From: Thomas Walker Lynch Date: Sat, 4 Jul 2026 04:23:44 +0000 (+0000) Subject: introductory chapter on computational naturalism X-Git-Url: https://git.reasoningtechnology.com/machine%20fig.png?a=commitdiff_plain;h=e31d37f3a1ee727718b8357f7d0700ab85947f0c;p=TM-2026 introductory chapter on computational naturalism --- diff --git a/document/TM-2026.html b/document/TM-2026.html index 3ef8d54..c20a394 100644 --- a/document/TM-2026.html +++ b/document/TM-2026.html @@ -15,7 +15,29 @@ @@ -24,12 +46,13 @@ - + + Introduction

@@ -42,11 +65,11 @@

- 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 failed 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 his 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 In 1924, a year before his death, he wrote unpublished diaries explicitly surrendering his life's work, declaring that logicism was a mistake and that mathematics must actually be derived from geometry.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.. + 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 failed 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 his 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..

- In 1903 Russell proposed a modification to set theory based on a hierarchy of types so as to modify and repair set theory. 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. + 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.

@@ -54,11 +77,11 @@

- A person might suggest defining S as the set of all definable mathematical objects, forming a universal set. If such a universal set S existed, the Axiom of Separation could be applied using the previous predicate to isolate \dot{R}. Because \dot{R} is a valid, definable set, it must reside within S by the very definition of a universal set. However, the logic evaluated earlier proved definitively that \dot{R} cannot be a member of S. The existence of a definable set \dot{R} that sits strictly outside of S contradicts the premise that S contains everything. Therefore, within any system governed by the Axiom of Separation, a universal set cannot exist. + 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 using the previous predicate to isolate \dot{R}. Because \dot{R} is a valid, definable set, it must reside within S by the very definition of a universal set. But notice that the logic evaluated earlier proved definitively that \dot{R} cannot be a member of S. Yet the existence of a definable set \dot{R} that sits strictly outside of S contradicts the premise that S contains everything. Therefore, within any system governed by the Axiom of Separation, a universal set cannot exist.

- Stepping back from the mechanics of set definition, a person can observe two competing approaches to establishing mathematical foundations. The first approach is constructive, making complex structures 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 to function, logic itself serves as the essential substrate. Consequently, a complete foundational study 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. + 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 to function, logic itself serves as the essential substrate. Consequently, a complete foundational study 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.

@@ -67,7 +90,7 @@

- 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 S, 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 S that proves G." If system S 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 S 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 S physically lacks the mechanical capacity to verify its own consistency, Hilbert's weaker finitistic subsystem is definitively incapable of accomplishing the task. Gödel's work establishes a hard mechanical boundary, asserting that truth and provability are distinct concepts in classical mathematics. + 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 S, 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 S that proves G." If system S 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 S 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 S physically 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.

@@ -136,12 +159,12 @@ 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 computer design abstraction stack + The computer design abstraction stack

The six levels

- There are a number of discernable levels to the computer design abstraction stack: + There are a number of discernible levels to the computer design abstraction stack:

    @@ -158,12 +181,12 @@

    - An architecture provides programmers with information that is valuable when designing the logic of programs. This includes programmers across the entire software stack, such as firmware engineers, driver developers, systems programmers, compiler writers, and application developers. Although applications programs might only be exposed to the virtual architecture presented by various standards, various libraries, and the programming language used. In addition to specifying the instructions (instruction set architecture), architecture includes describing the memory and hardware virtualization features, specifying the behavior of the interrupt subsystem, the method of doing I/O, DMA, the special registers and their effects, any architectural busses, and the standards to be followed for each if any. More recently, this also includes specifying how programs can make use of secure areas. The architecture is specified by an architect. + An architecture provides programmers with information that is valuable when designing the logic of programs. This includes programmers across the entire software stack, such as firmware engineers, driver developers, systems programmers, compiler writers, and application developers. Although applications programs might only be exposed to the virtual architecture presented by various standards, various libraries, and the programming language used. In addition to specifying the instructions (instruction set architecture), architecture includes describing the memory and hardware virtualization features, specifying the behavior of the interrupt subsystem, the method of doing I/O, DMA, the special registers and their effects, any architectural busses, and the standards to be followed for each if any. More recently, this also includes specifying how programs can make use of secure areas. The architecture is specified by an architect.

    - The classic text by Hamacher, Vranesic, and Zaky carefully defines the organizational level as sitting between architecture and implementation. - Organization is the register-transfer level description of the machine, which includes internal buses, external buses and the state machines that implement the protocols used, control units, interrupt structures, and ALU layout. Crucially, it is at this level that decisions regarding instruction-level parallelism are made, such as whether the processor will employ a scalar, superscalar, or VLIW design, the depth of its execution pipelines, the use of out-of-order execution, branch prediction strategies, and the specific hierarchy of hardware caches. It dictates the logical arrangement of hardware and the procedures that force the data to flow to satisfy the architectural constraints. Organization is sometimes called micro-architecture, and it is made by a design architect. + The classic text by Hamacher, Vranesic, and Zaky carefully defines the organizational level as sitting between architecture and implementation V. Carl Hamacher, Zvonko G. Vranesic, and Safwat G. Zaky, Computer Organization, 5th ed. (New York: McGraw Hill, 2002).. + Organization is the register transfer level description of the machine, which includes internal buses, external buses and the state machines that implement the protocols used, control units, interrupt structures, and ALU layout. Crucially, it is at this level that decisions regarding instruction level parallelism are made, such as whether the processor will employ a scalar, superscalar, or VLIW design, the depth of its execution pipelines, the use of out of order execution, branch prediction strategies, and the specific hierarchy of hardware caches. It dictates the logical arrangement of hardware and the procedures that force the data to flow to satisfy the architectural constraints. Organization is sometimes called micro architecture, and it is made by a design architect.

