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Text of the page (random words):
e has to go to some trouble to find the number one wishes to work on and keep it separate from other numbers 7 218 indeed as examples in turing machine examples post turing machine and partial functions show the work can be complicated minsky melzak lambek and shepherdson sturgis models cut the tape into many edit this section s style of writing may not reflect the encyclopedic tone used on wikipedia see wikipedia s guide to writing better articles for suggestions january 2024 learn how and when to remove this message initial thought leads to cutting the tape so that each is infinitely long to accommodate any size integer but left ended these three tapes are called post turing i e wang like tapes the individual heads move to the left for decrementing and to the right for incrementing in a sense the heads indicate the top of the stack of concatenated marks or in minsky 1961 3 and hopcroft and ullman 1979 16 171ff the tape is always blank except for a mark at the left end at no time does a head ever print or erase care must be taken to write the instructions so that a test for zero and a jump occur before decrementing otherwise the machine will fall off the end or bump against the end creating an instance of a partial function minsky 1961 3 and shepherdson sturgis 1963 7 prove that only a few tapes as few as one still allow the machine to be turing equivalent if the data on the tape is represented as a gödel number or some other uniquely encodable encodable decodable number this number will evolve as the computation proceeds in the one tape version with gödel number encoding the counter machine must be able to i multiply the gödel number by a constant numbers 2 or 3 and ii divide by a constant numbers 2 or 3 and jump if the remainder is zero minsky 1967 13 shows that the need for this bizarre instruction set can be relaxed to inc r jzdec r z and the convenience instructions clr r j r if two tapes are available however a simple gödelization is still required a similar result appears in elgot robinson 1964 17 with respect to their rasp model melzak s 1961 model is different clumps of pebbles go into and out of holes edit melzak s 1961 2 model is significantly different he took his own model flipped the tapes vertically called them holes in the ground to be filled with pebble counters unlike minsky s increment and decrement melzak allowed for proper subtraction of any count of pebbles and adds of any count of pebbles he defines indirect addressing for his model 2 288 and provides two examples of its use 2 89 his proof 2 290 292 that his model is turing equivalent is so sketchy that the reader cannot tell whether or not he intended the indirect addressing to be a requirement for the proof legacy of melzak s model is lambek s simplification and the reappearance of his mnemonic conventions in cook and reckhow 1973 18 lambek 1961 atomizes melzak s model into the minsky 1961 model inc and dec with test edit lambek 1961 4 took melzak s ternary model and atomized it down to the two unary instructions x x if possible else jump exactly the same two that minsky 1961 3 had come up with however like the minsky 1961 3 model the lambek model does execute its instructions in a default sequential manner both x and x carry the identifier of the next instruction and x also carries the jump to instruction if the zero test is successful elgot robinson 1964 and the problem of the rasp without indirect addressing edit a rasp or random access stored program machine begins as a counter machine with its program of instruction placed in its registers analogous to but independent of the finite state machine s instruction register at least one of the registers nicknamed the program counter pc and one or more temporary registers maintain a record of and operate on the current instruction s number the finite state machine s table of instructions is responsible for i fetching the current program instruction from the proper register ii parsing the program instruction iii fetching operands specified by the program instruction and iv executing the program instruction except there is a problem if based on the counter machine chassis this computer like von neumann machine will not be turing equivalent it cannot compute everything that is computable intrinsically the model is bounded by the size of its very finite state machine s instructions the counter machine based rasp can compute any primitive recursive function e g multiplication but not all mu recursive functions e g the ackermann function elgot robinson investigate the possibility of allowing their rasp model to self modify its program instructions 17 the idea was an old one proposed by burks goldstine von neumann 1946 1947 19 and sometimes called the computed goto melzak 1961 2 specifically mentions the computed goto by name but instead provides his model with indirect addressing computed goto a rasp program of