From: Thomas Walker Lynch Date: Sat, 4 Jul 2026 16:49:54 +0000 (+0000) Subject: recovering after file corruption, still a mess X-Git-Url: https://git.reasoningtechnology.com/RT-Manuscript_locator.js?a=commitdiff_plain;h=42f6818e689da17836c9e3ffd03a223ee3499986;p=TM-2026 recovering after file corruption, still a mess --- diff --git a/document/TM-2026.html b/document/TM-2026.html index c5879fe..95ce93c 100644 --- a/document/TM-2026.html +++ b/document/TM-2026.html @@ -985,7 +985,568 @@ 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.

+ Computational Analysis + +

+ 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. +

+ +

+ 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. +

+ +

+ 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. +

+ +

+ 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. +

+ +

+ 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. +

+ +

+ 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 + +

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

+ +

+ 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. +

+ +

+ 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. +

+ +

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

+ +

+ 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. +

+ +

+ 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. +

+ +

+ 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. +

+ +

+ 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 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. +

+ +

+ 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. +

+ +

+ 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? +

+ + 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. +

+ +

+ 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. +

+ +

+ 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. +

+ +

+ 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. +

+ +

+ 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. +

+ +

+ 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. +

+ +

+ 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. +

+ +

+ 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. +

+ +

+ 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: +

+ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
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 + +

Address defined area

+ +

We defined a tape as having a single feature, that of a leftmost cell, along with a linear neighbor topology. When a tape is mounted it gains a second feature, that of the cell the head is on. The leftmost cell was fixed in place, while any other single cell on the tape can be featured as the cell the head is on. When we introduced areas, we picked up two new features, that of the leftmost cell of the area, and the rightmost cell of the area. Again, those features could coincide with the former features. We noted that the features partitioned a tape thus defining more areas, and more features. +

+ +

+ When cells were defined we noted that they were sets, and that these sets had identity and could be referred to. We needed that to be the case so as to build the tape topology using neighbor properties. Had the tape been singly linked, perhaps we could have used the cells to represent themselves in the properties, but when we doubly linked it, a given cell had to appear in two places, as the right neighbor of its left neighbor, and the left neighbor of its right neighbor, so we stipulated the sets as being identifiable. Initially we did not say how they would be identified. This is common in mathematics, but we soon cleared this up by formalizing the concept of a symbol. +

+ +

+ When discussing symbols we noted they could be natural numbers, and even went so far as to point out that addresses were symbols, though we had not yet defined them. They are familiar to anyone involved in computing, so again, it did not present a serious problem. Here we have now formalized them. +

+ +

+ So we now have two means for identifying a feature. One means is to state its address, and the other is to put a tape machine head on it. +

+ +

As we noted in the section discussing area, an area has two distinguishing features, being that it has a leftmost cell, and a rightmost cell. That definition is topological. If we start with the leftmost cell of a tape, we are either already on the leftmost cell of a defined area, or we can step right to find it. At the point of finding it we know this leftmost cell is part of the area, then we are either already on the rightmost cell, or we can continue to step right on cells in the area until we find the rightmost cell in the area. The right neighbor of the rightmost cell in the area, and all cells to that right of that, are excluded from the area. +

+ +

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

+ +

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.

+ +

In an analogous manner to defining an address space for a tape, we can define an address space for an area. Accordingly, the leftmost cell of the area is assigned address zero, and the address increments cell by cell going to the right. A finite area will have a finite address space, with the address of the rightmost cell being the largest address in the address space. +

+ +

It is interesting that the extent of an area will be the same, whether it is calculated from the address space of the tape, or the address space of the area. When it is computed from the address space of the area, the extent will be identical to the largest address in the address space. We will use the Greek symbol omega, \\omega, to symbolize an extent. We can give it a subscript with the name of an area if the correspondence is not already clear. I chose \\omega makes sense here because \\omega is an inclusive bound, i.e. it is the rightmost letter included in the Greek alphabet. Thus it is the extent of the Greek alphabet. +