    @@ -171,11 +194,11 @@

    - The implementation instructs the manufacturing teams very specifically on what is to be built. For a microprocessor chip, this consists of the full wiring of the logic gates and transistors, instructions for cutting the lithography masks, the package to be used, and the production test programs to be run. The masks instructions for cutting the masks consists of the sizes and placement doping wells and gates, placement of contacts, where to run wires, etc. An implementation. The implementation is designed by design engineers, with the assistance of design synthesis tools and CAD tools. + The implementation instructs the manufacturing teams very specifically on what is to be built. For a microprocessor chip, this consists of the full wiring of the logic gates and transistors, instructions for cutting the lithography masks, the package to be used, and the production test programs to be run. The instructions for cutting the masks consist of the sizes and placement of doping wells and gates, the placement of contacts, and where to run wires. The implementation is designed by design engineers, with the assistance of design synthesis tools and CAD tools.

    - A realization is a physical box full of plastic, metal, fiberglass, and silicon, along with a smattering of exotic materials. A realization is made by manufacturing engineers, technicians, and product line workers, with the assistance of some of the most sophisticated machines ever built by humankind. + A realization is a physical box full of plastic, metal, fiberglass, and silicon, along with a smattering of exotic materials. A realization is made by manufacturing engineers, technicians, and product line workers, with the assistance of some of the most sophisticated machines ever built by humankind.

    @@ -197,7 +220,7 @@

    - In fact, architecture instructs organization. When we design an instruction set that has load instructions, it implies that there will be an instruction fetch, and thus an instruction bus. Furthermore the load data has to come from somewhere, so there will be data fetch and a data bus. Could both be the same bus? If not, then we have a "Harvard Architecture". The fact is, almost no one involved in computer design completely divorces architecture from organization. + In fact, architecture instructs organization. When an architect designs an instruction set that has load instructions, it implies that there will be an instruction fetch, and thus an instruction bus. Furthermore the load data has to come from somewhere, so there will be data fetch and a data bus. Could both be the same bus? If not, then we have a "Harvard Architecture". The fact is, almost no one involved in computer design completely divorces architecture from organization.

    @@ -207,7 +230,7 @@

    Where the Turing Machine fits in

    - The Turing Machine is a computation theory object that is suggestive of a simple architecture, and a computer organization. Any student who has had to do homework problems centered on Turing Machines, will have tracked the flow of data through the machine, i.e. worked at the register transfer level. However, a little work is needed to complete the architecture analog. The fundamentals are present, the read/write head, the tape, the procedure for using the tape, but other components are missing. The manipulation of symbols remains ungrounded. The tape is not well defined. The use of emptiness is non-architectural like. The tape transport is not articulated, though it is implied. The read buffer that is required, so the programmed controller can do a write without clobbering the read data needed for the next transition, is not identified as a component. As we proceed, we will likely discover other missing components. + The Turing Machine is a computation theory object that is suggestive of a simple architecture, and a computer organization. A person who has had to do homework problems centered on Turing Machines will have tracked the flow of data through the machine, i.e. worked at the register transfer level. However, a little work is needed to complete the architecture analog. The fundamentals are present, the read/write head, the tape, the procedure for using the tape, but other components are missing. The manipulation of symbols remains ungrounded. The tape is not well defined. The use of emptiness is non architectural like. The tape transport is not articulated, though it is implied. The read buffer that is required, so the programmed controller can do a write without clobbering the read data needed for the next transition, is not identified as a component. As we proceed, we will likely discover other missing components.

    Computation theoretic consequentiality

    @@ -217,7 +240,7 @@

    - When a transform applied to machine m_i produces machine m_{i.1}, and this latter machine gets the same results for the same computational inputs, and furthermore, if any computation theory analysis applied to m_{i.1} yields the same answer as it would when applied to m_i — we say that the transform is computation theoretic inconsequential. Otherwise, the transformation is said to be computation theoretic consequential. The remainder of this section defines these terms more precisely. + When a transform applied to machine m_i produces machine m_{i.1}, and this latter machine gets the same results for the same computational inputs, and furthermore, if any computation theory analysis applied to m_{i.1} yields the same answer as it would when applied to m_i — we say that the transform is computation theoretic inconsequential. Otherwise, the transformation is said to be computation theoretic consequential. The remainder of this section defines these terms more precisely.

    Definition of the same results transform property

    @@ -255,7 +278,7 @@

    - We can then assign a property to transform T called its doesn't change results property, as follows. If and only if: + We can then assign a property to transform T called its doesn't change results property, as follows. If and only if:

    @@ -313,7 +336,7 @@

    - then T is computation theoretic inconsequential for m_i. + then T is computation theoretic inconsequential for m_i.

    @@ -325,7 +348,7 @@

    - then we can say without qualification that T is computation theoretic inconsequential. Though still implied are the sets of machines and tapes. + then we can say without qualification that T is computation theoretic inconsequential. Though still implied are the sets of machines and tapes.

    @@ -338,7 +361,7 @@

    - Put more formally, given a set of instantiable objects and a collection of instances made from them, for these objects to be symbols, two structural conditions must be met. First, it must be possible to define an instance comparison operation, denoted =, that acts as an equivalence relation to partition the collection into discrete equivalence classes. There must be a one-to-one correspondence between the resulting equivalence classes and the original instantiable objects from which the member instances were derived. + Put more formally, given a set of instantiable objects and a collection of instances made from them, for these objects to be symbols, two structural conditions must be met. First, it must be possible to define an instance comparison operation, denoted =, that acts as an equivalence relation to partition the collection into discrete equivalence classes. There must be a one to one correspondence between the resulting equivalence classes and the original instantiable objects from which the member instances were derived.

    @@ -362,7 +385,7 @@

    - Given an original, all copies stemming from it will be equal to each other and to the original. By stemming from, this definition includes all direct copies and copies of copies. + Given an original, all copies stemming from it will be equal to each other and to the original. By stemming from, this definition includes all direct copies and copies of copies.

    @@ -370,7 +393,7 @@

    - Though symbol instances are integer-like in that copy and equality comparison operations can be used with them, symbol instances are strictly disallowed from being used with other integer operators. Symbols cannot be compared for greater-than or less-than; they cannot be incremented, added, nor subtracted, etc. + Though symbol instances are integer like in that copy and equality comparison operations can be used with them, symbol instances are disallowed from being used with other integer operators. Symbols cannot be compared for greater than or less than; they cannot be incremented, added, nor subtracted, etc.

