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Text of the page (random words):
able functions antedated but has not much influenced the extensive actual construction of digital computers these two aspects of theory and practice have been developed almost entirely independently of each other the main reason is undoubtedly that logicians are interested in questions radically different from those with which the applied mathematicians and electrical engineers are primarily concerned it cannot however fail to strike one as rather strange that often the same concepts are expressed by very different terms in the two developments wang 1954 1957 63 wang hoped that his paper would connect the two approaches indeed minsky confirms this that the first formulation of turing machine theory in computer like models appears in wang 1957 minsky 1967 200 minsky goes on to demonstrate turing equivalence of a counter machine with respect to the reduction of computers to simple turing equivalent models and vice versa minsky s designation of wang as having made the first formulation is open to debate while both minsky s paper of 1961 and wang s paper of 1957 are cited by shepherdson and sturgis 1963 they also cite and summarize in some detail the work of european mathematicians kaphenst 1959 ershov 1959 and péter 1958 the names of mathematicians hermes 1954 1955 1961 and kaphenst 1959 appear in the bibliographies of both sheperdson sturgis 1963 and elgot robinson 1961 two other names of importance are canadian researchers melzak 1961 and lambek 1961 for much more see turing machine equivalents references can be found at register machine mathematical theory edit with this encoding of action tables as strings it becomes possible in principle for turing machines to answer questions about the behaviour of other turing machines most of these questions however are undecidable meaning that the function in question cannot be calculated mechanically for instance the problem of determining whether an arbitrary turing machine will halt on a particular input or on all inputs known as the halting problem was shown to be in general undecidable in turing s original paper rice s theorem shows that any non trivial question about the output of a turing machine is undecidable a universal turing machine can calculate any recursive function decide any recursive language and accept any recursively enumerable language according to the church turing thesis the problems solvable by a universal turing machine are exactly those problems solvable by an algorithm or an effective method of computation for any reasonable definition of those terms for these reasons a universal turing machine serves as a standard against which to compare computational systems and a system that can simulate a universal turing machine is called turing complete an abstract version of the universal turing machine is the universal function a computable function which can be used to calculate any other computable function the utm theorem proves the existence of such a function efficiency edit without loss of generality the input of turing machine can be assumed to be in the alphabet 0 1 any other finite alphabet can be encoded over 0 1 the behavior of a turing machine m is determined by its transition function this function can be easily encoded as a string over the alphabet 0 1 as well the size of the alphabet of m the number of tapes it has and the size of the state space can be deduced from the transition function s table the distinguished states and symbols can be identified by their position e g the first two states can by convention be the start and stop states consequently every turing machine can be encoded as a string over the alphabet 0 1 additionally we convene that every invalid encoding maps to a trivial turing machine that immediately halts and that every turing machine can have an infinite number of encodings by padding the encoding with an arbitrary number of say 1 s at the end just like comments work in a programming language it should be no surprise that we can achieve this encoding given the existence of a gödel number and computational equivalence between turing machines and μ recursive functions similarly our construction associates to every binary string α a turing machine m α starting from the above encoding in 1966 f c hennie and r e stearns showed that given a turing machine m α that halts on input x within n steps then there exists a multi tape universal turing machine that halts on inputs α x given on different tapes in cn log n where c is a machine specific constant that does not depend on the length of the input x but does depend on m s alphabet size number of tapes and number of states effectively this is an o n log n displaystyle mathcal o left n log n right simulation using donald knuth s big o notation 5 the corresponding result for space complexity rather than time complexity is that we can simulate in a way that uses at most cn cells at any stage of the computation an o n