S={anbncn | n N} Def: A Turing machine (TM) accepts L if it accepts every x L and rejects every x L. In particular it must terminate.

Time complexity Def: The running time of T on input x is the number of δ transitions from the initial state to an accept/reject state.

Time-Complexity Definition : • Let t: N N be a complexity function Deterministic time : Det. Polynomial time : • EXP P Det Exponential time : • 33

Diagonalization Theorem Proof : : • P EXPTIME • We construct a language L EXPTIME, which is not accepted by any TM running in polynomial time: EXPTIME P 34

P vs EXPTIME Lemma : EXPTIME • L EXPTIME Proof : P • L can be decided in time poly(|x||x|) • By running the universal TM. Lemma : • L P Proof : • Assume a TM M that determines L in time cne run it on the string “<M>#1 c#1 e#” contradiction 35

Time Hierarchy Theorem Def: f: N N is time-constructible , if there exists a TM M Time(T(n)) s. t. M(x)=[T(|x|)]. Thm: If • f, g : N N are time-constructible, and • f(n)log f(n) = o(g(n)) Then: Time(f(n)) Time(g(n))

Growth Rate: rough classification time n! =2 O(n lg n) 2 n n 2 10 n input length 37

Basic split in time-complexity Maybe reasonable polynomial n. O(1) Totally unreasonable exponential 2 n. O(1) 38

Name the Class Hamiltonian Cycle Halting Problem Perfect Matching Minimum Spanning Tree EXP anb nc n P 39

The Church-Turing thesis “Everything computable is computable by a Turing machine. " Though not formally proven, today thesis has near-universal acceptance.

The Physical Church-Turing thesis All physically computable functions are Turing computable

The Strong Church-Turing thesis "A (probabilistic) Turing machine can efficiently simulate any realistic model of computation. “ "A probabilistic Turing machine can efficiently simulate any physical process. “

New Evidence that Quantum Mechanics is Hard to Simulate on Classical Computers I'll discuss new types of evidence that quantum mechanics is hard to simulate classically -- evidence that's more complexity-theoretic in character than (say) Shor's factoring algorithm, and that also corresponds to experiments that seem easier than building a universal quantum computer. Specifically: (1) I'll show that, by using linear optics (that is, systems of non-interacting bosonic particles), one can generate probability distributions that can't be efficiently sampled by a classical computer, unless P^#P = BPP^NP and hence the polynomial hierarchy collapses. The proof exploits an old observation: that computing the amplitude for n bosons to evolve to a given configuration involves taking the Permanent of an n-by-n matrix. I'll also discuss an extension of this result to samplers that only approximate the boson distribution. (Based on recent joint work with Alex Arkhipov) (2) Time permitting, I'll also discuss new oracle evidence that BQP has capabilities outside the entire polynomial hierarchy. (ar. Xiv: 0910. 4698)

“Can machines Think? ” Turing (1950): I PROPOSE to consider the question, 'Can machines think? ’. The question of whether it is possible for machines to think has a long history, which is firmly entrenched in the distinction between dualist and materialist views of the mind. From the perspective of dualism, the mind is non-physical (or, at the very least, has non-physical properties[6]) and, therefore, cannot be explained in purely physical terms. The materialist perspective argues that the mind can be explained physically, and thus leaves open the possibility of minds that are artificially produced. [7] Will there be digital computers that are indistinguishable from human beings?

What are we? • Was Alan Turing a computer mistreated by other computers? • Will there ever be a computer passing Turing’s test? • Can everything in our universe be captured as computation? • Is there free choice?

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