TM-2026

From: Thomas Walker Lynch Date: Mon, 6 Jul 2026 05:01:25 +0000 (+0000) Subject: . X-Git-Url: https://git.reasoningtechnology.com/machine%20fig.png?a=commitdiff_plain;h=c2e9c577aa66cc8b3e78ad45199ca041832ae967;p=TM-2026 . --- diff --git a/document/TM-2026.html b/document/TM-2026.html index dc9e2a2..48f405b 100644 --- a/document/TM-2026.html +++ b/document/TM-2026.html @@ -2,7 +2,7 @@ - TM-2026 + Computational Naturalism - - - - - - - - - - + - - Introduction + The search for a formal foundation for mathematics 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. 2 (Jena: Hermann Pohle, 1903), Appendix (Nachwort), 253.. @@ -130,10 +96,13 @@ 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 definitively 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.

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.

@@ -144,25 +113,27 @@

- To establish the missing connection to modern architecture, the volumes of the TTCA, starting in the following chapters, 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. + 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).

- 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. + 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.

- 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. + 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.

- 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 is that a running 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. + 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.

- 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). + 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 computer design abstraction stack

The six levels

@@ -865,7 +836,7 @@

-Area and partitions + Area and partitions

We call a subset of contiguous cells from a tape an area of said tape. A finite area of at least three cells will have a leftmost cell in the area, a rightmost cell in the area, with one or more interstitial cells. A leftmost cell in an area might have a left neighbor property, but the cell indicated by that property resides strictly outside the area. Or it is possible that the leftmost cell of an area is also the leftmost cell of the tape, and thus it lacks a left neighbor property. Any cell on a single ended tape will have a right neighbor property, but for a finite area the rightmost cell's right neighbor property will indicate a cell that falls outside the area. It is possible for an area to be open on the right, and thus be infinite. @@ -1317,7 +1288,6 @@

Consider the case of a partitioned tape. Then consider that we write a gasket, such that we have a higher level Turing Machine that considers each of the areas of the partition as a cell. So then, initially the Turing machine starts with its head on the leftmost area, stepping right steps to the right neighbor area etc. In this manner we abstract the concept of a cell to areas. A length then becomes a count of areas; however the size remains a count of the cells. Something similar is done in the C language, where the length of an array is a count of the elements in the array, but the size of the array is a count of bytes. A byte being an addressable unit in physical memory, and thus the underlying cell that everything is built up from.

- Zero length is a second order concept

Suppose we have a Turing Machine that is designed to mark an area based on some property of the symbols.

@@ -1326,7 +1296,12 @@

Once it finds such a cell it will write an area marker symbol to that cell, step right, and repeat writing area marker cells until it finds a cell that holds a symbol that lacks the area property. At which point the machine halts.

Once an area is marked, we can go back and run a length measuring machine that counts the sequence of marks. We will call this the length of the area.

Once an area is marked, we can go back and run a length measuring machine that counts the sequence of marks.

+ +

However we have a couple of problems, if there is no cell on the tape that has the special property. The area marking machine will step right without halting while looking for the marker. As a second problem, if the all the cells to right of the leftmost area cell, the area marker machine will never halt. +

+ +

Now suppose we employ a second order analysis.