From: Thomas Walker Lynch Date: Sat, 8 Aug 2026 14:01:47 +0000 (+0000) Subject: chapter 5 revision X-Git-Url: https://git.reasoningtechnology.com/%27%20%20%20window.RT.dirpr_library%20%20%20%27/Hindu-Arabic%20number%20fig%203.png?a=commitdiff_plain;h=6dbf68fb01e3c6253867c2234808e5d427810538;p=TM-2026 chapter 5 revision --- diff --git a/document/book/TM-2026.html b/document/book/TM-2026.html index d893755..4240f70 100644 --- a/document/book/TM-2026.html +++ b/document/book/TM-2026.html @@ -220,130 +220,6 @@ The Turing Machine transforms mathematics into computing

- To apply his proof to the Entscheidungsproblem, Turing carried the additional burden of establishing that Hilbert and Ackermann's intuitive concept of an effective procedure was functionally equivalent to a Turing Machine program. Turing addressed this issue directly in his 1936 paper. Over the following decades, the academic community evaluated and accepted his argument, cementing what is now known as the Church-Turing Thesis. This consensus supplied the necessary bridge between mathematics and modern computer science by formally equating the vague, historical notion of a human procedure with the rigorous, mechanical definition of an algorithm. -

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- For Turing's purposes working on the Entscheidungsproblem, establishing functional equivalence between algorithms and Turing Machine programs was sufficient. However, when the Turing Machine serves as a foundational model for computation theory, we are led to ask another question: whether the Turing Machine is representative of modern architectures, and to the extent it differs, how this would affect the applicability of computation theoretic results. -

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- In reading Alan Turing's 1936 paper, it is striking how modern the text feels, specifically because he discusses algorithms, stored programs, and the mechanical limits of computation. While his contemporaries largely built purely mathematical and logical frameworks, Turing uniquely tied computation theory directly to the abstraction of machines executing stored programs. Because physical hardware capable of executing stored memory programs had not yet been invented, this explicit architectural grounding makes Turing's work remarkably prescient. Still, Turing could not formally connect the Turing Machine to modern architectures, simply because those architectures did not yet exist. Here, by modern, I refer to architectures utilizing random access system memory, dedicated instruction fetch streams with dynamic branching, and discrete processing units. Though Charles Babbage's 1842 Analytical Engine touched on these concepts, they would wait until the 1940s to reemerge. The practical engineering context of 1936 was limited to calculating machines programmed via patch panels. Hence, for example, there is no explanation in his paper as to why a von Neumann architecture machine (1945) running a program would exhibit the computation theoretic results derived from a computation theory based on the Turing Machine (1936). -

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- Furthermore Turing's a-machine from his 1936 paper utilizes binary. George Boole's work (1847, 1854) was well established by then, so from a theoretical standpoint, it was a sensible simplification. However, utilizing binary within the context of a machine description effectively bridged the gap to the more practically minded engineers of the time. Alan Turing's paper arrived at the same time that switched telephone networks had reached a scale that made them difficult to maintain without systematic approaches. These networks were built upon electromechanical relays, which were decisively binary devices. At least seven men in addition to Alan Turing appear to have independently contemplated the intersection of Boolean algebra, logic, and physical computing: Victor Shestakov (1935, proposed mapping Boolean algebra to electromechanical relay circuits), Konrad Zuse (1936, adopted base 2 architecture to bypass the physical complexity of decimal mechanical gears), Akira Nakashima (1936, published the mathematical equivalence of Boolean algebra and two-terminal switching networks), Louis Couffignal (1936, proved calculating machines must shift to binary linkages to reduce physical friction), Claude Shannon (1937, published the definitive mathematical proof mapping Boolean algebra to electrical relays), George Stibitz (1937, constructed the first electromechanical binary adder), and John Vincent Atanasoff (1937, adopted binary to keep the vacuum tube count of electronic circuits physically viable). -

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- To complete the Turing Machine story then, we will transform the Turing Machine into a modern architecture in a stepwise fashion, while ensuring that at each step the modifications are inconsequential to computation theoretic existence proofs and complexity class results. We do run into some problems, so the architecture we derive will be a little different from those we currently build. Though the point here is, we could build it. -