displaystyle mathcal o n simulation 6 smallest machines edit when alan turing came up with the idea of a universal machine he had in mind the simplest computing model powerful enough to calculate all possible functions that can be calculated claude shannon first explicitly posed the question of finding the smallest possible universal turing machine in 1956 he showed that two symbols were sufficient so long as enough states were used or vice versa and that it was always possible to exchange states for symbols he also showed that no universal turing machine of one state could exist marvin minsky discovered a 7 state 4 symbol universal turing machine in 1962 using 2 tag systems other small universal turing machines have since been found by yurii rogozhin and others by extending this approach of tag system simulation if we denote by m n the class of utms with m states and n symbols the following tuples have been found 15 2 9 3 6 4 5 5 4 6 3 9 and 2 18 7 8 9 rogozhin s 4 6 machine uses only 22 instructions and no standard utm of lesser descriptional complexity is known however generalizing the standard turing machine model admits even smaller utms one such generalization is to allow an infinitely repeated word on one or both sides of the turing machine input thus extending the definition of universality and known as semi weak or weak universality respectively small weakly universal turing machines that simulate the rule 110 cellular automaton have been given for the 6 2 3 3 and 2 4 state symbol pairs 10 the proof of universality for wolfram s 2 state 3 symbol turing machine further extends the notion of weak universality by allowing certain non periodic initial configurations other variants on the standard turing machine model that yield small utms include machines with multiple tapes or tapes of multiple dimension and machines coupled with a finite automaton machines with no internal states edit if multiple heads are allowed on a turing machine then no internal states are required as states can be encoded in the tape for example consider a tape with 6 colours 0 1 2 0a 1a 2a consider a tape such as 0 0 1 2 2a 0 2 1 where a 3 headed turing machine is situated over the triple 2 2a 0 the rules then convert any triple to another triple and move the 3 heads left or right for example the rules might convert 2 2a 0 to 2 1 0 and move the head left thus in this example the machine acts like a 3 colour turing machine with internal states a and b represented by no letter the case for a 2 headed turing machine is very similar thus a 2 headed turing machine can be universal with 6 colours it is not known what the smallest number of colours needed for a multi headed turing machine are or if a 2 colour universal turing machine is possible with multiple heads it also means that rewrite rules are turing complete since the triple rules are equivalent to rewrite rules extending the tape to two dimensions with a head sampling a letter and its 8 neighbours only 2 colours are needed as for example a colour can be encoded in a vertical triple pattern such as 110 example of universal machine coding edit for those who would undertake the challenge of designing a utm exactly as turing specified see the article by davies in copeland 2004 103ff davies corrects the errors in the original and shows what a sample run would look like he claims to have successfully run a somewhat simplified simulation the following example is taken from turing 1936 for more about this example see turing machine examples turing used seven symbols a c d r l n to encode each 5 tuple as described in the article turing machine his 5 tuples are only of types n1 n2 and n3 the number of each m configuration instruction state is represented by d followed by a unary string of a s e g q3 daaa in a similar manner he encodes the symbols blank as d the symbol 0 as dc the symbol 1 as dcc etc the symbols r l and n remain as is after encoding each 5 tuple is then assembled into a string in order as shown in the following table current m configuration tape symbol print operation tape motion final m configuration current m configuration code tape symbol code print operation code tape motion code final m configuration code 5 tuple assembled code q1 blank p0 r q2 da d dc r daa daddcrdaa q2 blank e r q3 daa d d r daaa daaddrdaaa q3 blank p1 r q4 daaa d dcc r daaaa daaaddccrdaaaa q4 blank e r q1 daaaa d d r da daaaaddrda finally the codes for all four 5 tuples are strung together into a code started by and separated by i e daddcrdaa daaddrdaaa daaaddccrdaaaa daaaaddrda this code he placed on alternate squares the f squares leaving the e squares those liable to erasure empty the final assembly of the code on the tape for the u machine consists of placing two special symbols e one after the other then the code separated out on alternate squares and lastly the double colon symbol blanks shown here with for clarity ee d a d d c r d a a d a a d d r d a a a d a a a d d c c r d a a a a d a a a a d d r d a the u machine