From: Thomas Walker Lynch Date: Sat, 18 Jul 2026 17:29:40 +0000 (+0000) Subject: . X-Git-Url: https://git.reasoningtechnology.com/%27%20%20%20window.RT.dirpr_library%20%20%20%27/%27%20%20%20key%20%20%20%27?a=commitdiff_plain;h=38e36167e59387ff993424dd3a2e760495b4413a;p=TM-2026 . --- diff --git a/document/book/TM-2026.html b/document/book/TM-2026.html index 1d765c6..af67c23 100644 --- a/document/book/TM-2026.html +++ b/document/book/TM-2026.html @@ -501,7 +501,7 @@

If we view the empty symbol from within the definition of machine B, it is a category error. It is a sequence element not a container. Even if it were allowed that a position within a sequence were a container and thus could have the property of being empty, the symbol represents that there is no symbol in the very location it is found. Hence, to ascribe an appropriate meaning to the empty symbol, the entire system must be kept including the mapping to machine A, then the empty symbol found on machine B means that if machine A were used instead, the same location in the tape sequence would be an empty set. However, this feels unsatisfactory, because in all other respects, machine B is a perfectly well defined Turing Machine all by itself.

-

Going back to Turing's moniker of blank, does not bring the focus solely to machine B, because the original paper by Turing states the definition of blank meaning "no symbol", as was already discussed. Calling it blank is merely the same name colored by the engineering of wood pulp. However, there is a pleasing property of the blank, i.e. the empty symbol, that is valuable and unique to it. It prints as a space in diagrams given in text books, and Turing Machine simulators. There is an analogous symbolic system that also has this characteristic, and is being leveraged in these situations: the ASCII hex 20, called SP. It prints as a space leaving an area of the paper it is printed on blank. Perhaps a better name for the ersatz empty symbol is SP.

+

Going back to Turing's moniker of blank does limit the focus solely to machine B, because the original paper by Turing states the definition of blank meaning "no symbol", as was already discussed. Calling it blank is merely the same name colored by the engineering of wood pulp. However, there is a pleasing property of the blank, i.e. the empty symbol, that is valuable and unique to it. It prints as a space in diagrams given in text books and in the output of Turing Machine simulators. There is an analogous symbolic system that also has this characteristic, and is being leveraged in these situations: the ASCII hex 20, called SP. It prints as a space leaving an area of the paper it is printed on blank. Perhaps a better name for the ersatz empty symbol is SP.

There is more to consider about the former blank, former empty, now SP symbol. Computational theorists often describe a two-step process. First, a Turing Machine writes a string to be studied onto a tape initially filled with all SP symbols; second, the tape is moved to another Turing Machine whose purpose is to recognize a property of the string being studied. In this process, a tape is used as a communication medium, so it follows that communication theory can be applied to this messaging scenario. Messages sent on a channel must be formatted so they can be placed on the channel and later recovered. The specification for such structuring is called a communication protocol. The protocol implies a higher authority, as both the sender and the receiver must have knowledge of the same designed-in protocol. In a sense, a protocol is the meta-message sent before the message itself.

@@ -515,7 +515,7 @@

A common technique programmers use for embedding strings with terminators within other strings is to embed escape characters. This is another in-band protocol, and it is inefficient. The escape sequences grow exponentially with nesting levels. Also, such strings with embedded escape sequences cannot be interpreted without knowledge of the intended nesting level of the string author, which requires the interpreter of the string to know the author's intentions.

-

For the modified computational Turing machine presented later in this chapter, there will be a separate control alphabet of predefined symbols, as for the Hopcroft-Ullman interpretation. However, so as to support recursion, and because the current scheme does not achieve protocol guarantees, the predefined alphabet will be part of the alphabet Σ rather than distinct from it. Programmers then must explicitly design communication protocols that suit the problems they are working on, and those protocols become opaque for examination by a higher authority. Note that this does not exclude a programmer ssing SP as an EOM marker, rather it removes the embedding of that protocol from the Turing Machine definition.