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- The infinite tape is not as large of a hurdle as it might seem at first. For computational problems, the Turing Machine halts in a finite number of steps. Because the Turing Machine is limited to stepping the read/write head over one cell per machine execution step, in a finite number of steps, only a finite amount of tape is ever used. But for a given computation, how much tape is that? Resolving this by assuming more tape is simply attached when needed is analogous to cheating in a 'guess the bigger number' game by declaring, "My number is always one bigger than the given number, so I will tell you my guess after you state your number." Some mathematicians suggest that what is meant by infinity is precisely a rule of this sort. For engineers building physical hardware, however, to state that a resource starts finite and expands incrementally over time is a very different proposition than being asked to install infinite memory on a machine in the first place. -

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- In 1967, Marvin Minsky addressed this very topic in saying: "We need not think of the machine's tape as infinite. We imagine instead that the machine begins with a finite tape, but that, whenever an end is encountered, another unit of tape is attached." Marvin L. Minsky, Computation: Finite and Infinite Machines (Englewood Cliffs: Prentice-Hall, 1967), 167. In 1967, this was a perfectly natural thing to suggest, as computers utilized magnetic tape memory on manually mounted reels, and it was entirely possible for a computation to stop and request a new reel of tape to be mounted. Contemporary computer architectures do, in fact, achieve a similar effect through virtual memory. When physical RAM is depleted, the operating system pauses the active process and autonomously provisions apparent capacity by swapping memory pages out to auxiliary storage. However, this illusion of infinite tape remains bound by the physical limits of the secondary storage and the boundaries of the processor's address space. Once the available swap space is exhausted or the address space is saturated, the operating system abruptly terminates the process. -

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- Like a Turing Machine, a computer architecture is an abstraction. The box sitting on a person's desk is a realization of some computer architecture. To say a Turing Machine does something is to say that the Turing Machine was analyzed and the result of the analysis is that 'something'. A computer architecture can also be analyzed. A computer architecture is said to be Turing Complete when through analysis it is determined that it can do anything that a Turing Machine can do. The practical implications for a realization of a computer architecture are that running a program will only throw an error because a) the program logic told it to, b) the program has a flaw, or c) there is a mathematical fact standing in the way of execution. It is tolerable to call a computer architecture Turing Complete if it has the built-in ability to pause a program until a 'more memory' request is fulfilled. If there can be any other errors from a realization running a program, such as running out of address space or integer overflow, then the architecture is not Turing Complete. -

- - - - - - - The search that led to the Turing Machine - -

- In 1893 Gottlob Frege published an axiomatic construction of mathematics from set theory. Frege's grand objective was something he called Logicism, the philosophical thesis that all of mathematics can be derived entirely from pure logic. To bridge set theory and logic, Frege defined sets using a method known as set comprehension. Under this approach, a mathematician states a logical rule or property, and any object satisfying that logical statement automatically becomes a member of the set. Because the membership of a set is determined entirely by logical rules, the resulting sets, and the mathematics built upon them, are derived directly from logic. To implement this, his specific machinery relied upon unrestricted set comprehension, formalized as Basic Law V Gottlob Frege, Grundgesetze der Arithmetik, Vol. 1 (Jena: Hermann Pohle, 1893), §20.. -

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- At a conference in Paris in 1900, David Hilbert presented a list of pressing unsolved problems in mathematics. - Second on his list was "The Compatibility of the Arithmetical Axioms." Hilbert challenged mathematicians to find a means to demonstrate that "a finite number of logical steps based upon them [axioms] can never lead to contradictory results" David Hilbert, "Mathematical Problems," trans. Mary Winston Newson, Bulletin of the American Mathematical Society 8 (1902): 437–479. This can be found at https://www.gutenberg.org/cache/epub/71655/pg71655-images.html. The MathWorld article on this subject, https://mathworld.wolfram.com/HilbertsProblems.html, explains that Hilbert presented 10 problems at the conference, though the publication shows 23 problems, and a 24th problem was later discovered among Hilbert's unpublished notes. Based on the notes of this citation, it appears the second problem is the same on all of these lists. Also note, Hilbert discusses completeness specifically as an axiom for bounding on the sets, which appears to be distinct from the question of logical completeness for an axiomatic system.. -

