From: Thomas Walker Lynch Date: Mon, 10 Aug 2026 10:56:58 +0000 (+0000) Subject: . X-Git-Url: https://git.reasoningtechnology.com/Hindu-Arabic%20number%20fig%204.png?a=commitdiff_plain;h=refs%2Fheads%2Fcore-developer_branch;p=TM-2026 . --- diff --git a/document/book/TM-2026.html b/document/book/TM-2026.html index b18e04b..447ff2a 100644 --- a/document/book/TM-2026.html +++ b/document/book/TM-2026.html @@ -32,6 +32,7 @@ + Exordium @@ -191,7 +192,7 @@ - + The search that led to the Turing Machine @@ -273,12 +274,6 @@ 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 made an identification of this kind some months earlier Alonzo Church, "An Unsolvable Problem of Elementary Number Theory," American Journal of Mathematics 58, no. 2 (April 1936): 345–363, the identification at 356. A preliminary statement was presented to the American Mathematical Society on 1935-04-19. The name 'Church's thesis' is due to Kleene. Church first framed the identification in terms of his own λ-definability. Gödel told him that this was thoroughly unsatisfactory, and Church restated it in terms of Herbrand-Gödel general recursiveness, which is the form that reached print. The exchange survives in a letter from Church to Kleene of 1935-11-29, quoted in Stephen C. Kleene, "Origins of Recursive Function Theory," Annals of the History of Computing 3, no. 1 (1981): 52–67, at 59., @@ -290,16 +285,21 @@ Over the following decades, the academic community evaluated and accepted the 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.

+ + + + + The Turing Machine measured against modern architecture +

- 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. That question is addressed over the chapters that follow, and the answer arrived at is that neither side is quite fit to be compared to the other as it stands. -

+ 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 programs had not yet been invented, this explicit architectural grounding makes Turing's work remarkably prescient.

- 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 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 Analytical Engine, first described in 1837, 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). + 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. Charles Babbage had reached several of these ideas a century earlier with the Analytical Engine, and the accounts of it were in print and on library shelves in 1936, though Turing's paper does not mention themThe question is raised by Robin Gandy in his preface to the paper in Turing's collected works. Gandy argues that Turing cannot have read Babbage, on the ground that he would have said so had he done so. He allows that Turing might have seen the article on calculating machines in the eleventh edition of the Encyclopaedia Britannica, a copy of which Turing inherited from his father and consulted in later years, but observes that its treatment of the Analytical Engine is brief and dismissive, and would not have suggested to a reader that Babbage had conceived a universal machine. R. O. Gandy, preface to "On Computable Numbers, with an Application to the Entscheidungsproblem," in R. O. Gandy and C. E. M. Yates, eds., Mathematical Logic, The Collected Works of A. M. Turing (Amsterdam: North-Holland, 2001). Gandy directs the reader to his own study of the period for the background of ideas and for the contributions of Hilbert's school, of Church and his students, and of Post: R. O. Gandy, "The Confluence of Ideas in 1936," in Rolf Herken, ed., The Universal Turing Machine: A Half-Century Survey (Oxford: Oxford University Press, 1988), 55–111. The reasoning is an argument from silence. It moves from an absence in the text to an absence in the reading, and nothing short of a borrowing record or a letter could tell against it. What can be said is that the material was available. Menabrea's account, in the Countess of Lovelace's translation and with her notes, had been in print since 1843 in Taylor's Scientific Memoirs, and Babbage's own chapter on the engine had been in print since 1864. Babbage had held the Lucasian chair at Cambridge, where Turing was a Fellow of King's. Turing did not read these things, if he did not, by choice or by accident, and not for want of access. There is something peculiar in the shape of the defence. It secures Turing's independence by crediting him with a gap in his reading, which is an odd currency to pay originality in, and the transaction is made on Turing's behalf rather than by him. Gandy was Turing's student, his friend, and his literary executor, which is worth knowing when weighing the argument, though it is not an answer to it. It is also worth noting what he said once the question was live. By 1950 Turing discusses the Analytical Engine at some length, knows that Babbage held the Lucasian chair from 1828 to 1839, and is willing to call the engine a universal digital computer. Yet he reaches Lovelace's memoir there by way of Douglas Hartree's 1949 book, citing Hartree's page for the quotation rather than the memoir itself. That is consistent with his never having gone to the primary source, though it settles nothing about 1936. Alan M. Turing, "Computing Machinery and Intelligence," Mind 59, no. 236 (October 1950): 433–460, the Analytical Engine at 439 and Lovelace's objection at 450; Douglas R. Hartree, Calculating Instruments and Machines (Urbana: University of Illinois Press, 1949), 70; L. F. Menabrea, "Sketch of the Analytical Engine Invented by Charles Babbage," trans. Ada Augusta, Countess of Lovelace, with translator's notes, in Richard Taylor, ed., Scientific Memoirs, vol. 3 (London: Richard and John E. Taylor, 1843), 666–731; Charles Babbage, Passages from the Life of a Philosopher (London: Longman, Green, 1864), chap. VIII. The last consideration is the one that matters here, and it cuts the question down. Babbage would have been of no use to Turing. The Entscheidungsproblem needed a machine whose entire repertoire of behaviour could be catalogued and then diagonalized over. The Analytical Engine has far too much architecture for that. Reading it would have been a distraction rather than a shortcut, and a historian who establishes that Turing did read it will not thereby have taken anything away from him.. Whether he had read them is a question that deserves some care, and it is taken up in the note. The answer changes less than one might expect, because Babbage was designing a machine to compute with, whereas Turing needed a machine plain enough that everything it could ever do could be catalogued and reasoned over. The poverty of the Turing Machine is deliberate. Later in this chapter that same poverty returns as the machine's central defect when it is asked to serve as an architecture. The practical engineering context of 1936 was in any case 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).

- Also of interest, 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). + 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. That question is addressed over the chapters that follow, and the answer arrived at is that neither side is quite fit to be compared to the other as it stands.

@@ -325,7 +325,6 @@

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. The transformation does not run in one direction only. On the Turing Machine side, the controller has to stop being used as memory. On the modern side, the fixed widths an architecture stipulates, those of an address and of an Integer, are what have to give way. The two meet in the middle, and the machine we arrive at is less strange than that might suggest. It separates the control path from the data path; it holds an instruction table; and it has a small register file. It looks modern. The differences from what we currently build are real but few, and the point of the exercise is that we could build it.

- diff --git a/document/book/binary_cutting_on_the_floor.html b/document/book/binary_cutting_on_the_floor.html new file mode 100644 index 0000000..0a882f7 --- /dev/null +++ b/document/book/binary_cutting_on_the_floor.html @@ -0,0 +1,4 @@ +

+ George Boole's work (1847, 1854) was well established by then, so from a theoretical standpoint, it was a sensible simplification. The machine itself is not binary; its alphabet also carries symbols of the second kind, which it writes and erases as working marks. What reached the practically minded engineer was the narrower thing, a binary sequence sitting inside a description of a machine, and that was enough. 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). +

+