    Instance implementation

    @@ -380,11 +403,11 @@

    - In general, memory addresses are built-in symbol instances, hence within the context of a single process run, a program can make use of these symbols. However, this diminishes the size of the address space and leaves the memory at those addresses unused. A common hedge is then to use references into a dictionary, where the data looked up in the dictionary is the name of the symbol. + In general, memory addresses are built in symbol instances, hence within the context of a single process run, a program can make use of these symbols. However, this diminishes the size of the address space and leaves the memory at those addresses unused. A common hedge is then to use references into a dictionary, where the data looked up in the dictionary is the name of the symbol.

    - Such symbol names are non-structural strings, so they do not need to follow the rules of symbols. For example, a program written where references to strings were used as symbol instances, could give multiple, or all, strings the same name, and the program would function. Conventionally, the names are made to be distinct so as to avoid confusion. The hazard here is that a programmer will then conflate the string name with the symbol instance, and perform symbol operations with it. + Such symbol names are non structural strings, so they do not need to follow the rules of symbols. For example, a program written where references to strings were used as symbol instances, could give multiple, or all, strings the same name, and the program would function. Conventionally, the names are made to be distinct so as to avoid confusion. The hazard here is that a programmer will then conflate the string name with the symbol instance, and perform symbol operations with it.

    @@ -458,7 +481,7 @@ TapeAlphabet cell_2 = SYMBOL_A; /* Equality comparison over instances */ - if (cell_1 == cell_2) { + if(cell_1 == cell_2){ /* Evaluates to True */ } @@ -538,11 +561,11 @@ The Turing Machine as a computer architecture

    - In this interpretation of the Turing Machine, the architecture utilizes a single-ended tape, as done in Hopcroft and Ullman's book . If a computation strictly requires a two-way infinite tape, the single-ended tape machine can emulate it by interleaving the addresses: assigning odd-addressed cells to represent the right-going half, and even-addressed cells to represent the left-going half. This emulation requires taking two steps instead of one to advance in a given logical direction. When analyzing the time complexity of an algorithm, this overhead merely doubles the constant on the linear term, leaving the asymptotic order of complexity entirely unchanged. The outcomes of decider programs are unaffected. Therefore, utilizing a single-ended tape is an inconsequential variation of the two-way tape machine. + In this interpretation of the Turing Machine, the architecture utilizes a single ended tape, as done in Hopcroft and Ullman's book John E. Hopcroft and Jeffrey D. Ullman, Introduction to Automata Theory, Languages, and Computation (Reading: Addison Wesley, 1979).. If a computation requires a two way infinite tape, the single ended tape machine can emulate it by interleaving the addresses: assigning odd addressed cells to represent the right going half, and even addressed cells to represent the left going half. This emulation requires taking two steps instead of one to advance in a given logical direction. When analyzing the time complexity of an algorithm, this overhead merely doubles the constant on the linear term, leaving the asymptotic order of complexity entirely unchanged. The outcomes of decider programs are unaffected. Therefore, utilizing a single ended tape is an inconsequential variation of the two way tape machine.

    - Furthermore, this definition adopts the language of hardware specification rather than formal mathematics. Because this is strictly a terminology change with direct correspondences between operations, the resulting model remains isomorphic to the Hopcroft and Ullman definition. Adopting this architectural perspective facilitates the overarching goal of building downward toward a computer architecture, rather than strictly preparing for highly abstract mathematical analysis, though it certainly does not preclude such analysis. In this model, the structural components are read-only, while the memory components will potentially have data flowing through them. As detailed in the control procedures below, data flows dynamically at the command of the programmed controller. + Furthermore, this definition adopts the language of hardware specification rather than formal mathematics. Because this is merely a terminology change with direct correspondences between operations, the resulting model remains isomorphic to the Hopcroft and Ullman definition. Adopting this architectural perspective facilitates the goal of building downward toward a computer architecture, rather than upward to abstract mathematical analysis, though it certainly does not preclude such analysis. In this model, the fixed components are read only, while the memory components will potentially have data flowing through them. As detailed in the control procedures below, data flows dynamically at the command of the programmed controller.

    @@ -553,27 +576,27 @@

    The Turing Machine consists of:

      -
    • a read-only alphabet that instances can be made from
      • -
    • a read-only empty-symbol that instances can be made from
      • -
    • a constant tape consisting of read-writable cells
      • -
    • a read-only left from leftmost error symbol that instances can be made from
      • +
    • a read only alphabet that instances can be made from
      • +
    • a read only empty symbol that instances can be made from
      • +
    • a constant tape consisting of read writable cells
      • +
    • a read only left from leftmost error symbol that instances can be made from
    • a head bearing tape transport mechanism
      • -
    • a read-writable single symbol read data buffer
      • +
    • a read writable single symbol read data buffer
    • a constant state machine programmed controller with states that can be referenced
      • -
    • a constant initial-state reference
      • -
    • a constant halt-state reference
      • -
    • a read-writable current-state reference variable
      • -
    • a constant operation procedure where each step of the procedure can be referenced
      • -
    • a read-writable procedure step reference counter
      • -
    • a read-only clock to synchronize control and to cause counting until the halt state is reached.
      • -
    • a fixed hardware comparator to evaluate state and symbol equivalencies
      • +
    • a constant initial state reference
      • +
    • a constant halt state reference
      • +
    • a read writable current state reference variable
      • +
    • a constant operation procedure where each step of the procedure can be referenced
      • +
    • a read writable operation procedure step reference counter
      • +
    • a read only clock to synchronize control and to cause counting until the halt state is reached.
      • +
    • a fixed hardware comparator to evaluate state and symbol equivalencies
    • a reset button that activates logic that initializes the machine

    Each highlighted term is a short name for the associated item.

    - The distinct empty-symbol can be any symbol that is excluded from the alphabet. Only instances of alphabet symbols or the empty-symbol are permitted to be written to the tape. + The distinct empty symbol can be any symbol that is excluded from the alphabet. Only instances of alphabet symbols or the empty symbol are permitted to be written to the tape.

    @@ -581,7 +604,7 @@

    - Depending on the design, without reset being held during power up, a real machine can land in an illegal state that might not be resetable, and could conceivably be damaging. Hence, reset is normally held during power up, and it is reset, not power up, that causes a machine to land in a known initial state. + Depending on the design, without reset being held during power up, a real machine can land in an illegal state that might not be resettable, and could conceivably be damaging. Hence, reset is normally held during power up, and it is reset, not power up, that causes a machine to land in a known initial state.

