From: Thomas Walker Lynch Date: Mon, 6 Jul 2026 07:08:54 +0000 (+0000) Subject: . X-Git-Url: https://git.reasoningtechnology.com/%28%5B%5E?a=commitdiff_plain;h=79f24d6c55b423cad513bf2aa79b8a4e3f936bcf;p=TM-2026 . --- diff --git a/document/TM-2026.html b/document/TM-2026.html index 48f405b..52fce50 100644 --- a/document/TM-2026.html +++ b/document/TM-2026.html @@ -836,124 +836,58 @@

- 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. - 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 leads to nested areas. -

- -

Head partition

- -
    -
  1. The left side: a potentially empty finite set containing all of the cells to the left of the head cell.
    2. -
  2. Head: the head cell.
    3. -
  3. The right side: the infinite set extending rightward from the right neighbor of the head cell.
    4. -
- -

Leftmost/remaining partition

- -
    -
  1. Leftmost: the leftmost cell.
    2. -
  2. Remaining: the infinite set including the right neighbor of the leftmost cell, and all cells further to the right.
    3. -
- -

Active area partition

- -

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

- -
    -
  1. The left empty tail: this area is empty when the leftmost cell is nonempty. Otherwise, it consists of the leftmost cell and the empty cells, if any, to the right of the leftmost cell, up to the first alphabet cell.
    2. - -
  2. Active area: a finite area for computational problems, containing the cells extending from the leftmost alphabet cell up to and including the rightmost alphabet cell. It is possible that the leftmost alphabet cell and the rightmost alphabet cell will be the exact same cell.
    3. - -
  3. The right empty tail: the infinite set extending from the right neighbor of the rightmost cell of the active area, extending rightward.
    4. -
- -

Area implied partition

- -

- Given any area on a tape, or nested within another area, the potential for two additional areas is implied. -

- -
    -
  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 cell 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. -

- -

- The active area partition of a tape only works for tapes that have at least one alphabet cell. As soon as a machine does its first write of an alphabet symbol, it is known the tape has at least one alphabet symbol. However, what if a tape of unknown status, whether completely empty or containing an alphabet cell, is mounted on a tape machine, and it is desired that the machine recognize if the tape is empty or has an alphabet cell? This is the equivalent problem of looking for the leftmost cell of an active area. -

+ Computational Naturalism

- Recognizing that a tape is empty is generally impossible. 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 ever be rendered. + 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.

-

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 be unable 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 avoid leading back to the exact same case ambiguity, so the machine would forever step right without returning a decision. + We begin by defining the tape cell as a location in a physical memory, which provides us with arrays of charge configurations.

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

- If a tape is written by an initializing tape machine, unmounted, and then mounted on a second analyzing tape machine, the analyzing tape machine is starting with a populated tape. Similarly, a mathematician can, by decree, define an initial tape that holds predefined alphabet symbols. In these cases, the analyzing machine cannot use the signaling method described in the prior paragraph, as it was excluded from controlling all the writes of alphabet symbols to the tape. The only solution to this problem is for the initializing machine and the analyzing machine to use a shared communication protocol for signaling the end of the active area. + 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.

- By definition, a computation must finish in a finite number of steps. It follows that the active area when a Turing Machine halts a computation will always be finite. Consequently, if a Turing Machine initially starts working with a tape that was computed by another Turing Machine, the length of the input will be finite. Still, that input will be in the active area, so the receiving Turing Machine will need to read control data left on the tape under a common communication protocol to be able to find the bounds of the input area. + On top of this we can define the Peano Machine, a counter, and then use that machine as the definition of Natural Numbers.

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

-

In band and out of band control

-

- Because of the impossibility of recognizing certain tape features, when a tape is written by one Turing Machine, then used by another, there must be some sort of system for messaging control. There are two approaches for mixing data and control together: one is in band signaling, while the other is out of band signaling. + An axiomatic proof is then a decider that is built up from subroutine calls to the axioms. We might then quantify over all possible compositions of our subroutines in analysis and ask if it is possible that a contradiction decider would return Y or N.

- 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 where a recognizer, i.e., merely examining the data, is completely incapable 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 afterthought, a sort of hack. + 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.

