Turing machine equivalents

The following models are single tape Turing machines but restricted with (i) restricted tape symbols { mark, blank }, and/or (ii) sequential, computer-like instructions, and/or (iii) machine-actions fully atomised.

Emil Post in an independent description of a computational process, reduced the symbols allowed to the equivalent binary set of marks on the tape { "mark", "blank"=not_mark }. He changed the notion of "tape" from 1-way infinite to the right to an infinite set of rooms each with a sheet of paper in both directions. He atomised the Turing 5-tuples into 4-tuples—motion instructions separate from print/erase instructions. Although his 1936 model is ambiguous about this, Post's 1947 model did not require sequential instruction execution.

His extremely simple model can emulate any Turing machine, and although his 1936 Formulation 1 does not use the word "program" or "machine", it is effectively a formulation of a very primitive programmable computer and associated programming language, with the boxes acting as an unbounded bitstring memory, and the set of instructions constituting a program.

In an influential paper, Hao Wang reduced Post's "formulation 1" to machines that still use a two-way infinite binary tape, but whose instructions are simpler – being the "atomic" components of Post's instructions – and are by default executed sequentially (like a "computer program"). His stated principal purpose was to offer, as an alternative to Turing's theory, one that "is more economical in the basic operations". His results were "program formulations" of a variety of such machines, including the 5-instruction Wang W-machine with the instruction-set

{ SHIFT-LEFT, SHIFT-RIGHT, MARK-SQUARE, ERASE-SQUARE, JUMP-IF-SQUARE-MARKED-to xxx }

and his most-severely reduced 4-instruction Wang B-machine ("B" for "basic") with the instruction-set

{ SHIFT-LEFT, SHIFT-RIGHT, MARK-SQUARE, JUMP-IF-SQUARE-MARKED-to xxx }

which has not even an ERASE-SQUARE instruction.

Many authors later introduced variants of the machines discussed by Wang:

Minsky evolved Wang's notion with his version of the (multi-tape) "counter machine" model that allowed SHIFT-LEFT and SHIFT-RIGHT motion of the separate heads but no printing at all.[3] In this case the tapes would be left-ended, each end marked with a single "mark" to indicate the end. He was able to reduce this to a single tape, but at the expense of introducing multi-tape-square motion equivalent to multiplication and division rather than the much simpler { SHIFT-LEFT = DECREMENT, SHIFT-RIGHT = INCREMENT }.

Davis, adding an explicit HALT instruction to one of the machines discussed by Wang, used a model with the instruction-set

{ SHIFT-LEFT, SHIFT-RIGHT, ERASE, MARK, JUMP-IF-SQUARE-MARKED-to xxx, JUMP-to xxx, HALT }

and also considered versions with tape-alphabets of size larger than 2.

In keeping with Wang's project to seek a Turing-equivalent theory "economical in the basic operations", and wishing to avoid unconditional jumps, a notable theoretical language is the 4-instruction language P" introduced by Corrado Böhm in 1964 – the first "GOTO-less" imperative "structured programming" language to be proved Turing-complete.

In practical analysis, various types of multi-tape Turing machines are often used. Multi-tape machines are similar to single-tape machines, but there is some constant k number of independent tapes.

If the action table has at most one entry for each combination of symbol and state then the machine is a "deterministic Turing machine" (DTM). If the action table contains multiple entries for a combination of symbol and state then the machine is a "non-deterministic Turing machine" (NDTM). The two are computationally equivalent, that is, it is possible to turn any NDTM into a DTM (and vice versa), although they usually have different runtimes. This can be proved via construction.

An oblivious Turing machine is a Turing machine where, for each input length, movement of the various heads is a fixed function of time, independent of the input. In other words, there is a predetermined sequence in which the various tapes are scanned, advanced, and written to. The actual values that are written to the tape at any step can still be different for each input of that length. Pippenger and Fischer showed that any computation that can be performed by a multi-tape Turing machine in n steps can be performed by an oblivious two-tape Turing machine in ${\displaystyle O(n\log n)}$ steps.[4]

Oblivious machines correspond in a step-wise linear fashion with combinational logic circuits, when the complexity of the transition table is taken as constant. It is thus possible to realize computations as circuit problems in size ${\displaystyle O(n\log n)}$ and depth ${\displaystyle O(n)}$ (see Circuit complexity). This improves upon the original ${\displaystyle O(n^{3})}$ result by Cook and Levin.

The primitive model register machine is, in effect, a multitape 2-symbol Post–Turing machine with its behaviour restricted so its tapes act like simple "counters".