s action table state transition table is responsible for decoding the symbols turing s action table keeps track of its place with markers u v x y z by placing them in e squares to the right of the marked symbol for example to mark the current instruction z is placed to the right of x is keeping the place with respect to the current m configuration daa the u machine s action table will shuttle these symbols around erasing them and placing them in different locations as the computation progresses ee d a d d c r d a a z d a a x d d r d a a a d a a a d d c c r d a a a a d a a a a d d r d a turing s action table for his u machine is very involved a number of other commentators notably penrose 1989 provide examples of ways to encode instructions for the universal machine as does penrose most commentators use only binary symbols i e only symbols 0 1 or blank mark penrose goes further and writes out his entire u machine code penrose 1989 71 73 he asserts that it truly is a u machine code an enormous number that spans almost 2 full pages of 1 s and 0 s for readers interested in simpler encodings for the post turing machine the discussion of davis in steen steen 1980 251ff may be useful asperti and ricciotti described a multi tape utm defined by composing elementary machines with very simple semantics rather than explicitly giving its full action table this approach was sufficiently modular to allow them to formally prove the correctness of the machine in the matita proof assistant programming turing machines edit various higher level languages are designed to be compiled into a turing machine examples include laconic and turing machine descriptor 11 12 see also edit alternating turing machine von neumann universal constructor an attempt to build a self replicating turing machine kleene s t predicate a similar concept for µ recursive functions turing completeness references edit martin davis the universal computer the road from leibniz to turing 2017 arora and barak 2009 theorem 1 9 boldface replacing script turing 1936 in davis 1965 127 128 an example of turing s notion of s d is given at the end of this article in particular burks goldstine von neumann 1946 preliminary discussion of the logical design of an electronic computing instrument reprinted in bell and newell 1971 arora and barak 2009 theorem 1 9 arora and barak 2009 exercises 4 1 rogozhin 1996 kudlek and rogozhin 2002 neary and woods 2009 neary and woods 2009b shtetl optimized blog archive the 8000th busy beaver number eludes zf set theory new paper by adam yedidia and me www scottaaronson com 3 may 2016 retrieved 29 december 2016 laconic esolang esolangs org retrieved 29 december 2016 general references arora sanjeev barak boaz 2009 complexity theory a modern approach cambridge university press isbn 978 0 521 42426 4 section 1 4 machines as strings and the universal turing machine and 1 7 proof of theorem 1 9 original paper turing a m 1936 on computable numbers with an application to the entscheidungsproblem pdf seminal papers hennie f c stearns r e 1966 two tape simulation of multitape turing machines journal of the acm 13 4 533 doi 10 1145 321356 321362 s2cid 2347143 implementation kamvysselis kellis manolis 1999 scheme implementation of a universal turing machine self published formal verification asperti andrea ricciotti wilmer 2015 a formalization of multi tape turing machines pdf theoretical computer science elsevier 603 23 42 doi 10 1016 j tcs 2015 07 013 issn 0304 3975 other references copeland jack ed 2004 the essential turing seminal writings in computing logic philosophy artificial intelligence and artificial life plus the secrets of enigma oxford uk oxford university press isbn 0 19 825079 7 davis martin 1980 what is computation in steen lynn arthur ed mathematics today twelve informal essays new york vintage books random house isbn 978 0 394 74503 9 davis martin 2000 engines of logic mathematicians and the origin of the computer 1st ed new york ny w w norton company isbn 0 393 32229 7 pb goldstine herman h von neumann john planning and coding of the problems for an electronic computing instrument institute for advanced study rep 1947 ed princeton bell c gordon newell allen 1971 computer structures readings and examples reprinted ed new york mcgraw hill book company pp 92 119 isbn 0 07 004357 4 herken rolf 1995 the universal turing machine a half century survey springer verlag isbn 3 211 82637 8 knuth donald e 1973 the art of computer programming vol 1 fundamental algorithms second ed addison wesley publishing company the first of knuth s series of three texts kudlek manfred rogozhin yurii 2002 a universal turing machine with 3 states and 9 symbols in werner kuich grzegorz rozenberg arto salomaa eds developments in language theory 5th international conference dlt 2001 wien austria july 16 21 2001 revised papers lecture notes in computer science vol 2295 springer pp 311 318 doi 10 1007 3 540 46011 x_27 isbn 978 3 540 43453 5 minsky marvin 1962 size and structure of univ...