- 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 lack 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. + Russell's Paradox will then be expressed as a Turing Machine that can be analyzed in the second order, but cannot be analyzed in the first order. That is, the paradox exists merely in the first order as it will never halt when run. However, it is not a paradox in the second order. It is, of course, through second order analysis that we are able to describe why Russell's Set description does not resolve in the first order.

-

Abstract areas and partitions

-

- A tape area and partitioning can be an abstraction defined by a function rather than merely by a leftmost and rightmost cell. Such areas can have different topologies than those of the base tape. A familiar example for most computer scientists is utilizing software to create the appearance of a two dimensional array over a linear memory. + With the language of Computational Naturalism it is possible to restate every statement ever made by any mathematician; we might say that the mere fact that a mathematician was able to state something qualifies its membership into Zermelo's S set.

- Accordingly, suppose there is a three tape Universal Turing Machine gasket machine that holds the definition of a base machine on a first tape. It calls the base machine as a subroutine to access the base machine's tape (the second tape), and it uses its own tape to organize the tape abstraction. Then this outer machine can present to its user a variety of transforms of the base machine's tape. + Given our knowledge that a universal halting problem analyzer that decides if a Turing Machine program halts does not exist, it is desirable to have rules that guide our writing only Turing Machine programs that are known to halt. This can be done through construction, as proposed by Russell, or through axioms of separation, mapping, and choice, as proposed by Zermelo and others. But then we know from the completeness and correctness theorems, that when such guiding rules are applied, there will exist Turing Machine programs that do halt, but whose definition cannot be constructed, nor surmised using the said axioms.

- As an example, a gasket machine could partition the base machine's tape into two areas, one consisting of the odd addressed cells, and the other of the even addressed cells. Though these two areas consist of noncontiguous cells on the base tape, when viewed through the gasket machine, they appear perfectly contiguous. In this case, each area will have a leftmost cell and remain open on the right. + This brings us back to the reference from the Zermelo discussion in the introduction. Specifically, the question posed is if our finding through second order analysis that Russell's paradoxical set formulation will not run in the first order and define a set, does this mean that we are merely using S to state that we are tossing out sets that cannot be defined? The answer is it is not quite this simple. Had Frege said, 'we merely dismiss such sets', we would not have the formalization for the second order analysis. Throwing out Russell's paradox simply because we ran it and it never halted is not a practical approach. Instead, we omit it specifically from S because analyzing R reveals that it fails to define a first order halting machine, and S, by definition, only holds first order halting machines. Zermelo's language is precise and formal. However, it is this bothersome nuance, now articulated here, which caused us to scratch our heads when reading Zermelo the first time.

- When the base machine tape cells behind an abstract area are physically contiguous, we say that the abstract area is compact. In the odd even tape partition example, the abstract areas lack compactness. + Russell's set formulation, R, can be analyzed to determine that it would not halt if it were run. We still keep it as a second order object, and have even given it a name, R. However, this begs the question, are there machines that cannot even be analyzed in the second order to ascertain if they would return a result in first order analysis, i.e., when they are run? If Gödel has a say here, a person would wager that such machines exist. But then, is there an option for analysis in the third order? Is Turing's halting proof a third order analysis as it reasons about running the second order analyzer? Or is it merely a recursive application of second order analysis?

Computational Analysis @@ -999,186 +933,169 @@ 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.

- Computational Naturalism + Addresses + +

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).

+ +

If we were to run the Natural Numbers Machine and watch as it writes to the tape, we would watch as the Natural Numbers are printed one after another, '·s·ss·sss·ssss· ...'. Here we are using the middle dot as a terminator symbol. As the Natural Numbers Machine never halts, we cannot use the Natural Number Machine to initialize a tape, but we can analyze the machine. When the leftmost cell holds a terminator, we say it has the value 'zero'. We call '·s·' the number 'one'. Each set of 's' adjacent symbols surrounded by the terminators, and zero, is said to be a Natural Number. +

+ +

To say that one Natural Number A is smaller than Natural Number B is to say that A would occur on the Natural Number Machine tape to the left of B, if the machine were to be run. Similarly, if B were said to be greater than A, that would mean B would occur further to the right. To increment a Natural Number is to find its right neighbor. To decrement a number is to find its left neighbor. +

- 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. + Now suppose we have a single Natural Number on a tape, say 'sss·', then we can define an Increment machine that, when initialized with a tape, writes an additional 's' and a new terminator. For example, when initialized with 'sss·' and then run, it produces 'ssss·'. In an analogous manner, we can define a Decrement machine. +

+ +

As such, we can assign a Natural Number to each member of the tape sequence through the following procedure. Given a machine, say P, we lock to it a second Address Machine, say A_P. When P's tape is first mounted, at the same time A_P is mounted with a tape that has only a terminator symbol, '·'. When P steps right, A_P writes an 's', steps right, and writes the terminator. For each step left of P, the machine A_P steps left and writes the terminator symbol.