+ +

Length

+ +

If cells in an area are transacted, the cost of the area is the cost of a cell multiplied by the count of cells. Vincent Atanasoff probably found himself needing to know such a count when ordering capacitors. The count of cells in an area is also known as the area's length. We will use the Greek symbol Ï¡, pronounced as sampi, to refer to the length. The length of an area, the count of its cells, will always be one greater than its extent, Ï¡ = \\omega + 1. This symbol makes sense here, as the Greek number system fell short of letters, so the symbol Ï¡ was tacked on to the end of the alphabet, but did not belong to the alphabet. +

+ +

The count of cells in an area, the length of an area, and the cardinality of the address space for an area are all the same number, Ï¡. The extent of an area, \\omega, is an address in an area's address space, whereas the cardinality of an area, Ï¡, falls completely outside it. This has implications. Because extent is an address, extent and addresses can always use the same number representation. In contrast, there is no such guarantee for cardinality, Ï¡. Take for example an area that contains 256 cells. The addresses run from zero to 255, and all can be represented with an 8 bit binary number. However, the number 256 does not fit. It is for this reason that code will have fewer end case problems when expressing the size of objects with extents, rather than with lengths. +

+ +

Is the cardinality of an open on the right area a Natural Number?

+ +

So we find an interesting situation with the cardinality of an address space for an area that is open on the right. It goes like this.

+ +

How cardinality is computed

+ +
    +
  1. We set Turing Machine P with its head on the leftmost cell of an area. We mount the initial tape, '·', on the A machine.
    2. +
  2. We step P and simultaneously run the A machine. Stepping stops when P reaches the rightmost cell of the area. At this point, the tape on the A machine holds the address space extent.
    3. +
  3. The A machine is run an additional time. The output on the tape is defined to be the cardinality of the address space, aka the cardinality.
    4. +
+ +

Lemma 1, the A machine produces Natural Numbers

+ +

This follows from its definition; it is literally the increment from the Natural Numbers Machine.

+ +

Lemma 2, cardinality is a natural number

+ +

Cardinality is produced by repeatedly calling the A machine, and the A machine produces Natural Numbers.

+ +

Lemma 3, cardinality is not in the address space

+ +

At the end of the second step in the procedure for producing the cardinality, the tape of the A machine held the extent of the address space. Then in step 3, A was called again, thus leaving a number one larger than the extent on its tape. The extent is the largest number in the address space, hence cardinality is larger than all numbers in the address space. Thus cardinality is not in the address space.

+ +

Lemma 4, the address space of an open on the right area is identical to the Natural Numbers.

+ +

Composing the A machine with an unterminated loop call, where each result is written to a tape with a terminator between entries, results in the same machine as the Natural Numbers Machine.

+ +

The apparent contradiction.

+ +

By Lemma 2, cardinality is a Natural Number. By Lemma 3 cardinality is not in the address space. By Lemma 4 the address space is identical to the Natural Numbers.

+ +

Resolution

+ +

The resolution lies in the computational reality of Step 2. For an area that is open on the right, the stepping of machine P never halts. Because Step 2 never terminates, Step 3 is never executed. The A machine never runs that final, additional time. Therefore, the cardinality of an open area is never actually produced by the machine. In the language of Computational Naturalism, Lemma 2 is false for an infinite area; the cardinality of an open on the right area is excluded from being a Natural Number because a Natural Number Machine cannot reach it in the first order.

+ +

So then can we add a property to cardinality, such that a second order analysis could use this property to continue downstream analysis? In short we could say that cardinality has no first order value, but it has a second order one. This is analogous to inventing a new type of number, a complex number with a second component. I.e., there is no 'real' solution, but there is an 'imaginary' one. Or analogous to error algebra, where a number value is replaced with a rule on how to handle downstream operations when it is given as an input. +

+ +

Such a value would be a new Turing Machine, one that composes a call to the never halting Natural Number machine followed by an increment operation. It cannot be run, but it perfectly explains the situation to an analyst. Perhaps we name this machine \\aleph_0. +

+ +

What if extent was used instead of cardinality?

+ +

+ Had extent been used instead of cardinality, we would lack the final increment step in the three step computing procedure. However, step 2 still cannot complete. Rather than a value, the result of the second order analysis would be a machine that produces ever larger Natural Numbers. We can call this machine \\aleph_{-1}. +

+ +

+ Now here is an interesting result: +

+ + + \\aleph_{0} - \\aleph_{-1} = 1 + +

+ Neither machine can be run. However we can compose the two machines, then simplify the composition. When we do so, the ever larger Natural Number machines annihilate each other, and the increment machine remains. The increment machine can be run, so we end up with an output value. +

+ + +

Size

+ +

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

+ +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. +

+ +

+ 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. +

+ +

+ 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. +

+ +

+ 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. +

+ +

+ 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. +

+ +

+ 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. +

+ +

+ 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. +

+ +

+ 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. +

+ +

+ 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 BC. 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: +

+ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
Memory TierReal World LatencyScaled DelayScaled Arrival EraHistorical Context
L1 Cache Hit1 ns3 days0043 03 18 BCThree days after the assassination.
DRAM (Main memory)100 ns300 days0042 01 09 BCNearly 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 memory subsystems that have a high statistical chance of executing in the lower levels of the memory abstraction stack. There is not a single chapter on computation theory in Hennessy and Patterson's Computer Architecture: A Quantitative Approach, for example. 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
+ + Address defined area, distance, extent, length, and size + +

We defined a tape as having a single feature, that of a leftmost cell, along with a linear neighbor topology. When a tape is mounted it gains a second feature, that of the cell the head is on. The leftmost cell was fixed in place, while any other single cell on the tape can be featured as the cell the head is on. When we introduced areas, we picked up two new features, that of the leftmost cell of the area, and the rightmost cell of the area. Again, those features could coincide with the former features. We noted that the features partitioned a tape thus defining more areas, and more features. +

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+ When cells were defined we noted that they were sets, and that these sets had identity and could be referred to. We needed that to be the case so as to build the tape topology using neighbor properties. Had the tape been singly linked, perhaps we could have used the cells to represent themselves in the properties, but when we doubly linked it, a given cell had to appear in two places, as the right neighbor of its left neighbor, and the left neighbor of its right neighbor, so we stipulated the sets as being identifiable. Initially we did not say how they would be identified. This is common in mathematics, but we soon cleared this up by formalizing the concept of a symbol. +

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+ When discussing symbols we noted they could be natural numbers, and even went so far as to point out that addresses were symbols, though we had not yet defined them. They are familiar to anyone involved in computing, so again, it did not present a serious problem. Here we have now formalized them. +

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+ So we now have two means for identifying a feature. One means is to state its address, and the other is to put a tape machine head on it. +

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As we noted in the section discussing area, an area has two distinguishing features, being that it has a leftmost cell, and a rightmost cell. That definition is topological. If we start with the leftmost cell of a tape, we are either already on the leftmost cell of a defined area, or we can step right to find it. At the point of finding it we know this leftmost cell is part of the area, then we are either already on the rightmost cell, or we can continue to step right on cells in the area until we find the rightmost cell in the area. The right neighbor of the rightmost cell in the area, if there is one, is excluded from the area. +

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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. +

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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.

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In an analogous manner to defining an address space for a tape, we can define an address space for an area. Accordingly, the leftmost cell of the area is assigned address zero, and the address increments cell by cell going to the right. A finite area will have a finite address space, with the address of the rightmost cell being the largest address in the address space. +

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It is interesting that the extent of an area will be the same, whether it is calculated from the address space of the tape, or the address space of the area. When it is computed from the address space of the area, the extent will be identical to the largest address in the address space. We will use the Greek symbol omega, ω, to symbolize an extent. We can give it a subscript with the name of an area if the correspondence is not already clear. I chose ω because ω is the rightmost letter included in the Greek alphabet. Thus it is the extent of the Greek alphabet. +

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If cells in an area are transacted, the cost of the area is the cost of a cell multiplied by the count of cells. Vincent Atanasoff probably found himself needing to know such a count when ordering capacitors. The count of cells in an area is also known as the area's length. We will use the Greek symbol ϡ, pronounced as sampi, to refer to the length. The length of an area, the count of its cells, will always be one greater than its extent, ϡ = ω + 1. This symbol makes sense here, as the Greek number system fell short of letters, so the symbol ϡ was tacked on to the end of the alphabet, but did not belong to the alphabet. +