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- In 1901 Bertrand Russell found a well-formed set formulation using Frege's set theory that did not correspond to a set. As Frege's work was based on this set theory, this called into question his entire work. Russell pointed out that it was possible to define a set of all sets that do not contain themselves. However this was a paradox, because if said set contained itself, it shouldn't, and if it didn't it should. Thus the formulation fails to define a set because the logical condition cannot be satisfied Bertrand Russell, The Principles of Mathematics (Cambridge: Cambridge University Press, 1903), Chapter X, 'The Contradiction'.. Russell communicated this to Frege in a letter dated 1902-06-16, shortly before Frege's second volume was going to print Bertrand Russell to Gottlob Frege, June 16, 1902, reprinted in Jean van Heijenoort, From Frege to Gödel: A Source Book in Mathematical Logic (Cambridge: Harvard University Press, 1967), 124–125. Gottlob Frege, Grundgesetze der Arithmetik, Vol. 2 (Jena: Hermann Pohle, 1903), Appendix (Nachwort), 253. Frege writes: 'Hardly anything more unfortunate can befall a scientific writer than to have one of the foundations of his edifice shaken after the work is finished.'. Frege hurriedly authored an appendix (the Nachwort) admitting his system was compromised Frege was a quiet, rigid man who had spent decades building his logical fortress in almost total academic obscurity. Frege was personally devastated by Russell's letter. Shortly after, he suffered the loss of his wife, fell into severe depression, and his academic output almost entirely ceased. In 1924, a year before his death, he wrote unpublished diaries explicitly surrendering his life's work, declaring that logicism was a mistake and that mathematics must actually be derived from geometry. Note I. Grattan-Guinness, The Search for Mathematical Roots, 1870–1940 (Princeton: Princeton University Press, 2000). For an analysis of Frege's intellectual decline, personal tragedies, and his unpublished 1924–1925 diaries where he formally surrenders the logicist program, see Chapter 7.. -

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- In 1903 Russell proposed a hierarchy of types to repair this foundational vulnerability. At the base were sets of individuals, then sets based on individuals or sets of individuals, etc. This looks a lot like how types work in modern software Bertrand Russell, The Principles of Mathematics (Cambridge: Cambridge University Press, 1903), Appendix B: The Doctrine of Types.. In this manner, it is not possible to write a paradoxical set definition. Russell and Alfred North Whitehead then engineered an entirely new, massive scaffolding utilizing this type system to pursue Frege's original objective of deriving mathematics from logic, publishing their results in three volumes between 1910 and 1913 Alfred North Whitehead and Bertrand Russell, Principia Mathematica (Cambridge: Cambridge University Press, 1910–1913).. Russell's system can be cumbersome due to requiring a large construction to be assembled in place of what otherwise might have been a simple rule. -

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- In 1908 Ernst Zermelo published an alternative system designed to avoid the known paradoxical statements of the time, even though absolute consistency remained unproven. In Zermelo's set theory, a mathematician first starts with an existing set, and then applies the Axiom of Separation using definite properties to partition out subsets Ernst Zermelo, "Untersuchungen über die Grundlagen der Mengenlehre I," Mathematische Annalen 65 (1908): 261–281.. To see how this works, consider the expression \{x \mid P(x)\}. Under unrestricted comprehension, a logician is permitted to define the predicate P(x) as x ∉ x. This produces Russell's Paradox, so the set fails to be defined. In contrast, consider the same predicate, though restricted by Zermelo's Axiom of Separation over a predefined set S, written as \dot{R} = \{x \mid x ∈ S ∧ x ∉ x\}. The only thing a person needs to know about S here is that it has already been successfully defined. So let us ask, is \dot{R} in \dot{R}? If we assume \dot{R} is a member of S, evaluating the second term forces the familiar fatal loop: if \dot{R} is in \dot{R}, it shouldn't be, and if it isn't, it should be. Thus if we assume that \dot{R} is in S, then \dot{R} can not be defined, but by definition, S is defined, and thus its members are defined. As we arrived at a contradiction, the original assumption must be false, i.e. it is wrong to assume that \dot{R} is in S. As \dot{R} is definitively not a member of S, the first term of the set comprehension rule, x ∈ S, is false, and the paradox vanishes. -

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- A person might suggest defining S as the set of all definable mathematical objects, forming a universal set. However, if such a universal set S existed, the Axiom of Separation could be applied as per the proof in the prior paragraph to show \dot{R} is not in S. However, as \dot{R} is a valid, definable set, it must reside within S by the very definition of a universal set. This contradicts the premise that S contains everything. Therefore, within any system governed by the Axiom of Separation, a universal set cannot exist. -