- We begin by defining the tape cell as a location in a physical memory, which provides us with arrays of charge configurations. + The Natural Number found on A_P is then called the address of the cell the head is on for machine P. As each increment and decrement of the address is a constant time operation, keeping the address of the cell the head is on is a computationally inconsequential action.

- We then define the symbol in computational terms, as done in the prior section. This begins with memory addresses, represented as charge arrays, acting as primitive symbols, and extends to the symbol factory, copy operations, and instance comparison. + An address space is the set of addresses that would be placed into correspondence to cells if we were to step across all those cells while assigning an address to each cell the head is on. Address space is a second order concept. It is because assigning an address to a cell can be a second order concept that we often take it for granted.

- 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. + It is interesting to contemplate if the Natural Number at the basis of the Address Machine instead used Arabic Representation. In that case, an increment or decrement would be a linear time operation due to the carry. Consider a constant time operation on the base machine P, say it steps to the right 5280 cells and writes a mile marker. It would still be a constant time operation no matter the time taken for the address increment. So then consider an operation that is linear time, say a n, where n is the length of the input. Say, for example, it stepped n cells to the right and wrote a marker. If the increment time grows linearly with each step, then what was time n without addresses becomes time n^2 with them, or polynomial time. So indeed using Arabic Representation instead of unary for the address would be computationally consequential for all but constant time machines, if addresses were kept track of in the first order.

- On top of this we can define the Peano Machine, a counter, and then use that machine as the definition of Natural Numbers. + As of the time of this writing, real machines keep the illusion of constant time Arabic Representation for addresses due to fixed width clocked operations. When the width is fixed, and that fixed number is small enough that interconnect delay does not dominate over computation element delay, adder time tends to be logarithmic. The same is true of the related problem of address decoding. Thus an add can be performed within 'one clock tick'. Native address arithmetic operations on modern machines are always done in this manner. Perhaps there exists a special problem, such as pointer swizzling, where address arithmetic would come under software control.

- Where Gödel reduced logic to natural numbers, we go the other direction to expand upon logic from natural numbers. + The model used here to define an address is analogous to keeping a pointer into memory, and then using that pointer value as the address. Each step then increments or decrements the pointer. On a real machine, to access memory requires sending that pointer on a trip through a virtual memory system perhaps, and then through an address decoder. In the Turing Machine model, the head directly indicates a location, so it is more akin to the output of the memory decoder, though unlike the output of a memory decoder, it is stateful, i.e., a persistent value that can be moved incrementally. The analogy with the Turing Machine model holds due to the fiction of a constant decode time, as described in the prior paragraph.

- An axiomatic proof is then a decider that is built up from subroutine calls to the axioms. We might then quantify over all possible compositions of our subroutines in analysis and ask if it is possible that a contradiction decider would return Y or N. + At a higher level, that virtual memory system level, the memory architecture begins to look more like that of a Turing Machine. The translation lookaside buffer provides stateful location context, and the neighbor relationship between pages might be taken into account for performance reasons. However, once a program starts performing at virtual memory page fetch times instead of local system memory access times, we say that it is page thrashing and know it will become too slow to wait on, no matter its computation complexity class.

+ + Area and partitions +

- 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. + 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. + 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 leads to nested areas.