+

For the modified computational Turing machine presented later in this chapter, there will be a separate control alphabet of predefined symbols, as for the Hopcroft-Ullman interpretation. However, so as to support recursion, and because the current scheme does not achieve protocol guarantees, the predefined alphabet will be part of the alphabet Σ rather than distinct from it. Programmers then must explicitly design communication protocols that suit the problems they are working on, and those protocols become opaque for examination by a higher authority. Note that this does not preclude a programmer from using SP as an EOM marker, rather it removes the embedding of that protocol from the Turing Machine definition.

The unspecified symbol

@@ -556,7 +556,7 @@
  1. Fixed global default (δF_0): Selects the rule that matches the value in the gate register. There is only one rule in this set; it compares the gate register to the unspecified symbol, then transitions to the unspecified_decision state.
  2. Programmed Conditional (δ_1): Selects the transition rule that matches the current state and the value of the gate register.
    3. -
  3. Programmed Default (δ_2): If no programmed conditional next state rule is found, selects the default transition rule that matches the current state.
    4. +
  4. Programmed Default (δ_2): If no conditional next state rule is found, selects the default transition rule that matches the current state.
  5. Global Fallback (δ_3): An unconditional transition of last resort if no prior layer provides a valid next state.
@@ -564,11 +564,11 @@

There is a mechanical procedure for converting a Moore Machine into a Mealy Machine, and the reverse. Two such converted machines are equally expressive. Making the read operation an explicit action, instead of having it implied by a state transition, increases the number of states in a controller, and consequently the number of steps that must be taken. However, the new machine can do in two steps anything the former machine could do in one; consequently, this change affects the multiplier constant on the linear term of the step count formula but does not change the computation complexity class determined from such a step count. The layers of next-state functions can be collapsed into one layer, where the missing arguments are filled in with all possible unused values. This potentially increases the number of state transition arcs that must be specified, but the state transition logic and the number of states remain unchanged. Partitioning the machine definition by separation of concerns does not change the total specification. Hence, these modifications are computationally inconsequential.

-

The TTCA Turing Machine fixed part

+

The TTCA Turing Machine fixed part

- MF = (QF, ΣF, AF, δF_0) + MF = (QF, ΣF, AF, δF_0, HF)
@@ -617,6 +617,10 @@

In any state if the g register holds the symbol \mathtt{ΣF{·}unspecified}, then the next state will be \mathtt{QF{·}unspecified\_decision}, independent of all other defined next state transitions.

+

The set of predefined halting states (only one):

+
+ HF = \{QF{·}\mathtt{unspecified\_decision}\} +

The TTCA Turing Machine variables

@@ -637,7 +641,7 @@
- MP = (QP, ΣP, λP, δ_1, δ_2, δ_3) + MP = (QP, ΣP, λP, δ_1, δ_2, δ_3, HP)
@@ -676,11 +680,17 @@

This is the transition of last resort. It is unconditional, the next state becomes q_1.

+

A set of programmer defined halting states:

+
+ HP +
+ +

The TTCA Turing Machine in total

- M = (Q, Σ, AF, λP, δ) + M = (Q, Σ, AF, λP, δ, H)
@@ -699,15 +709,18 @@ δ = [δF_0 | δ_1 | δ_2 | δ_3] - +

The complete set of halting states, uniting the fixed predefined states and the programmed states:

+
+ H = HF \cup HP +

TTCA computation theoretic TM executor directions

-

To execute the programmed TTCA Turing Machine, a person must maintain the current machine variables MV and evaluate the programmed logic in an infinitely repeating two-phase cycle. Because this is a Moore-style architecture, the execution of an action is isolated from the evaluation of the next state transition.

+

To execute the programmed TTCA Turing Machine, a person must maintain the current machine variables MV and evaluate the programmed logic in an repeating two-phase cycle, until a halting state is reached. Because this is a Moore-style architecture, the execution of an action is isolated from the evaluation of the next state transition.

Initialization

-

Before the first cycle begins, a tape selected and mounted. The read/write head is positioned over the leftmost tape cell. The machine variables are initialized as follows:

+

Before the first cycle begins, a tape is selected and mounted. The read/write head is positioned over the leftmost tape cell. The machine variables are initialized as follows: