From 860f53ef2fb581df20743e4fa6dbfcfc9557861f Mon Sep 17 00:00:00 2001 From: Thomas Walker Lynch Date: Sat, 4 Jul 2026 16:02:32 +0000 Subject: [PATCH] doc --- document/TM-2026.html | 263 ++++++++++-------------------------------- 1 file changed, 62 insertions(+), 201 deletions(-) diff --git a/document/TM-2026.html b/document/TM-2026.html index 5821baa..c5879fe 100644 --- a/document/TM-2026.html +++ b/document/TM-2026.html @@ -50,7 +50,7 @@ - + Introduction @@ -689,105 +689,6 @@ A person can also say that the head indexes the head cell. This utilizes the classical mechanical definition, where an index is a mark for aligning gears. This physical meaning contrasts with an index integer used for addressing an array. The architectural definition of the Turing Machine developed here relies strictly upon the topological properties of the tape, independent of the definition of Natural Numbers. Addresses, which do rely on Natural Numbers, are discussed further on in this volume. This represents a minor divergence from Alan Turing's original paper, as he took it as a given that numbers naturally paired with the squares. We explicitly establish that pairing only after deriving Natural Numbers using the Turing Machine itself.

-

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 left most cell in the area, a right most cell in the area, with one or more interstitial cells. A leftmost cell in an area might have a left neighbor property, but the cell indicated by that property is not in the area. Or it is possible that the leftmost cell of an area is also the leftmost cell of the tape, and thus it does not have a left neighbor property. Any cell on a single ended tape will have a right neighbor property, but for a a finite area the rightmost cell's right neighbor property will indicate a cell that is not in the area. It is possible for an area to be open on the right, and thus not be finite. - In this manner we distinguish between tape cell types, and area cell types. -

- -

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

- -

Head partition

- -
    -
  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 non empty 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 not empty. 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 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, there is implied the potential for two more areas. -

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

The impossibility of recognizing an empty tape

- -

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

- -

- 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. -

- -

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

- -

The impossibility of recognizing the rightmost cell of the active area

- -

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

- -

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

- -

- 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 not starting with an empty tape. Similarly, a mathematician can, by decree, define an initial tape that is not empty. In these cases, the analyzing machine cannot use the signaling method described in the prior paragraph, as it was not in control of 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. -

- -

- 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. -

- -

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

- -

In band and out of band control

- -

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

- -

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

- -

The programmed controller

@@ -963,170 +864,130 @@ After the reset button is released, the machine begins stepping. If the program is a computation, the machine will eventually halt. If the machine eventually halts, then we know the associated program was a computation. Otherwise we do not know. Any amount of time we wait where the machine has not halted, we will not know that it will ever halt. Hence, we cannot in general use 'running a Turing Machine' as a means to determine if a given program is computational. (We could instead try to answer the question 'is it computational' through analysis, but there too, Turing has shown that in general that will not work either.)

- Computational Analysis -

- 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. -

+Area and partitions

- 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. + 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.

-

- 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. +

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.

-

- 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. -

+

Head partition

-

- 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. -

+
    +
  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

- 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. + 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

+

- 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. + 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

- 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. + 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.

- 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. + 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.

- 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. + 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.

-Computational Naturalism +

The impossibility of recognizing the rightmost cell of the active area

- 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. + 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.

+

In band and out of band control

+

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

- 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. + 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.

- 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. + 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.

- 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. + 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.

- 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? + 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.

- Addresses and cells - -

Natural Number, address

- -

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

- -

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

- -

As such we can assign a Natural Number to each member of the tape sequence. We will call the machine that does this the address machine.

- -

Since multiple tape Turing Machines are equivalent to a single tape Turing Machines, we begin with the simplification and say that our address machine has two tapes. One we will call the main tape. We will assign addresses to the elements on the main tape.

- -

The second tape we will call the address tape. It is initially empty.

- -

The controller makes use of our increment routine for unary numbers which we described earlier.

- -

We begin with the main tape head on the leftmost element. The address tape reads zero, this is the address for the leftmost element.

- -

The controller then steps the main tape head right one element, and it runs the increment routine on the address tape. The address tape then reads ‘s’, i.e. one. This is the address for the right neighbor of the leftmost element.

- -

We then repeat this procedure while assigning addresses to each successive element, two, three, four, etc. We call the set of addresses made in this manner the ‘Natural Numbers’.

- -

Though we can never run this machine to the point where it halts, we may analyze the Natural Number machine and reason about it.

- -

As one example, we can take a first given Turing Machine with it’s head somewhere on a tape, encode that machine and its state, and also encode an address machine, and provide these two machines to a third machine, one called address-of.

- -

The address-of machine can then run the address machine while stepping the head left on the first given machine.

- -

At the point where the next left step would cause a left of leftmost error, the address will appear on the address tape of the address machine.

- -

Distance

- -

The distance between two elements is defined to be the difference in their corresponding addresses.

- -

Area and length

- -

An area is a subsequence of cells on a tape. An area has permission to be defined using two addresses. The first address being that of the leftmost cell in the area, and the second address being that of the rightmost cell in the area.

- -

The length of an area is one greater than the distance between the address of the leftmost cell of the area, and that of the rightmost cell of the area.

- -

Hence the smallest length that an area has permission to have is 1. This occurs when the address to the leftmost cell is the same as the address to the rightmost cell, i.e. when there is only one cell.

- -

The concept of area has permission to be abstracted to include a set of cells that share a property.

- -

This might be a property of the values in the cells, or a property of the addresses of cells.

- -

An abstract area can be colored by a property detecting machine.

- -

Such a machine would start at the leftmost cell, check the value in that cell for the given property and mark it accordingly. It would then step right and repeat.

- -

For such a marker machine to be computational there will have to be a leftmost cell in the area, and upon finding a leftmost cell in the area, a rightmost cell will have to exist. Without either of these a marker machine will never halt.

- -

Even if a marker machine might not halt, it still might be useful for analysis.

- -

Such marker machines that color based only the value found in each cell are context free.

- -

Abstract areas have permission to also be defined based on functions of cell values that include context.

- -

When the cells in an abstract area are contiguous, we say that the area is compact.

- -

An example of an area that is excluded from being compact is that of the odd addressed cells.

- -

The odd and even addressed cells of a tape form two non-overlapping abstract areas.

- -

Given a cell in either of these areas, its direct neighbors will not be in the same area.

+ -

Zero length is a second order concept

+ Zero length is a second order concept

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

-- 2.20.1