+

Head partition

+ +
    +
  1. The left side: a potentially empty finite set containing all of the cells to the left of the head cell.
    2. +
  2. Head: the head cell.
    3. +
  3. The right side: the infinite set extending rightward from the right neighbor of the head cell.
    4. +
+ +

Leftmost/remaining partition

+ +
    +
  1. Leftmost: the leftmost cell.
    2. +
  2. Remaining: the infinite set including the right neighbor of the leftmost cell, and all cells further to the right.
    3. +
+ +

Active area partition

+

- Russell's Paradox will then be expressed as a Turing Machine that can be analyzed in the second order, but cannot be analyzed in the first order. That is, the paradox exists merely in the first order as it will never halt when run. However, it is not a paradox in the second order. It is, of course, through second order analysis that we are able to describe why Russell's Set description does not resolve in the first order. + A nonempty tape, one with at least one cell holding an alphabet symbol, can be partitioned into the following areas:

+
    +
  1. The left empty tail: this area is empty when the leftmost cell is nonempty. Otherwise, it consists of the leftmost cell and the empty cells, if any, to the right of the leftmost cell, up to the first alphabet cell.
    2. + +
  2. Active area: a finite area for computational problems, containing the cells extending from the leftmost alphabet cell up to and including the rightmost alphabet cell. It is possible that the leftmost alphabet cell and the rightmost alphabet cell will be the exact same cell.
    3. + +
  3. The right empty tail: the infinite set extending from the right neighbor of the rightmost cell of the active area, extending rightward.
    4. +
+ +

Area implied partition

+

- With the language of Computational Naturalism it is possible to restate every statement ever made by any mathematician; we might say that the mere fact that a mathematician was able to state something qualifies its membership into Zermelo's S set. + Given any area on a tape, or nested within another area, the potential for two additional areas is implied.

+ +
    +
  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 cell of the given area. For a nested area, this could be empty or finite.
    4. +
+ + +

The impossibility of recognizing an empty tape

- Given our knowledge that a universal halting problem analyzer that decides if a Turing Machine program halts does not exist, it is desirable to have rules that guide our writing only Turing Machine programs that are known to halt. This can be done through construction, as proposed by Russell, or through axioms of separation, mapping, and choice, as proposed by Zermelo and others. But then we know from the completeness and correctness theorems, that when such guiding rules are applied, there will exist Turing Machine programs that do halt, but whose definition cannot be constructed, nor surmised using the said axioms. + 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.

- This brings us back to the reference from the Zermelo discussion in the introduction. Specifically, the question posed is if our finding through second order analysis that Russell's paradoxical set formulation will not run in the first order and define a set, does this mean that we are merely using S to state that we are tossing out sets that cannot be defined? The answer is it is not quite this simple. Had Frege said, 'we merely dismiss such sets', we would not have the formalization for the second order analysis. Throwing out Russell's paradox simply because we ran it and it never halted is not a practical approach. Instead, we omit it specifically from S because analyzing R reveals that it fails to define a first order halting machine, and S, by definition, only holds first order halting machines. Zermelo's language is precise and formal. However, it is this bothersome nuance, now articulated here, which caused us to scratch our heads when reading Zermelo the first time. + The active area partition of a tape only works for tapes that have at least one alphabet cell. As soon as a machine does its first write of an alphabet symbol, it is known the tape has at least one alphabet symbol. However, what if a tape of unknown status, whether completely empty or containing an alphabet cell, is mounted on a tape machine, and it is desired that the machine recognize if the tape is empty or has an alphabet cell? This is the equivalent problem of looking for the leftmost cell of an active area.

- Russell's set formulation, R, can be analyzed to determine that it would not halt if it were run. We still keep it as a second order object, and have even given it a name, R. However, this begs the question, are there machines that cannot even be analyzed in the second order to ascertain if they would return a result in first order analysis, i.e., when they are run? If Gödel has a say here, a person would wager that such machines exist. But then, is there an option for analysis in the third order? Is Turing's halting proof a third order analysis as it reasons about running the second order analyzer? Or is it merely a recursive application of second order analysis? + Recognizing that a tape is empty is generally impossible. 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 ever be rendered.

- Addresses +

The impossibility of recognizing the rightmost cell of the active area

-

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).

+

+ 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 be unable 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 avoid leading back to the exact same case ambiguity, so the machine would forever step right without returning a decision. +

-

If we were to run the Natural Numbers Machine and watch as it writes to the tape, we would watch as the Natural Numbers are printed one after another, '·s·ss·sss·ssss· ...'. Here we are using the middle dot as a terminator symbol. As the Natural Numbers Machine never halts, we cannot use the Natural Number Machine to initialize a tape, but we can analyze the machine. When the leftmost cell holds a terminator, we say it has the value 'zero'. We call '·s·' the number 'one'. Each set of 's' adjacent symbols surrounded by the terminators, and zero, is said to be a Natural Number. +

+ 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.