    @@ -589,50 +612,59 @@

    - Note that we packaged a tape transport along with the head. In this volume we will talk about sending step commands to the head, but of course a head doesn't step, rather a tape transport mechanism steps. Hence, we when we talk about 'stepping the head' implied is that the head remains stationary, and the tape transport mechanism moves the tape. As the tape is normally very long and held on spools, even if we could move the head it would not be very effective. + Note that we packaged a tape transport along with the head. In this volume we will talk about sending step commands to the head, but of course a head doesn't step, rather a tape transport mechanism steps. Hence, when we talk about 'stepping the head' implied is that the head remains stationary, and the tape transport mechanism moves the tape. As the tape is normally very long and held on spools, even if we could move the head it would not be very effective.

    - The constant operation procedure should not be conflated with the Turing Machine program. For a microcode controlled machine, the procedure will be found in microcode memory, and it will be execute as though a program. Each line of the procedure, when read, results in a set of bits being connected to the machine's control lines. Some of those control lines will control what the procedure does, and some will extend out into the data path and be used to configure execution units and gate data on to busses. + The constant operation procedure should not be conflated with the Turing Machine program. For a microcode controlled machine, the procedure will be found in microcode memory, and it will be executed as though a program. Each line of the procedure, when read, results in a set of bits being connected to the machine's control lines. Some of those control lines will control what the procedure does, and some will extend out into the data path and be used to configure execution units and gate data on to busses.

    - For a hardwired machine, the operation program will be expressed with logic gates and flip-flops (single bit memory registers). Whether a machine is microcode controlled, or hardware controlled, is a question of implementation. The values on the control lines remain the same independent of those implementation decisions, so those decisions are inconsequential to our architecture discussion. + For a hardwired machine, the operation program will be expressed with logic gates and flip flops (single bit memory registers). Whether a machine is microcode controlled, or hardware controlled is a question of implementation. The values on the control lines remain the same independent of those implementation decisions, so those decisions are inconsequential to our architecture discussion.

    - The Turing Machine architecture specifies an infinite tape, which can neither be implemented, nor realized. Though we will successful address this issue in a later chapter. + The Turing Machine architecture specifies an infinite tape, which can neither be implemented, nor realized. We will introduce a computationally inconsequential modification in a later chapter that causes the tape to be finite.

    The Turing Machine tape cell

    - A cell is the square from Alan Turing's 1936 paper . A cell is a foundational mathematical object that has no meaning apart from its distinct identity and its capacity to hold specific properties. Due to its distinct identity, a cell can be referenced. + A property is a pair, where the components are called the name and the value. A name is an instance of a symbol and it must uniquely identify the property within its context. The value is a variable that can be written then read back. +

    + +

    + A cell is the square from Alan Turing's 1936 paper 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.. Mathematically, a cell is a distinct identifiable set, with one to three property members depending on the type of cell.

    - A leftmost cell possesses exactly two compulsory properties: a right neighbor and data. A rightmost cell also possesses exactly two compulsory properties: a left neighbor and data. An interstitial cell possesses three compulsory properties: a left neighbor, a right neighbor, and data. While an island cell has a single compulsory property, that of data. + A leftmost cell is a set of two properties named 'right neighbor' and 'data'. A rightmost cell is a set of two properties named 'left neighbor' and 'data'. An interstitial cell is a set holding three properties: a 'left neighbor', a 'right neighbor', and 'data'. While an island cell owns a single compulsory property, that of 'data'.

    - Said neighbor properties are singular. When the left neighbor property is present, there is exactly one left neighbor, and when the right neighbor property is present, there is exactly one right neighbor. Said data property is restricted to being either an instance of an alphabet symbol, or alternatively, an instance of the empty-symbol. + The value of a cell neighbor property is limited to being the identity of a cell. The value of a cell data property can be an instance of an alphabet symbol, or alternatively, an instance of the empty symbol.

    +

    The Turing Machine tape

    - The Turing Machine tape is a set containing exactly one leftmost cell and an infinite number of interstitial cells. For each cell in the set, called cell A, if cell A has a right neighbor that is cell B, then cell B's left neighbor is cell A. Similarly, if cell A has a left neighbor of cell B, then cell B's right neighbor is cell A. Furthermore, every cell in the set must be reachable by traversing right neighbors starting from the leftmost cell. + The Turing Machine tape is a set containing exactly one leftmost cell and an infinite number of interstitial cells. For each cell in the set, called cell A, if cell A has a right neighbor that is cell B, then cell B's left neighbor is cell A. Similarly, if cell A has a left neighbor of cell B, then cell B's right neighbor is cell A. Furthermore, any cell in the set must be reachable by traversing right neighbors starting from the leftmost cell, in a finite number of steps.

    - Once a tape is defined, properties cannot be added nor removed from the constituent cells. Further, the neighbor properties definitions are fixed. Cells don't move, new cells cannot be added, and cells existing on the tape cannot be removed. We can say that a Turing Machine tape has a constant, fixed, linear topology. + To say that a tape is infinite, and to simultaneously require that any cell can be reached in a finite number of steps, might seem to be contradictory. However, these locutions are compatible in their meaning. By saying the tape is infinite, we are saying that after reaching a cell through a finite number of right neighbor hops, there will always be further cells to the right. So though any cell can be reached in finite hops starting at the leftmost cell, all of the cells cannot be reached.

    - Nor can the data property be removed; however, unlike neighbor properties, the data property value can be changed while the Turing Machine is running. + In conventional computation theory, once a tape is defined the cell neighbor properties values are fixed. Cells don't move, new cells cannot be added, and cells existing on the tape cannot be removed. This fits the definition of a space, so we can say that a Turing Machine tape has a constant, fixed, linear topology. This also matches the reality of hardware memories. On the other hand, it does not track well with general memory containers such as linked lists where destructive operations are often permitted.