instructions that modifies the goto address in a conditional or unconditional jump program instruction but this does not solve the problem unless one resorts to gödel numbers what is necessary is a method to fetch the address of a program instruction that lies far beyond above the upper bound of the finite state machine instruction register and table example a counter machine equipped with only four unbounded registers can e g multiply any two numbers m n together to yield p and thus be a primitive recursive function no matter how large the numbers m and n moreover less than 20 instructions are required to do this e g 1 clr p 2 jz m done 3 outer_loop jz n done 4 cpy m temp 5 inner_loop jz m outer_loop 6 dec m 7 inc p 8 j inner_loop 9 outer_loop dec n 10 j outer_loop halt however with only 4 registers this machine has not nearly big enough to build a rasp that can execute the multiply algorithm as a program no matter how big we build our finite state machine there will always be a program including its parameters which is larger so by definition the bounded program machine that does not use unbounded encoding tricks such as gödel numbers cannot be universal minsky 1967 13 hints at the issue in his investigation of a counter machine he calls them program computer models equipped with the instructions clr r inc r and rpt a times the instructions m to n he doesn t tell us how to fix the problem but he does observe that the program computer has to have some way to keep track of how many rpt s remain to be done and this might exhaust any particular amount of storage allowed in the finite part of the computer rpt operations require infinite registers of their own in general and they must be treated differently from the other kinds of operations we have considered 13 214 but elgot and robinson solve the problem 17 they augment their p 0 rasp with an indexed set of instructions a somewhat more complicated but more flexible form of indirect addressing their p 0 model addresses the registers by adding the contents of the base register specified in the instruction to the index specified explicitly in the instruction or vice versa swapping base and index thus the indexing p 0 instructions have one more parameter than the non indexing p 0 instructions example inc r base index effective address will be r base index where the natural number index is derived from the finite state machine instruction itself hartmanis 1971 edit by 1971 hartmanis has simplified the indexing to indirection for use in his rasp model 20 indirect addressing a pointer register supplies the finite state machine with the address of the target register required for the instruction said another way the contents of the pointer register is the address of the target register to be used by the instruction if the pointer register is unbounded the ram and a suitable rasp built on its chassis will be turing equivalent the target register can serve either as a source or destination register as specified by the instruction note that the finite state machine does not have to explicitly specify this target register s address it just says to the rest of the machine get me the contents of the register pointed to by my pointer register and then do xyz with it it must specify explicitly by name via its instruction this pointer register e g n or 72 or pc etc but it doesn t have to know what number the pointer register actually contains perhaps 279 431 cook and reckhow 1973 describe the ram edit cook and reckhow 1973 18 cite hartmanis 1971 20 and simplify his model to what they call a random access machine ram i e a machine with indirection and the harvard architecture in a sense we are back to melzak 1961 2 but with a much simpler model than melzak s precedence edit minsky was working at the mit lincoln laboratory and published his work there his paper was received for publishing in the annals of mathematics on 15 august 1960 but not published until november 1961 3 while receipt occurred a full year before the work of melzak 2 and lambek 4 was received and published received respectively may and 15 june 1961 and published side by side september 1961 that i both were canadians and published in the canadian mathematical bulletin ii neither would have had reference to minsky s work because it was not yet published in a peer reviewed journal but iii melzak references wang and lambek references melzak leads one to hypothesize that their work occurred simultaneously and independently almost exactly the same thing happened to shepherdson and sturgis 21 their paper was received in december 1961 just a few months after melzak and lambek s work was received again they had little at most 1 month or no benefit of reviewing the work of minsky they were careful to observe in footnotes that papers by ershov 22 kaphengst 10 and péter 9 had recently appeared 21 219 these were published much earlier but appeared in the german language in german journals so issues of accessibility present