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The Church- Turing Thesis contributed articles DOI:10.1145/3198448 new computing paradigms that do In its original form, the Church-Turing thesis breach the Church-Turing barrier, in which the uncomputable becomes com- concerned computation as Alan Turing putable, in an upgraded sense of “com- and Alonzo Church used the term in 1936— putable”? Before addressing these ques- human computation. tions, we first look back to the 1930s to consider how Alonzo Church and Alan Turing formulated, and sought to jus- BY B. JACK COPELAND AND ORON SHAGRIR tify, their versions of CTT. With this nec- essary history under our belts, we then turn to today’s dramatically more pow- erful versions of CTT. The Church- History of the Thesis Turing stated what we will call “Turing’s thesis” in various places and with vary- ing degrees of rigor. The following for- Turing Thesis: mulation is one of his most accessible. Turing’s thesis. “L.C.M.s [logical com- puting machines, Turing’s expression for Turing machines] can do anything Logical Limit that could be described as … ‘purely me- chanical’.”38 Turing also formulated his thesis in terms of numbers. For example, he or Breachable said, “It is my contention that these op- erations [the operations of an L.C.M.] include all those which are used in the computation of a number.”36 and Barrier? “[T]he ‘computable numbers’ include all numbers which would naturally be regarded as computable.”36 Church (who, like Turing, was work- ing on the German mathematician David Hilbert’s Entscheidungsproblem) advanced “Church’s thesis,” which he expressed in terms of definability in his lambda calculus. THE CHURCH-TURING THESIS (CTT) underlies tantalizing Church’s thesis. “We now define the open questions concerning the fundamental place notion … of an effectively calculable of computing in the physical universe. For example, key insights is every physical system computable? Is the universe ˽ The term “Church-Turing thesis” is used essentially computational in nature? What are the today for numerous theses that diverge significantly from the one Alonzo Church implications for computer science of recent speculation and Alan Turing conceived in 1936. about physical uncomputability? Does CTT place a ˽ The range of algorithmic processes studied in modern computer science fundamental logical limit on what can be computed, far transcends the range of processes a a computational “barrier” that cannot be broken, no “human computer” could possibly carry out. ˽ There are at least three forms of matter how far and in what multitude of ways computers the “physical Church-Turing thesis”— develop? Or could new types of hardware, based perhaps modest, bold, and super-bold—though, at the present stage of physical inquiry, on quantum or relativistic phenomena, lead to radically it is unknown whether any of them is true. 66 COMMUNICATIONS OF THE ACM | JANUARY 2019 | VOL. 62 | NO. 1 Is everything in the physical universe computable? Hubble Space Telescope view of the Pillars of Creation in the Eagle Nebula. function of positive integers by iden- nition,” Turing quickly proved that Church’s thesis have distinct meanings tifying it with the notion of a recursive λ-definability and his own concept of and so are different theses, since they function of positive integers (or of a computability (over positive integers) are not intensionally equivalent. A lead- λ-definable function of positive inte- are equivalent. Church’s thesis and Tur- ing difference in their meanings is that gers).”5 ing’s thesis are thus equivalent, if atten- Church’s thesis contains no reference Church chose to call this a definition. tion is restricted to functions of positive to computing machinery, whereas Tur- American mathematician Emil Post, on integers. (Turing’s thesis, more gen- ing’s thesis is expressed in terms of the the other hand, referred to Church’s the- eral than Church’s, also encompassed “Turing machine,” as Church dubbed it sis as a “working hypothesis” and criti- computable real numbers.) However, in his 1937 review of Turing’s paper. cized Church for masking it in the guise it is important for a computer scientist It is now widely understood that of a definition.33 to appreciate that despite this exten- Turing introduced his machines with IMAGE BY NASA, ESA, AND THE HUBBLE HERITAGE TEAM (STSCI/AURA) AND THE HUBBLE HERITAGE ESA, NASA, BY IMAGE Upon learning of Church’s “defi- sional equivalence, Turing’s thesis and the intention of providing an idealized JANUARY 2019 | VOL. 62 | NO. 1 | COMMUNICATIONS OF THE ACM 67 contributed articles description of a certain human activ- each going far beyond CTT-O. First, we Yanofsky in terms of equivalence class- ity—numerical computation; in Tur- look more closely at the algorithmic es of programs, while Moshe Vardi has ing’s day computation was carried out form of thesis, as stated to a first approx- speculated that an algorithm is both