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Turing machine

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1 Introduction

Wikipedia article

2 Variants

There are several variants of defining the concept of a Turing machine. These varants can be classified across several aspects. I shall classify possible variants in terms of two questions:

  • how the machine behaves at transitions
  • how the the running of the machine terminates

2.1 A classifying aspect

2.1.1 Transition

I make a distinction between sum-like and product-like transition behavior:

product-like transition
means that the head moves and writes at each transition (this may have a funny consequence: sometimes, we have to undo the last moving step of the head at the end of running -- i.e. there may remain a superfluous last moving that has to be undone). Maybe that is why some of these variants extend the moving possibilities: besides left, right there is also an idle move.
sum-like transition
means that the head moves or writes at each transition (never both). [Mon:MathLog] describes this approach. I like this approach better, because it looks more ecomical conceptually (we do not have to undo any superfluous final movings, nor do we need to introduce idle moves) and because it looks nicer for me even conceptually.

2.1.2 Halting

Halting can be done in several ways -- a (second) special state (final state), or a special action (termination action) etc... Here, I avoid the concept of halting state. The only special state is the starting state, but I do not overload the concept of state in any other way -- halting is achieved in another way

2.2 A construct called

Sum-like transition and avoiding the concept halting state -- they look like two independent, orthogonal aspects. But a construct (called
in a -- maybe deprecated -- Hugs library module) can implement both aspect in one construct.
 data Maybe2 a b = Nothing2 | Just2 a b

2.3 Moves

There are variants of the Turing machine concept also accross other problems, e.g. how the head and the tape moves related from each other: which is regarded as moving, which is immobile.

Using the facts that

  • the head is
    • finite (even if the states are included)
    • and even limited, even constant (having fixed a concrete Turing machine): it does not grow during a run
  • the tape is infinite, or (in some approaches to Turing machine concept) finite but unlimited, maintaining dynamically its approporiate size

I prefer a convention taken from a physical view: I regard the head as being mobile even if it is made from the heaviest transuranium metal or a black hole, and tape being (or growing) immobile even if it is made of the finest membrane.

3 Turing machine definition language as a programming language

I use term Turing machine definition language for what we generally mean when talking on the language of Turing simulator softwares: a language which can represent infinite -- even all possible -- concrete Turing machines, so it is able to serve as an (of course Turing complete, in the most literal meaning) programming language

3.1 Syntax

I was dreaming of a syntax reflecting the conceptual structure of the main concepts -- maybe reflecting even the logic of the Haskell implementation! In generally, I do not like special names (like start is for starting state, everything else...) -- I like the concept of tags much better -- I mean the concept of tags as shown in classical examples -- e.g. direct sum or

3.1.1 Verbose syntax Example

Incrementing a number represented in the tally (in other words, unary) notation:

starting : blank : move right with starting;
starting : letter 1 : write letter 1 with becoming processing;
becoming processing : blank  : write letter 1 with becoming extended;
becoming processing : letter 1 : move right with becoming processing;
becoming extended : blank : stop;
becoming extended : letter 1 : stop; Tagging structure

As we can see, this Turing machine definition language language has an algebraically motivated tagging structure.

This tagging structure makes such programs (i.e Turing machine definitions) more readable, that make use of different namespaces for symbols and states.

As symbols and states are different concept, most Turing machine definition languages allow us to use te same symbol for a state and a symbol -- they are simply in different namespaces. This is a natural consequence of the essential concept of Turing machine. [Mon:MathLog] leverages namespaces in an extreme way: everything -- states, symbols, even actions -- are coded as natural numbers.

The tagging structure enables us also to define the set of states and symbols in an easy way: no burnt-in presupposition is required (e.g. restricting the set of symbols for characers). Special keyword for declaring sets of states and sets of symbols are also superfluous. Algebraic background of the tagging structure

Keywords can be classified according their corresponding type and tag coordinates, and coloring scheme can reflect this. States, symbols
States and symbols are rather different concepts in the theory of Turing machines, but they both have (maybe incidentally) a common feature. It is wort of distinguishing a special state (the starting state) -- and the same is true at symbols (blank symbol). This has a cosequence: the tagging structure is the same for these two rather different concepts, and this common tagging structure can be covered by
Just Nothing
Concept (type) Symbol letter blank
State becoming starting Result of transition
Representation of halting, and in non-halting cases maintaing new state and action -- this has also a tagging structure. It is
Just2 Nothing2
Concept (type) Result of transition with stop

3.1.2 Concise syntax Example
0 : _ : -> / 0
0 : '1 : !'1 / @processing
@processing : _ : !'1 / @extended
@processing : '1 : -> / @processing
@extended : _ : .
@extended : '1 : . Motivation of signs
  • 0 signing starting state comes from David S. Woodruff's Turing machine interpreter TM
  • Exclamation mark signing the concept of writing comes from the Z model-based formal specification language, where exclamation mark is used to mark output variables
  • Repeated colons signing compound cases come from the concept of currying.
  • At-character signing state comes from its literal meaning in natural languages (in everyday English language, as a preposition for expressing being at places times or states) -- also reflected in its use in e-mail addresses
  • Apostrophe signing a (non-blank) letter comes from using apostrophes in programming languages for character literals and from using apostrophes (or upper corners) for quotations in mathematical logic
  • Slash-character signing the concept of compound action (allowing the possibility of halting instead) comes from the theory of automata
  • Dot signing termination is hard to explain. In Prolog, dot is a terminating symbol, but in another sense: it terminates syntactical units (rules and facts), not processes!

3.2 Implementation

3.2.1 Representations of main concepts Language
 module Language where
 import Util (Maybe2 (Nothing2, Just2))
 type Turing state symbol = Turing0 (Maybe state) (Maybe symbol)
 type Turing0 state symbol = state -> symbol -> Maybe2 (Action symbol) state
 data Action symbol = Write symbol | Move Bool Discernable non-emptyness: special state, special symbol
Most conceptual frameworks of Turing machine concept allow us a big liberty at chosing the set of states and symbols -- but empty set is not allowed. Of course Haskell types are never empty, but non-emptyness provided by
is not enough. We need discernable non-emptyness. This restriction is reflected directly in the representation of language concepts: we require the presence of a special symbol and a special state. The trick of
solves the problem of default state and symbol. Default symbol is called blank and default state is called starting. The presence of special symbol, special state is reflected directly in the representation of language concepts -- and both are represented by

This way allows us a big liberty at chosing the set of states and symbols -- we do not have to restrict the user's freedom to use any types for both symbols and states. Strings, integers, enumarations, characters, booleans... Halting

Another -- maybe analogous -- problem is the representation of halting.

There may be alternative solutions: introduction of concepts like halting state or halting action, etc... I felt that
(Hugs has this data type in a library module) is a conceptually esthetic solution. Tape

At first I wrote a circular program to make double linked list to implement a bidirectionally infinite tape. It is superfluous, there is a more simple solution:

 module Tape where
 import Util (Stream (Cons))
 data Tape a = Tp {left, right :: Stream a, cell :: a}
 move :: Bool -> Tape a -> Tape a
 put :: a -> Tape a -> Tape a
 get :: Tape a -> a

... Utility module
 module Util where
 data Maybe2 a b = Nothing2 | Just2 a b
 data Stream a = Cons a (Stream a)

3.3 Extending language

A pattern language, or some other extending possibilities, to avoid redundances. Maybe a macro language as at David S. Woodruff's Turing machine interpreter TM. Other syntactic sugar.

4 Bibliography

Monk, J. Donald: Mathematical Logic. Springer-Verlag, New York * Heidelberg * Berlin, 1976.