-

To say that one Natural Number A is smaller than Natural Number B is to say that A would occur on the Natural Number Machine tape to the left of B, if the machine were to be run. Similarly, if B were said to be greater than A, that would mean B would occur further to the right. To increment a Natural Number is to find its right neighbor. To decrement a number is to find its left neighbor. +

+ If a tape is written by an initializing tape machine, unmounted, and then mounted on a second analyzing tape machine, the analyzing tape machine is starting with a populated tape. Similarly, a mathematician can, by decree, define an initial tape that holds predefined alphabet symbols. In these cases, the analyzing machine cannot use the signaling method described in the prior paragraph, as it was excluded from controlling all the writes of alphabet symbols to the tape. The only solution to this problem is for the initializing machine and the analyzing machine to use a shared communication protocol for signaling the end of the active area.

- Now suppose we have a single Natural Number on a tape, say 'sss·', then we can define an Increment machine that, when initialized with a tape, writes an additional 's' and a new terminator. For example, when initialized with 'sss·' and then run, it produces 'ssss·'. In an analogous manner, we can define a Decrement machine. + By definition, a computation must finish in a finite number of steps. It follows that the active area when a Turing Machine halts a computation will always be finite. Consequently, if a Turing Machine initially starts working with a tape that was computed by another Turing Machine, the length of the input will be finite. Still, that input will be in the active area, so the receiving Turing Machine will need to read control data left on the tape under a common communication protocol to be able to find the bounds of the input area.

-

As such, we can assign a Natural Number to each member of the tape sequence through the following procedure. Given a machine, say P, we lock to it a second Address Machine, say A_P. When P's tape is first mounted, at the same time A_P is mounted with a tape that has only a terminator symbol, '·'. When P steps right, A_P writes an 's', steps right, and writes the terminator. For each step left of P, the machine A_P steps left and writes the terminator symbol. +

+ 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

+

- The Natural Number found on A_P is then called the address of the cell the head is on for machine P. As each increment and decrement of the address is a constant time operation, keeping the address of the cell the head is on is a computationally inconsequential action. + Because of the impossibility of recognizing certain tape features, when a tape is written by one Turing Machine, then used by another, there must be some sort of system for messaging control. There are two approaches for mixing data and control together: one is in band signaling, while the other is out of band signaling.

- An address space is the set of addresses that would be placed into correspondence to cells if we were to step across all those cells while assigning an address to each cell the head is on. Address space is a second order concept. It is because assigning an address to a cell can be a second order concept that we often take it for granted. + 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 where a recognizer, i.e., merely examining the data, is completely incapable 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 afterthought, a sort of hack.

- It is interesting to contemplate if the Natural Number at the basis of the Address Machine instead used Arabic Representation. In that case, an increment or decrement would be a linear time operation due to the carry. Consider a constant time operation on the base machine P, say it steps to the right 5280 cells and writes a mile marker. It would still be a constant time operation no matter the time taken for the address increment. So then consider an operation that is linear time, say a n, where n is the length of the input. Say, for example, it stepped n cells to the right and wrote a marker. If the increment time grows linearly with each step, then what was time n without addresses becomes time n^2 with them, or polynomial time. So indeed using Arabic Representation instead of unary for the address would be computationally consequential for all but constant time machines, if addresses were kept track of in the first order. + 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 lack 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.

+

Abstract areas and partitions

+

- As of the time of this writing, real machines keep the illusion of constant time Arabic Representation for addresses due to fixed width clocked operations. When the width is fixed, and that fixed number is small enough that interconnect delay does not dominate over computation element delay, adder time tends to be logarithmic. The same is true of the related problem of address decoding. Thus an add can be performed within 'one clock tick'. Native address arithmetic operations on modern machines are always done in this manner. Perhaps there exists a special problem, such as pointer swizzling, where address arithmetic would come under software control. + A tape area and partitioning can be an abstraction defined by a function rather than merely by a leftmost and rightmost cell. Such areas can have different topologies than those of the base tape. A familiar example for most computer scientists is utilizing software to create the appearance of a two dimensional array over a linear memory.