    - Initially the Turing Machine tape is filled with empty symbols. A Turing Machine does not have the ability to fill an empty tape in a finite number of steps, so no Turing machine can compute and erase an entire tape. The initial tape must be made by other means, and at this point in our discussion we punt this issue, as all others have done thus far, and define an initial tape to have empty symbols in all of its cells. + Nor can a cell data property be removed; however, unlike for cell neighbor property values, the cell data property value can be changed while the Turing Machine is running. In fact some people would say this is the whole point of running a Turing Machine. +

    + +

    + Initially the Turing Machine tape is filled with empty symbols. However, as we noted above, a Turing Machine cannot visit all the cells on a tape, so a Turing Machine cannot erase a tape in advance for another Turing Machine to use. In a later chapter we will provide a computationally inconsequential alternative, but for now we will do as all have done before us, and decree that initial empty tapes are available by definition.

    @@ -640,13 +672,13 @@

    - Mathematically, a Turing Machine tape can be expressed as a path graph. However, a tape mode and a path graph model imply different ontological contexts. The neighbor property of a Turing Machine tape cell specifically informs a clock-driven atomic step function where to place the machine head next. The machine only has defined meaning at the state points on of the programmed controller. In contrast, a path graph exists in the wider context of graph theory. A path graph has edges and each edge can be focused on, said to be traversed over, and given general properties. These are things we explicitly excluded in the tape definition. If we were to move the tape from the context of the machine and into the more abstract mathematical context by modeling the tape with a path graph, then we would do so for the purpose of analyzing the tape, which is a higher order and more abstract activity. However, in the current exercise we are not reaching towards the more abstract; rather, we are reaching in the other direction, towards machine architecture. This is one of the reasons that Alan Turing's use of a tape, rather than a path graph, so elegantly suggested that computation theory applied to real machines and was not merely limited to the Entscheidungsproblem. + Mathematically, a Turing Machine tape can be expressed as a path graph. However, a tape model and a path graph model imply different ontological contexts. The neighbor property of a Turing Machine tape cell specifically informs a clock driven atomic step function where to place the machine head next. The machine only has defined meaning at the state points on the programmed controller. In contrast, a path graph exists in the wider context of graph theory. A path graph has edges and each edge can be focused on, said to be traversed over, and given general properties. These are things we explicitly excluded in the tape definition. If we were to move the tape from the context of the machine and into the more abstract mathematical context by modeling the tape with a path graph, then we would do so for the purpose of analyzing the tape, which is a higher order and more abstract activity. However, in the current exercise we are not reaching towards the more abstract; rather, we are reaching in the other direction, towards machine architecture.

    The Turing Machine tape head

    - The tape head consists of a reference to exactly one of the tape cells and a set of four functions: read, write, step-left, and step-right. In addition, the tape head can throw an error, left-of-leftmost, if the Turing Machine attempts to step left from the leftmost cell. When stepping, the cell reference in the head is updated based on the neighbor properties of the currently referenced cell. The cell referenced by the head is called the cell the head is on, or more simply, the head cell. + The tape head consists of a reference to exactly one of the tape cells and a set of four functions: read, write, step-left, and step-right. In addition, the tape head can throw an error, left-of-leftmost, if the Turing Machine attempts to step left from the leftmost cell. When stepping, the cell reference in the head is updated based on the neighbor properties of the currently referenced cell. The cell referenced by the head is called the cell the head is on, or more simply, the head cell, or more generally as the indicated cell.

    @@ -656,7 +688,11 @@

    Area and partitions

    - We call a subset of contiguous cells from a tape an area of said tape. A tape partition is a set of areas that completely span a tape. For any partition of a single-ended Turing Machine tape, at least one of the areas will necessarily be infinite. + 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 left most cell in the area, a right most 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 is not in the area. Or it is possible that the leftmost cell of an area is also the leftmost cell of the tape, and thus it does not have a left neighbor property. Any cell on a single ended tape will have a right neighbor property, but for a a finite area the rightmost cell's right neighbor property will indicate a cell that is not in the area. It is possible for an area to be open on the right, and thus not be finite. + In this manner we distinguish between tape cell types, and area cell types. +

    + +

    A tape partition is a set of areas that completely span a tape. For any partition of a single ended Turing Machine tape, at least one of the areas will necessarily be infinite. An area can also be partitioned, which would lead to nested areas.

    Head partition

    @@ -677,7 +713,7 @@

    Active area partition

    - A non-empty tape, one with at least one cell holding an alphabet symbol, can be partitioned into the following areas: + A non empty tape, one with at least one cell holding an alphabet symbol, can be partitioned into the following areas:

      @@ -688,10 +724,23 @@
    1. The right empty tail: The infinite set extending from the right neighbor of the rightmost cell of the active area, extending rightward.
    -

    The impossibility of recognizing an empty tape

    +

    Area implied partition

    + +

    + Given any area on a tape, or nested within another area, there is implied the potential for two more areas. +

    + +
      +
    1. The left side: a potentially empty finite set containing all of the cells to the left of the given area.
      2. +
    2. The given area.
      3. +
    3. The right side: for a tape, the infinite set extending rightward from the rightmost of the given area. For a nested area, this could be empty or finite.
      4. +
    + + +

    The impossibility of recognizing an empty tape

    - Recognition is a process where a Turing Machine decides if a pattern is present on a tape solely by reading symbols found on the tape. No meta-information, such as a message communicating something about the area being examined or the nature of the program that wrote the symbols, can be taken into account. + Recognition is a process where a Turing Machine decides if a pattern is present on a tape solely by reading symbols found on the tape. No meta information, such as a message communicating something about the area being examined or the nature of the program that wrote the symbols, can be taken into account.

    @@ -699,17 +748,17 @@

    - Recognizing that a tape is empty is not possible. Suppose it were attempted, and a machine started scanning the tape rightward from the leftmost cell; for every cell that was empty, the machine would have to scan further rightward to check for an alphabet cell. If the tape is truly empty, the recognizer would never stop scanning, so no decision would be rendered. + Recognizing that a tape is empty is not generally possible. Suppose it were attempted, and a machine started scanning the tape rightward from the leftmost cell; for every cell that is discovered to be empty, the machine would have to scan further rightward to check for an alphabet cell. If the tape is truly empty, the recognizer would never stop scanning, so no decision would be rendered.