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The count of cells in an area, the length of an area, and the cardinality of the address space for an area are all the same number, ϡ. The extent of an area, ω, is an address in an area's address space, whereas the cardinality of an area, ϡ, falls completely outside it. This has implications. Because extent is an address, extent and addresses can always use the same number representation. In contrast, there is no such guarantee for cardinality, ϡ. Take for example an area that contains 256 cells. The addresses run from zero to 255, and all can be represented with an 8 bit binary number. However, the number 256 does not fit. It is for this reason that it might make more sense to use extents than to use lengths. +

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It is considered by some to be profound that the cardinality of a bounded set of Natural Numbers is excluded from the set. If we generalize to an area that is open on the right, for the cardinality of the address space to fall outside the address space would mean that the cardinality of Natural Numbers falls outside the Natural Numbers. It seems peculiar that cardinality, which is a count, is found to fall outside the numbers we use for counting. Though at this point we are asking for an outer bounding value on an infinite set. +

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Is the cardinality of an open on the right area a Natural Number?

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So we find an interesting situation with the cardinality of an address space for an area that is open on the right. It goes like this.

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How cardinality is computed

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  1. We set Turing Machine P with its head on the leftmost cell of an area. We mount the initial tape, '·', on the A machine.
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  2. We step P and simultaneously run the A machine. Stepping stops when P reaches the rightmost cell of the area. At this point, the tape on the A machine holds the address space extent.
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  3. The A machine is run an additional time. The output on the tape is defined to be the cardinality of the address space, aka the cardinality.
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Lemma 1, the A machine produces Natural Numbers

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This follows from its definition; it is literally the increment from the Natural Numbers Machine.

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Lemma 2, cardinality is a natural number

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Cardinality is produced by repeatedly calling the A machine, and the A machine produces Natural Numbers.

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Lemma 3, cardinality is not in the address space

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At the end of the second step in the procedure for producing the cardinality, the tape of the A machine held the extent of the address space. Then in step 3, A was called again, thus leaving a number one larger than the extent on its tape. The extent is the largest number in the address space, hence cardinality is larger than all numbers in the address space. Thus cardinality is not in the address space.

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Lemma 4, the address space of an open on the right area is identical to the Natural Numbers.

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Composing the A machine with an unterminated loop call, where each result is written to a tape with a terminator between entries, results in the same machine as the Natural Numbers Machine.

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The apparent contradiction.

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By Lemma 2, cardinality is a Natural Number. By Lemma 3 cardinality is not in the address space. By Lemma 4 the address space is identical to the Natural Numbers.

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Resolution

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The resolution lies in the computational reality of Step 2. For an area that is open on the right, the stepping of machine P never halts. Because Step 2 never terminates, Step 3 is never executed. The A machine never runs that final, additional time. Therefore, the cardinality of an open area is never actually produced by the machine. In the language of Computational Naturalism, Lemma 2 is false for an infinite area; the cardinality of an open on the right area is excluded from being a Natural Number because a Natural Number Machine cannot reach it in the first order.

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So then can we add a property to cardinality, such that a second order analysis could use this property to continue downstream analysis? In short we could say that cardinality has no first order value, but it has a second order one. This is analogous to inventing a new type of number, a complex number with a second component. I.e., there is no 'real' solution, but there is an 'imaginary' one. Or analogous to error algebra, where a number value is replaced with a rule on how to handle downstream operations when it is given as an input. +

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Such a value would be a new Turing Machine, one that composes a call to the never halting Natural Number machine followed by an increment operation. It cannot be run, but it perfectly explains the situation to an analyst. Perhaps we name this machine ℵ_0. +

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What if extent was used instead of cardinality?

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+ Had extent been used instead of cardinality, we would lack the final increment step in the three step computing procedure. However, step 2 still cannot complete. Rather than a value, the result of the second order analysis would be a machine that produces ever larger Natural Numbers. We can call this machine ℵ_{−1}. +

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+ Now here is an interesting result: +

+ + + ℵ_{0} − ℵ_{−1} = 1 + + +

+ Neither machine can be run. However we can compose the two machines, then simplify the composition. When we do so, the ever larger Natural Number machines annihilate each other, and the increment machine remains. The increment machine can be run, so we end up with an output value. +

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Size

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

Zero length is a second order concept