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- The authority to remove Russell's Paradox set formulation comes from the set S. If we know its definition, then the authority comes through that definition. However, if we merely stipulate that S must be defined, then we are expressing our authority through S by declaring, "Undefined sets are not allowed." In the explanation above, it is only after discovering a set is undefined that we conclude it is not a member of S. I sometimes wonder how mathematics might have evolved had Frege simply taken that approach. We take this question up again in chapter , Computational Naturalism, and discover there is a deeper issue. -

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- Stepping back from the mechanics of set definition, a person can observe two competing approaches to establishing mathematical foundations. The first approach is constructive, building complex systems by assembling them upward from fundamental primitives. The second approach relies on islands of meaning, carving out valid spaces from the abstract void using precise rules and axioms, exactly as Zermelo did. Because both methodologies rely entirely upon a rigorous framework of deduction, logic itself serves as the essential substrate. Consequently, a complete study of the foundation of mathematics requires the examination of three distinct subjects: the primitives used for construction, the rules that bound the theoretical islands, and the underlying logic that evaluates them both. -

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- In 1928 David Hilbert and Wilhelm Ackermann published a textbook on mathematical logic, Grundzüge der theoretischen Logik David Hilbert and Wilhelm Ackermann, Grundzüge der theoretischen Logik (Berlin: Springer, 1928). This first edition has not been translated into English.. A feature of this book is its attention to procedures to follow for mechanically determining truth of statements. They called the problem solved by such a procedure the Entscheidungsproblem. In the first chapter they review the procedure for solving the Entscheidungsproblem in the propositional logic. For the first order predicate calculus they define the problem as, "Universal validity concerns the following question: How can one determine, for any given logical expression that contains no individual signs [constants], whether the expression represents a true assertion for arbitrary substitutions for the occurring variables, or not?" Ibid., 72–73.. They review some special cases with solutions, including one published earlier by Ackermann, but then throw down the gauntlet by saying, - "A general solution to the Entscheidungsproblem, regardless of whether a person considers the first or the second formulation, is not yet available." Ibid., 81. "Eine allgemeine Lösung des Entscheidungsproblems, mag man nun die erste oder die zweite Fassung nehmen, liegt bis jetzt noch nicht vor." The term Entscheidungsproblem literally translates to 'decision problem'. However, there are many types of decision problems, and later we will meet a class of Turing Machine programs called deciders, so it appears to be best to keep the original German. As we will see later Alan Turing also did this.. -

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- In 1931 Kurt Gödel published his incompleteness theorems Kurt Gödel, "Über formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme I," Monatshefte für Mathematik und Physik 38 (1931): 173–198.. By mapping formal logic into arithmetic, he demonstrated that any consistent formal system sufficiently powerful to perform basic arithmetic, let us call it system F, will inevitably contain well-formed formulas that are mathematically true yet cannot be proven within the system itself For the definitive English translation, see Jean van Heijenoort, From Frege to Gödel: A Source Book in Mathematical Logic, 1879–1931 (Cambridge: Harvard University Press, 1967), 596–616.. Gödel achieved this by engineering a specific formula that evaluates to the claim: "G: There exists no sequence of valid logical steps within system F that proves G." If system F is consistent, it cannot output a proof for G; thus, the claim G makes is factually accurate, rendering it true but mechanically unprovable. Furthermore, Gödel demonstrated that system F cannot output a proof of its own consistency. This result fractured David Hilbert's 1900 vision of utilizing a weaker, strictly "finitistic" logical subsystem to definitively prove that the axioms of arithmetic are entirely free of contradictions David Hilbert, "Mathematical Problems," Bulletin of the American Mathematical Society 8 (1902): 437–479.. If the full, powerful system F lacks the mechanical capacity to verify its own consistency, Hilbert's weaker finitistic subsystem is definitively incapable of accomplishing the task. Gödel's work established a hard mechanical boundary, asserting that truth and provability are distinct concepts in classical mathematics. -