    -

    The impossibility of recognizing the rightmost cell of the active area

    +

    The impossibility of recognizing the rightmost cell of the active area

    In general, a Turing Machine cannot step across a tape reading cells to recognize the rightmost cell of the active area, or equivalently, the leftmost cell of the right empty tail. Suppose a recognizer attempted this by starting in the active area and stepping right, and the machine discovered an empty cell. The machine would not be able to distinguish between the case of said empty cell being embedded within the active area (meaning more alphabet cells lie further to the right), or the case where said cell is genuinely the leftmost cell of the right tail. To resolve the ambiguity, the machine would be obligated to continue stepping right. Yet, there would never come a time where finding another empty cell would not lead back to the exact same case ambiguity, so the machine would forever step right without returning a decision.

    - It follows that if knowledge of the end of the active area is needed, this information must be encoded as a message. For example, a special symbol can be reserved in the alphabet specifically to serve as the end of active area marker. Each time a machine steps beyond the current end of active area marker and does a write, it writes the marker in the right neighbor cell, and goes back and erases the old mark. This method is related to communications theory and the science of signaling. Here, the active area marker is an out-of-band control signal. + It follows that if knowledge of the end of the active area is needed, this information must be encoded as a message. For example, a special symbol can be reserved in the alphabet specifically to serve as the end of active area marker. Each time a machine steps beyond the current end of active area marker and does a write, it writes the marker in the right neighbor cell, and goes back and erases the old mark. This method is related to communications theory and the science of signaling. Here, the active area marker is an out of band control signal.

    @@ -724,21 +773,21 @@ When an input tape is provided as a general mathematical object, either decreed by definition or perhaps abstracted from 'what a Turing Machine computation would produce in the limit of step count', then the input can be either finite or infinite.

    -

    In-band and out-of-band control

    +

    In band and out of band control

    - In-band control occurs when control signals or structural metadata are mixed directly into the same channel and alphabet as the data payload. In-band signaling leads to ambiguities between what is control and what is data. As we saw, there are cases a recognizer, i.e. merely examining the data, is not capable of resolving even the simplest of control questions. A conventional approach for resolving these ambiguities makes use of escape sequence schemes that grow in length as the levels of communication grow. This has always been an after thought, a sort of hack. + In band control occurs when control signals or structural metadata are mixed directly into the same channel and alphabet as the data payload. In band signaling leads to ambiguities between what is control and what is data. As we saw, there are cases a recognizer, i.e. merely examining the data, is not capable of resolving even the simplest of control questions. A conventional approach for resolving these ambiguities makes use of escape sequence schemes that grow in length as the levels of communication grow. This has always been an after thought, a sort of hack.

    - In contrast, out-of-band control communicates structural information through a strictly separate channel or by utilizing symbols definitively excluded from the programmer-visible data alphabet. The rightmost tape marker is an out-of-band mechanism because it utilizes an expanded hardware tape alphabet strictly reserved for machine management, guaranteeing it can never be conflated with the user's data. Modern architectures often do not have the luxury of inventing new symbols to serve as control rather than data. Another out-of-band signaling technique is to structure the data into channels; such structure is called formatting. We find formatting on hard drives, in frame-based and packet-based communication channels, and in data structures. + In contrast, out of band control communicates structural information through a strictly separate channel or by utilizing symbols definitively excluded from the programmer visible data alphabet. The rightmost tape marker is an out of band mechanism because it utilizes an expanded hardware tape alphabet strictly reserved for machine management, guaranteeing it can never be conflated with the user's data. Modern architectures often do not have the luxury of inventing new symbols to serve as control rather than data. Another out of band signaling technique is to structure the data into channels; such structure is called formatting. We find formatting on hard drives, in frame based and packet based communication channels, and in data structures.

    The programmed controller

    - There are two distinct centers of logical control for the Turing Machine. The one we are most familiar with from the many descriptions of the Turing Machine in papers is called the 'state controller'; however, this term becomes confusing when we realize there are additional controllers involved. Hence, we call this component the programmed controller. It is customized for each different problem the Turing Machine will work on, while the other controllers give the Turing Machine its fixed characteristics. + There are two distinct centers of logical control for the Turing Machine. The one we are most familiar with from the many descriptions of the Turing Machine in papers is called the 'state controller'; however, this term becomes confusing when we realize there are additional controllers involved. Hence, we call this component the programmed controller. It is customized for each different problem the Turing Machine will work on, while the other controllers give the Turing Machine its fixed characteristics.

    Components

    @@ -759,7 +808,7 @@

    - Each state corresponds to a symbol. Here, instances of the state symbols appear in a different context than that of the data alphabet symbols or the empty-symbol, and thus they do not need to be distinct from them. In real machines, state symbol instances are unsigned integers. + Each state corresponds to a symbol. Here, instances of the state symbols appear in a different context than that of the data alphabet symbols or the empty symbol, and thus they do not need to be distinct from them. In real machines, state symbol instances are unsigned integers.

    @@ -795,7 +844,7 @@

    - Each defined row of the instruction table specifies a state-specific instruction to be given to the tape transport unit. The instruction will be one from the set: { no-op, step-left, step-right, and write(value) }, where the value parameter for a write instruction is coded directly as part of the instruction. + Each defined row of the instruction table specifies a state specific instruction to be given to the tape transport unit. The instruction will be one from the set: { no-op, step-left, step-right, and write(value) }, where the value parameter for a write instruction is coded directly as part of the instruction.

    Control logic

    @@ -840,14 +889,14 @@

    - For a real machine, the step pulse will be derived from the machine clock. The clock will have a constant period, so there is a constant duration of time that will be the same for each pass through the execution procedure. Thus, if we replace the step count with a count of clock ticks, we will get the same decider and complexity results as we would have from step counts. This fits the definition we have been using for inconsequential. + For a real machine, the step pulse will be derived from the machine clock. The clock will have a constant period, so there is a constant duration of time that will be the same for each pass through the execution procedure. Thus, if we replace the step count with a count of clock ticks, we will get the same decider and complexity results as we would have from step counts. This fits the definition we have been using for inconsequential.