by rote workers called “computers,” or, imation by Lewis and Papadimitriou29: abstract-state machine and recursor. It sometimes, “computors”; see, for exam- “[W]e take the Turing machine to be a is also debated whether an algorithm ple, Turing.37 The Church-Turing thesis precise formal equivalent of the intuitive must be physically implementable. Mos- is about computation as the term was notion of ‘algorithm’.” chovakis and Vasilis Paschalis (among used in 1936—human computation. others) adopt a concept of algorithm “so Church’s term “effectively calculable What Is an Algorithm? wide as to admit ‘non-implementable’ function” was intended to refer to func- The range of algorithmic processes algorithms,”30 while other approaches tions that are calculable by an idealized studied in modern computer science do impose a requirement of physical im- human computer; and, likewise, Tur- far transcends the range of processes plementability, even if only a very mild ing’s phrase “numbers which would a Turing machine is able to carry out. one. David Harel, for instance, writes: naturally be regarded as computable” The Turing machine is restricted to, say, [A]ny algorithmic problem for which we was intended to refer to those numbers changing at most one bounded part at can find an algorithm that can be pro- that could be churned out, digit by digit, each sequential step of a computation. grammed in some programming lan- by an idealized human computer work- As Yuri Gurevich pointed out, the con- guage, any language, running on some ing ceaselessly. cept of an algorithm keeps evolving: “We computer, any computer, even one that Here, then, is our formulation of have now parallel, interactive, distrib- has not been built yet but can be built the historical version of the Church- uted, real-time, analog, hybrid, quan- … is also solvable by a Turing machine. Turing thesis, as informed by Turing’s tum, etc. algorithms.”22 There are en- This statement is one version of the so- proof of the equivalence of his and zymatic algorithms, bacterial foraging called Church/Turing thesis.”23 Church’s theses: algorithms, slime-mold algorithms, and Steering between these debates— CTT-Original (CTT-O). Every function more. The Turing machine is incapable and following Harel’s suggestion that that can be computed by the idealized of performing the atomic steps of algo- the algorithms of interest to computer human computer, which is to say, can rithms carried out by, say, an enzymatic science are always expressible in pro- be effectively computed, is Turing-com- system (such as selective enzyme bind- gramming languages—we arrive at the putable. ing) or a slime mold (such as pseudopod following program-oriented formula- Some mathematical logicians view extension). The Turing machine is simi- tion of the algorithmic thesis: CTT-O as subject ultimately to either larly unable to duplicate (as opposed to CTT-Algorithm (CTT-A). Every algo- mathematical proof or mathemati- simulate) John Conway’s Game of Life, rithm can be expressed by means of a cal refutation, like open mathematical where—unlike a Turing machine—ev- program in some (not necessarily cur- conjectures, as in the Riemann hypoth- ery cell updates simultaneously. rently existing) Turing-equivalent pro- esis, while others regard CTT-O as not A thesis aiming to limit the scope gramming language. amenable to mathematical proof but of algorithmic computability to Turing There is an option to narrow CTT-A supported by philosophical arguments computability should thus not state by adding “physically implementable” and an accumulation of mathematical that every possible algorithmic process before “program,” although in our view evidence. Few logicians today follow can be performed by a Turing machine. this would be to lump together two dis- Church in regarding CTT-O as a defini- The way to express the thesis is to say tinct computational issues that are bet- tion. We subscribe to Turing’s view of the extensional input-output function ter treated separately. the status of CTT-O, as we outline later. ια associated with an algorithm α is al- The evolving nature and open-end- In computer science today, algo- ways Turing-computable; ια is simply edness of the concept of an algorithm is rithms and effective procedures are, of the extensional mapping of α’s inputs matched by a corresponding open-end- course, associated not primarily with to α’s outputs. The algorithm the Tur- edness in the concept of a programming humans but with machines. (Note, while ing machine uses to compute ια might language. But this open-endedness not- some expositors might distinguish be- be very different from α itself. A ques- withstanding, CTT-A requires that all tween the terms “algorithm” and “ef- tion then naturally arises: If an algo- algorithms be bounded by Turing com- fective procedure,” we use the terms in- rithmic process need not be one a Tur- putability.
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