- The model used here to define an address is analogous to keeping a pointer into memory, and then using that pointer value as the address. Each step then increments or decrements the pointer. On a real machine, to access memory requires sending that pointer on a trip through a virtual memory system perhaps, and then through an address decoder. In the Turing Machine model, the head directly indicates a location, so it is more akin to the output of the memory decoder, though unlike the output of a memory decoder, it is stateful, i.e., a persistent value that can be moved incrementally. The analogy with the Turing Machine model holds due to the fiction of a constant decode time, as described in the prior paragraph. + Accordingly, suppose there is a three tape Universal Turing Machine gasket machine that holds the definition of a base machine on a first tape. It calls the base machine as a subroutine to access the base machine's tape (the second tape), and it uses its own tape to organize the tape abstraction. Then this outer machine can present to its user a variety of transforms of the base machine's tape.

- At a higher level, that virtual memory system level, the memory architecture begins to look more like that of a Turing Machine. The translation lookaside buffer provides stateful location context, and the neighbor relationship between pages might be taken into account for performance reasons. However, once a program starts performing at virtual memory page fetch times instead of local system memory access times, we say that it is page thrashing and know it will become too slow to wait on, no matter its computation complexity class. + As an example, a gasket machine could partition the base machine's tape into two areas, one consisting of the odd addressed cells, and the other of the even addressed cells. Though these two areas consist of noncontiguous cells on the base tape, when viewed through the gasket machine, they appear perfectly contiguous. In this case, each area will have a leftmost cell and remain open on the right.

- Let us put this into perspective. Suppose in ancient Roman times that a clock tick for a computer was scaled to be one day long. Under this scale, a single nanosecond of real world execution time equates to three days. Suppose a program initiated a read request for a location in memory on the Ides of March, the date when Caesar was assassinated on -0043-03-15. The following table provides the historical date that the variable value would finally be loaded into the processor, depending on the memory tier being accessed: + When the base machine tape cells behind an abstract area are physically contiguous, we say that the abstract area is compact. In the odd even tape partition example, the abstract areas lack compactness.

- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
Memory TierReal World LatencyScaled DelayScaled Arrival EraHistorical Context
L1 Cache Hit1 ns3 days-0043-03-18Three days after the assassination.
DRAM (Main memory)100 ns300 days-0042-01-09Nearly a year later, during the Liberators' civil war.
NVMe SSD Page Swap15 µs45,000 days0080123 years later, exactly as the Colosseum is completed in Rome.
SATA SSD Page Swap100 µs300,000 days0778821 years later, during the reign of Charlemagne and the Frankish Empire.
Magnetic HDD Page Swap10 ms30,000,000 days82092Tens of thousands of years in the future, long after current human civilizations are dust.
-

This is why computer architects spend almost all of their effort designing computers that execute as many instructions per cycle as possible within a memory subsystems that has a high statistical chance of executing in the lower levels of the memory abstraction stack. As a demonstration of this, there is not a single chapter on computation theory in Hennessy and Patterson's Computer Architecture: A Quantitative Approach. John L. Hennessy and David A. Patterson, Computer Architecture: A Quantitative Approach, 6th ed. (Cambridge: Morgan Kaufmann, 2017).

- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
Chapter 1Fundamentals of Quantitative Design and Analysis
Chapter 2Memory Hierarchy Design
Chapter 3Instruction-Level Parallelism and Its Exploitation
Chapter 4Data-Level Parallelism in Vector, SIMD, and GPU Architectures
Chapter 5Thread-Level Parallelism
Chapter 6Warehouse-Scale Computers
Chapter 7Domain-Specific Architectures
Chapter 8The Future of Computing
Metrics @@ -1205,7 +1122,7 @@

With addresses we can now define an area with two addresses, two natural numbers, the address of the leftmost cell, and that of the rightmost cell. All natural numbers greater than or equal to the address of the leftmost cell, or less than or equal to the address of the rightmost cell, are addresses of cells that are in the area. This feels more satisfactory for most of us, as now we are talking about arithmetic rather than graph topology. Though should the topology of the tape be generalized, this could become limiting. It reminds me of Frege's admonition that perhaps math should be built on top of geometry.

-

Distance and Extent

+

Distance and extent

The distance between two cells is the absolute difference in their addresses. When we talk about the distance between the leftmost cell of an area and the rightmost cell of an area, we call this number the area's extent.

@@ -2067,3 +1984,114 @@ + +