    An alternative: stored program and sequencer

    - The Universal Turing Machine, proposed by Alan Turing, introduced a profound architectural inversion: relocating the defining state tables from hardwired logic, or manually configured patch panels, directly onto the tape itself. This enables replacing the custom programmed controller with a fixed controller that derives its behavior dynamically from the tape data. Consequently, a single, immutable hardware architecture can simulate the execution of any conceivable Turing Machine. + The Universal Turing Machine, proposed by Alan Turing, introduced a profound architectural inversion: relocating the defining state tables from hardwired logic, or manually configured patch panels, directly onto the tape itself. This enables replacing the custom programmed controller with a fixed controller that derives its behavior dynamically from the tape data. Consequently, a single, immutable hardware architecture can simulate the execution of any conceivable Turing Machine.

    @@ -855,19 +904,19 @@

    - We can optimize this representation. Instead of storing the state tables verbatim, we can list a sequence of instructions directly on the tape. To achieve this, the architecture expands to support two distinct categories of instructions: the physical tape transport instructions we defined previously, and a newly introduced category of control instructions. The programmed controller is then replaced with a fixed hardware controller called a sequencer. + We can optimize this representation. Instead of storing the state tables verbatim, we can list a sequence of instructions directly on the tape. To achieve this, the architecture expands to support two distinct categories of instructions: the physical tape transport instructions we defined previously, and a newly introduced category of control instructions. The programmed controller is then replaced with a fixed hardware controller called a sequencer.

    - Because the original state tables allowed for non-linear execution paths, the instruction sequence on the tape cannot always execute in a straight line. Therefore, the architect must include at least two control instructions: a halt instruction and a test-and-branch instruction. The sequencer starts at the first instruction in the program, perhaps at the leftmost cell on the tape, and evaluates it. If it is a control instruction, the sequencer acts upon it directly to alter the flow of execution or stop the machine. Otherwise, if it is a physical instruction for the head unit, the sequencer passes it down to the tape transport. + Because the original state tables allowed for non linear execution paths, the instruction sequence on the tape cannot always execute in a straight line. Therefore, the architect must include at least two control instructions: a halt instruction and a test and branch instruction. The sequencer starts at the first instruction in the program, perhaps at the leftmost cell on the tape, and evaluates it. If it is a control instruction, the sequencer acts upon it directly to alter the flow of execution or stop the machine. Otherwise, if it is a physical instruction for the head unit, the sequencer passes it down to the tape transport.

    - Because we have not yet derived Natural Numbers or memory addresses in this architecture, a test-and-branch instruction cannot jump to a numerical address. Instead, it must operate topologically. A topological branch instruction simply commands the sequencer to scan the tape for a specific target symbol, and resume executing instructions from that physical location. + Because we have not yet derived Natural Numbers or memory addresses in this architecture, a test and branch instruction cannot jump to a numerical address. Instead, it must operate topologically. A topological branch instruction simply commands the sequencer to scan the tape for a specific target symbol, and resume executing instructions from that physical location.

    - As noted in the prior section, an instruction consists of an instruction code and potentially an argument. There are many choices that can be made in instruction set design. Among those choices, almost all will be inconsequential from a computation-theoretic point of view, but almost all will introduce strict efficiency trade-offs in physical hardware. + As noted in the prior section, an instruction consists of an instruction code and potentially an argument. There are many choices that can be made in instruction set design. Among those choices, almost all will be inconsequential from a computation theoretic point of view, but almost all will introduce strict efficiency trade offs in physical hardware.

    Machine control

    @@ -891,64 +940,111 @@

    Main:

      -
    1. Based on the value of the reset line coming from the rest button: -
        -
      1. true: Reset -
      2. Based on the value of the halt line coming from the comparator: -
          -
        1. true: freeze here until reset is asserted, and go to step 1. -
        2. false: on each clock tick send a step pulse to the programmed controller or sequencer, depending on which is used. -
        -
      +
    2. evaluate the value of the reset line coming from the reset button
      3. +
    3. if the reset line is true, execute the Reset procedure
      4. +
    4. if the reset line is false, evaluate the halt line coming from the comparator
      5. +
    5. if the halt line is true, freeze execution until reset is asserted, then return to step 1
      6. +
    6. if the halt line is false, send a step pulse to the programmed controller (or sequencer) on each clock tick

    - We can read this procedure with the caveat, "if we could realize such a machine, this is what we would do. Later, these directions can be modified and applied to the machine variation that has an expanding tape. + A person can read this procedure with the caveat, "if we could realize such a machine, this is what we would do." Later, these directions can be modified and applied to the machine variation that has an expanding tape. +

    + +

    + To start the machine we must first select a tape. Common choices are an empty tape, a tape with data on it the machine is to decide matches a given language pattern, or a tape with a Turing Machine on it to be analyzed. After the tape is selected it is mounted on the Turing Machine, then the reset button is hit. +

    + +

    + After the reset button is released, the machine begins stepping. If the program is a computation, the machine will eventually halt. If the machine eventually halts, then we know the associated program was a computation. Otherwise we do not know. Any amount of time we wait where the machine has not halted, we will not know that it will ever halt. Hence, we cannot in general use 'running a Turing Machine' as a means to determine if a given program is computational. (We could instead try to answer the question 'is it computational' through analysis, but there too, Turing has shown that in general that will not work either.)

    + Computational Analysis +

    - To start the machine we must first select a tape. Common choices are an empty tape, a tape with data on it the machine is to decide matches a given language pattern, a tape with a Turing Machine on it to be analyzed. After the tape is selected it is mounted on the Turing Machine, then the reset button is hit. + In mathematics, analysis is the rigorous study of limits, continuity, rates of change, and bounds. It encompasses several specialized branches. Real analysis studies the behavior of real numbers, sequences, and continuous functions. Complex analysis extends these principles to functions of complex variables. Functional analysis examines vector spaces where the elements themselves are functions. Numerical analysis focuses on the design of algorithms to yield approximate solutions for continuous mathematical problems. Harmonic analysis studies the representation of functions or signals as the superposition of basic waves, such as Fourier series. Across all these branches, analysis provides a formal framework for evaluating mathematical objects.