By the time of Melzak, Lambek, and Minsky the notion of a "computer program" produced a different type of simple machine with many left-ended tapes cut from a Post–Turing tape. In all cases the models permit only two tape symbols { mark, blank }.[3]

Some versions represent the positive integers as only a strings/stack of marks allowed in a "register" (i.e. left-ended tape), and a blank tape represented by the count "0". Minsky eliminated the PRINT instruction at the expense of providing his model with a mandatory single mark at the left-end of each tape.[3]

In this model the single-ended tapes-as-registers are thought of as "counters", their instructions restricted to only two (or three if the TEST/DECREMENT instruction is atomised). Two common instruction sets are the following:

(1): { INC ( r ), DEC ( r ), JZ ( r,z ) }, i.e.

{ INCrement contents of register #r; DECrement contents of register #r; IF contents of #r=Zero THEN Jump-to Instruction #z}

(2): { CLR ( r ); INC ( r ); JE ( r_i, r_j, z ) }, i.e.

{ CLeaR contents of register r; INCrement contents of r; compare contents of r_i to r_j and if Equal then Jump to instruction z}

Although his model is more complicated than this simple description, the Melzak "pebble" model extended this notion of "counter" to permit multi- pebble adds and subtracts.

Melzak recognised a couple serious defects in his register/counter-machine model:[5] (i) Without a form of indirect addressing he would not be able to "easily" show the model is Turing equivalent, (ii) The program and registers were in different "spaces", so self-modifying programs would not be easy. When Melzak added indirect addressing to his model he created a random access machine model.

(However, with Gödel numbering of the instructions Minsky offered a proof that with such numbering the general recursive functions were indeed possible; he offers proof that μ recursion is indeed possible[3]).

Unlike the RASP model, the RAM model does not allow the machine's actions to modify its instructions. Sometimes the model works only register-to-register with no accumulator, but most models seem to include an accumulator.

van Emde Boas divides the various RAM models into a number of sub-types:[2]

• SRAM, the "successor RAM" with only one arithmetic instruction, the successor (INCREMENT h). The others include "CLEAR h", and an IF equality-between-register THEN jump-to xxx.
• RAM: the standard model with addition and subtraction
• MRAM: the RAM augmented with multiplication and division
• BRAM, MBRAM: Bitwise Boolean versions of the RAM and MRAM
• N****: Non-deterministic versions of any of the above with an N before the name

The RASP is a RAM with the instructions stored together with their data in the same 'space' – i.e. sequence of registers. The notion of a RASP was described at least as early as Kiphengst. His model had a "mill"—an accumulator, but now the instructions were in the registers with the data—the so-called von Neumann architecture. When the RASP has alternating even and odd registers—the even holding the "operation code" (instruction) and the odd holding its "operand" (parameter), then indirect addressing is achieved by simply modifying an instruction's operand.[6]

The original RASP model of Elgot and Robinson had only three instructions in the fashion of the register-machine model,[7] but they placed them in the register space together with their data. (Here COPY takes the place of CLEAR when one register e.g. "z" or "0" starts with and always contains 0. This trick is not unusual. The unit 1 in register "unit" or "1" is also useful.)

{ INC ( r ), COPY ( r_i, r_j ), JE ( r_i, r_i, z ) }

The RASP models allow indirect as well as direct-addressing; some allow "immediate" instructions too, e.g. "Load accumulator with the constant 3". The instructions may be of a highly restricted set such as the following 16 instructions of Hartmanis.[8] This model uses an accumulator A. The mnemonics are those that the authors used (their CLA is "load accumulator" with constant or from register; STO is "store accumulator"). Their syntax is the following, excepting the jumps: "n, <n>, <<n>>" for "immediate", "direct" and "indirect"). Jumps are via two "Transfer instructions" TRA—unconditional jump by directly "n" or indirectly "< n >" jamming contents of register n into the instruction counter, TRZ (conditional jump if Accumulator is zero in the same manner as TRA):

{ ADD n , ADD < n >, ADD << n >>, SUB n, SUB < n >, SUB << n >>, CLA n, CLA < n >, CLA << n >>, STO < n >, STO << n >>, TRA n, TRA < n >, TRZ n, TRA < n >, HALT }

A relative latecomer is Schönhage's Storage Modification Machine or pointer machine. Another version is the Kolmogorov-Uspensky machine, and the Knuth "linking automaton" proposal. (For references see pointer machine). Like a state-machine diagram, a node emits at least two labelled "edges" (arrows) that point to another node or nodes which in turn point to other nodes, etc. The outside world points at the center node.