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- In 1936-03, Alonzo Church leveraged Gödel's foundational papers to directly answer the Entscheidungsproblem Alonzo Church, "A Note on the Entscheidungsproblem," The Journal of Symbolic Logic 1, no. 1 (March 1936): 40–41; correction, The Journal of Symbolic Logic 1, no. 3 (September 1936): 101–102., resting that argument on the unsolvability result he published the following month Alonzo Church, "An Unsolvable Problem of Elementary Number Theory," American Journal of Mathematics 58, no. 2 (April 1936): 345–363.. Working independently, Alan Turing had arrived at his own mechanical solution, and upon seeing Church's work, Turing rushed to submit his manuscript, which was received on 1936-05-28 Alan M. Turing, "On Computable Numbers, with an Application to the Entscheidungsproblem," Proceedings of the London Mathematical Society s2 42, no. 1 (1936): 230–265. Received May 28, 1936, read November 12, 1936.. On 1936-08-28 he appended a proof that his mechanical architecture was mathematically equivalent to Church's lambda calculus. -

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- Alan Turing used an abstraction of a computing machine, also described as a clerk working at a desk with pen and squares on paper while following a procedure, to prove that no primary 'analyzer' program can universally decide whether a second analyzed program will halt when it is run Ibid.. An answer to this halting problem (specifically asserting "The studied machine halts" or "The studied machine does not halt") would indeed be a statement in first order logic. Thus, by showing no analyzer can universally make such a determination, Turing proved that no decider could exist for the Entscheidungsproblem. -

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- Turing employed an enumerative diagonal argument to establish this result. A simpler proof by contradiction that is commonly used today was first published by Christopher Strachey in 1965 Christopher Strachey, "An Impossible Program," The Computer Journal 7, no. 4 (January 1965): 313. In his letter, Strachey explicitly attributed the distilled logic to an existing "well known piece of folklore among programmers.". To begin the proof, assume a person builds a perfect decider program, H(p, i), that evaluates any given program p executing with input i, then outputs 'Y' if p(i) halts, and 'N' if it does not halt. Next, a person writes a malicious program, M(x), that incorporates H as a subroutine. When M receives an input program x, it evaluates H(x, x) to determine how program x behaves when given itself as input. If H(x, x) outputs 'Y', M enters an infinite loop; if H(x, x) outputs 'N', M immediately halts. -

- - - - - M( x ){ - if( H(x ,x) == 'Y' ) while( true ); - else if( H(x ,x) == 'N' ) return; - } - - Code . The diagonal machine that defeats a supposed halting decider - - -

- The evil part occurs when we give program M(x) itself as input, M(M). Program M calls its subroutine and asks H(M, M) what M will do. If H outputs 'Y', it is wrong, because M loops infinitely. If H outputs 'N', then it is wrong, because M halts. The decider H is forced into an inescapable failure, proving that no universal decider can exist. -

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- While Gödel, Church, and Turing established the primary boundaries of computation, they did not work in a vacuum. During this period, the broader academic community worked to synthesize the definitive mechanics of effective calculability. Jacques Herbrand and Gödel formalized general recursive functions between 1931 and 1934 Kurt Gödel, "On Undecidable Propositions of Formal Mathematical Systems," mimeographed lecture notes, Institute for Advanced Study, Princeton, 1934.. Emil Post independently defined "Finite Combinatory Processes" in 1936, outlining a theoretical architecture functionally identical to Turing's model Emil L. Post, "Finite Combinatory Processes—Formulation 1," The Journal of Symbolic Logic 1, no. 3 (September 1936): 103–105.. Stephen Kleene subsequently unified these disparate threads, proving the strict mathematical equivalence of Church's lambda calculus and Herbrand-Gödel recursive functions Stephen C. Kleene, "General Recursive Functions of Natural Numbers," Mathematische Annalen 112 (1936): 727–742.. Turing supplied the remaining leg himself, showing his machines equivalent to the lambda calculus Alan M. Turing, "Computability and λ-definability," The Journal of Symbolic Logic 2, no. 4 (December 1937): 153–163.. -

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- The academic community was thus equipped with three mathematically equivalent foundations for computation theory: recursive functions, the lambda calculus, and the Turing Machine. While all three frameworks remain active subjects of study, Turing's model is unique in providing practical intuition through the abstraction of physical machines and programs. This made it the foundation of choice for computation theory textbooks by Stephen Kleene Stephen C. Kleene, Introduction to Metamathematics (Amsterdam: North-Holland, 1952)., Martin Davis Martin Davis, Computability and Unsolvability (New York: McGraw-Hill, 1958)., and Marvin Minsky Marvin L. Minsky, Computation: Finite and Infinite Machines (Englewood Cliffs: Prentice-Hall, 1967)., leading to the modern standard presentations by authors such as John Hopcroft and Jeffrey Ullman John E. Hopcroft and Jeffrey D. Ullman, Introduction to Automata Theory, Languages, and Computation (Reading: Addison-Wesley, 1979)., as well as Harry Lewis and Christos Papadimitriou Harry R. Lewis and Christos H. Papadimitriou, Elements of the Theory of Computation (Englewood Cliffs: Prentice-Hall, 1981).. -