    - After the reset button is released, the machine begins stepping. If the program is a computation, the machine will eventually halt. If the machine eventually halts, then we know the associated program was computation. Otherwise we do not know. Any amount of time we wait where the machine has not halted, we will not know that it will ever halt. Hence, we can not in general use 'running a Turing Machine' as a means to determine if a given program is computational. (We could instead try to answer the question 'is it computational' through analysis, but there too, Turing has shown that in general that will not work either.) + In computation theory, computational analysis is defined as the static evaluation of a formal system or machine definition to deduce its absolute boundaries and properties. This process remains entirely distinct from dynamically executing the machine to yield a computed result.

    +

    + This analytical perspective has been present from the very beginning of the field. Alan Turing's formulation of the halting problem relies explicitly on one machine examining another. To properly evaluate the limits of this analytical capacity, the examining machine must necessarily be given definitions of programs that loop infinitely. Consequently, the foundational proofs of computer science formally establish non computational programs as legitimate objects of analytical study. +

    +

    + We call a program that examines another program so as to deduce properties of its results an analyzer. The program or machine definition being subjected to this evaluation is called the studied program or studied machine. +

    +

    + Turing Machines that halt in a finite number of steps for any finite input within a stipulated domain are said to be computational over that domain. +

    -

    That includes sets. When two current state sets have the same members, we say they are instances of the same state. When they have different members we say they are different states.

    +

    + By definition, first order analysis is the running of programs, and it has its place. + Generally, when we want to know what output a computational machine will produce, the fastest route to this knowledge is to run the program. Most programs written to solve problems are most effectively run to solve those problems rather than analyzed to deduce what they will produce. +

    +

    + Now suppose we quantify the inputs to a computational machine over a domain and want to know a property of the machine. Perhaps, say, that it always produces an even number. If the quantification is over a large set, then it might be faster to study the machine than it would be to run the machine on every input in the domain while checking its output. +

    - Orders of analysis +

    + Further suppose that the quantification is over an infinite input space. Then the only option for answering a question about the properties of results from such a machine is to analyze it. This is second order analysis, also called simply analysis. +

    -

    Turing Machines that halt in finite number of steps for any finite input withing a stipulated domain are said to be computational over that domain. It is also the case that for any given non-computational Turing Machine over a given domain there will exist at least one input for which the machine will never halt.

    +

    + Famously, we know that universally an analyzer cannot determine if a machine is computational. This knowledge was derived by reasoning about the properties of a hypothetically existing analyzer machine. This is a third order analysis activity. +

    -

    By analyzing a machine we might learn that some machine, say M, is conditionally computational on a given set of inputs, say I.

    +

    + As an analyzer does not run the machine being studied, it is not required to be a machine that halts. Suppose we have a machine that produces an infinite sequence of digits to a tape without halting. A limit analyzer could examine that machine and, in some cases, determine if it has asymptotic behavior. For example, recognizing that appending a binary fractional sequence of 0.1111... indefinitely evaluates in the limit to 1.0. In this manner, the use of analyzers facilitates using computation theory for deriving higher order mathematics. +

    -

    Then in case of the strongest delineation, computation will not halt for members of I, or it might be the case that we don't know about the inputs in I, or that I analysis shows that I can be broken into two sets, inputs for which computation will not stop for, and those for which analysis does not provide an answer.

    +Computational Naturalism -

    To analyze a given First Turing Machine we do not execute it, but instead we place its definition on the tape as input for a second Turing Machine to analyze.

    +

    + Given that we have an architectural definition for a Turing Machine, and will modify this in a later chapter to define a realizable Turing Complete architecture, it is possible to invert the foundation of mathematics. Instead of mathematics preceding computation, we posit that given a Turing Machine exists, all of mathematics is an interpretation of what can be done with it. +

    -

    We might also add on the input tape to the first Turing Machine another Turing Machine that defines the language input to the first Turing Machine, otherwise we have to assume that any input is possible.

    +

    + We begin by defining the tape cell as a location in a physical memory, which provides us with arrays of charge configurations. +

    -

    We then run this analysis Turing Machine so as to learn about the first one. (It is often the case that this second Turing Machine which performs the analysis is in a person's brain. Perhaps the person is writing a proof about the Turing Machine that is being studied.)

    +

    + We then define the symbol in computational terms, as done in the prior section. This begins with memory addresses, represented as charge vectors, acting as primitive symbols, and extends to the symbol factory, copy operations, and instance comparison. +

    -

    Analysis can be applied to both computational and non-computational machines. In many cases it is possible to learn something useful about a non-computational machine, one which we cannot evaluate, by analyzing it.

    +

    + Logic is then defined on top of relay switch logic, as Shannon and others have already done. A machine that requires all inputs to be the '1' symbol to produce a '1' symbol output is a conjunction machine, and so forth. +

    + +

    + On top of this we can define the Peano Machine, a counter, and then use that machine as the definition of Natural Numbers. +

    -

    The term 'analysis' can be applied more generally. Accordingly, 'first order' analysis is the same as computation. ‘Second order’ analysis is what was described in the prior paragraph.

    +

    + Where Gödel reduced logic to natural numbers, we go the other direction to expand upon logic from natural numbers. +

    -

    When the term 'analysis' is used without further qualification, the order is implied by context, but typically we are referring to second order analysis.

    +

    + Frege's set theory is then the analysis of a logic program against an enumeration of inputs to choose if a proposed symbol is in a set. Perhaps executing such a program is left to first order analysis, or perhaps execution is not practical, and evaluation is left to second order analysis. +

    -

    It is possible to perform higher orders of analysis. Here is a question that belongs to the next higher order: does there exist a Turing Machine A that can analyze a Turing Machine B to decide if B will always halt? We are standing above looking down and asking a question about machines of the second order.

    +

    + Russell's Paradox is then defined as a Turing Machine that can be analyzed in the second order, but cannot be analyzed in the first order. +

    -
    +

    + With this language it is possible to restate every statement ever made by any mathematician; the mere fact that a mathematician was able to state it qualifies its membership into Zermelo's S set. +

    Addresses and cells

    Natural Number, address

    -

    We can define a Turing Machine that is identical to the recursive definition of Natural Numbers as given by Peano.

    +

    We can define a Turing Machine that is identical to the recursive definition of Natural Numbers as given by Peano. Giuseppe Peano, Arithmetices principia, nova methodo exposita (Turin: Fratres Bocca, 1889).

    As the Natural Numbers never end, we cannot run this machine to a halting point, but we can analyze it.