- - - - - - The Turing Machine transforms mathematics into computing - -

To apply his proof to the Entscheidungsproblem, Turing carried the additional burden of establishing that Hilbert and Ackermann's intuitive concept of an effective procedure was functionally equivalent to a Turing Machine program. Alonzo Church had already made an identification of this kind, some months earlier, defining effective calculability by identifying it with general recursiveness, or equivalently with λ-definability in his own lambda calculus Alonzo Church, "An Unsolvable Problem of Elementary Number Theory," American Journal of Mathematics 58, no. 2 (April 1936): 356. A preliminary statement was presented to the American Mathematical Society in April 1935. The name 'Church's thesis' is due to Kleene.. What Church gave was a definition, an identification of one formal notion with another. What Turing gave was an argument: he analyzed what a person does when he computes with pencil and paper, and then built a machine to match. Church acknowledged the difference in his 1937 review of Turing's paper, the review in which he also coined the phrase 'Turing machine', granting that computability by such a machine had the advantage of "making the identification with effectiveness in the ordinary (not explicitly defined) sense evident immediately", without preliminary theorems Alonzo Church, review of A. M. Turing, "On Computable Numbers, with an Application to the Entscheidungsproblem," The Journal of Symbolic Logic 2, no. 1 (March 1937): 42–43.. Gödel, who had found Church's proposal unsatisfactory, accepted the identification only upon seeing Turing's.

@@ -363,7 +239,6 @@ Furthermore Turing restricted the figures printed by his a-machine to 0 and 1, so that the sequences it computes are binary. George Boole's work (1847, 1854) was well established by then, so from a theoretical standpoint, it was a sensible simplification. However, utilizing binary within the context of a machine description effectively bridged the gap to the more practically minded engineers of the time. Alan Turing's paper arrived at the same time that switched telephone networks had reached a scale that made them difficult to maintain without systematic approaches. These networks were built upon electromechanical relays, which were decisively binary devices. At least seven men in addition to Alan Turing appear to have independently contemplated the intersection of Boolean algebra, logic, and physical computing: Victor Shestakov (1935, proposed mapping Boolean algebra to electromechanical relay circuits), Konrad Zuse (1936, adopted base 2 architecture to bypass the physical complexity of decimal mechanical gears), Akira Nakashima (1936, published the mathematical equivalence of Boolean algebra and two-terminal switching networks), Louis Couffignal (1936, argued calculating machines must shift to binary linkages to reduce physical friction), Claude Shannon (1937, published the definitive mathematical proof mapping Boolean algebra to electrical relays), George Stibitz (1937, constructed the first electromechanical binary adder), and John Vincent Atanasoff (1937, adopted binary to keep the vacuum-tube count of electronic circuits physically viable).

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To complete the Turing Machine story, then, we will transform the Turing Machine into a modern architecture in a stepwise fashion, while ensuring that at each step the modifications are inconsequential to computation theoretic existence proofs and complexity class results. We do run into some problems, so the architecture we derive will be a little different from those we currently build. Though the point here is that we could build it.

@@ -380,7 +255,6 @@ Like a Turing Machine, a computer architecture is an abstraction. The box sitting on a person's desk is a realization of some computer architecture. To say a Turing Machine does something is to say that the Turing Machine was analyzed and the result of the analysis is that 'something'. A computer architecture can also be analyzed. A computer architecture is said to be Turing Complete when, through analysis, it is determined that it can do anything that a Turing Machine can do. The practical implication for a realization of such an architecture is that running a program will fault only because a) the program logic told it to, b) the program has a flaw, or c) there is a mathematical fact standing in the way of execution. It is tolerable to call a computer architecture Turing Complete if it has the built-in ability to pause a program until a 'more memory' request is fulfilled, as that is a limit of the realization rather than of the architecture. However, if the architecture itself stipulates a bound that a program can reach, such as a fixed address width or a fixed integer width, then every realization of it must fail on some program that a Turing Machine would carry to completion, and the architecture is not Turing Complete.

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