themselves the final paper of shepherdson and sturgis did not appear in a peer reviewed journal until 1963 7 and as they note in their appendix a the systems of kaphengst 1959 10 ershov 1958 22 and péter 1958 9 are all so similar to what results were obtained later as to be indistinguishable to a set of the following produce 0 i e 0 n increment a number i e n 1 n i e of performing the operations which generate the natural numbers 7 246 copy a number i e n m to change the course of a computation either comparing two numbers or decrementing until 0 indeed shepherson and sturgis conclude the various minimal systems are very similar 7 246 by order of publishing date the work of kaphengst 1959 10 ershov 1958 22 péter 1958 were first 9 see also edit counter machine counter machine model pointer machine random access machine random access stored program machine turing machine universal turing machine turing machine examples wang b machine post turing machine description plus examples one instruction set computer algorithm algorithm characterizations halting problem busy beaver stack machine wdr paper computer bibliography edit background texts the following bibliography of source papers includes a number of texts to be used as background the mathematics that led to the flurry of papers about abstract machines in the 1950s and 1960s can be found in van heijenoort 1967 23 an assemblage of original papers spanning the 50 years from frege 1879 24 to gödel 1931 25 davis ed the undecidable 1965 26 carries the torch onward beginning with gödel 1931 25 through gödel s 1964 postscriptum 27 71 the original papers of alan turing 1936 28 1937 and emil post 1936 6 are included in the undecidable the mathematics of church rosser and kleene that appear as reprints of original papers in the undecidable is carried further in kleene 1952 29 a mandatory text for anyone pursuing a deeper understanding of the mathematics behind the machines both kleene 1952 29 and davis 1958 30 are referenced by a number of the papers for a good treatment of the counter machine see minsky 1967 chapter 11 models similar to digital computers he calls the counter machine a program computer 13 a recent overview is found at van emde boas 1990 31 a recent treatment of the minsky 1961 3 lambek 1961 4 model can be found boolos burgess jeffrey 2002 32 they reincarnate lambek s abacus model to demonstrate equivalence of turing machines and partial recursive functions and they provide a graduate level introduction to both abstract machine models counter and turing and the mathematics of recursion theory beginning with the first edition boolos burgess 1970 33 this model appeared with virtually the same treatment the papers the papers begin with wang 1957 12 and his dramatic simplification of the turing machine turing 1936 28 kleene 1952 29 davis 1958 30 and in particular post 1936 6 are cited in wang 1957 12 in turn wang is referenced by melzak 1961 2 minsky 1961 3 and shepherdson sturgis 1961 1963 21 7 as they independently reduce the turing tapes to counters melzak 1961 2 provides his pebble in holes counter machine model with indirection but doesn t carry the treatment further the work of elgot robinson 1964 17 define the rasp the computer like random access stored program machines and appear to be the first to investigate the failure of the bounded counter machine to calculate the mu recursive functions this failure except with the draconian use of gödel numbers in the manner of minsky 1961 3 leads to their definition of indexed instructions i e indirect addressing for their rasp model elgot robinson 1964 17 and more so hartmanis 1971 20 investigate rasps with self modifying programs hartmanis 1971 20 specifies an instruction set with indirection citing lecture notes of cook 1970 34 for use in investigations of computational complexity cook and his graduate student reckhow 1973 18 provide the definition of a ram their model and mnemonic convention are similar to melzak s but offer him no reference in the paper the pointer machines are an offshoot of knuth 1968 35 1973 and independently schönhage 1980 36 for the most part the papers contain mathematics beyond the undergraduate level in particular the primitive recursive functions and mu recursive functions presented elegantly in kleene 1952 29 and less in depth but useful nonetheless in boolos burgess jeffrey 2002 32 all texts and papers excepting the four starred have been witnessed these four are written in german and appear as references in shepherdson sturgis 1963 7 and elgot robinson 1964 17 shepherdson sturgis 1963 7 offer a brief discussion of their results in shepherdson sturgis appendix a the terminology of at least one paper kaphengst 1959 10 seems to hark back to the burke goldstine von neumann 1946 1947 19 analysis of computer architecture author year reference turing machine counter machine ram rasp pointer machine indirect addressing self modifying program goldstine von neumann 1947 19 kleene 1952 29 hermes 1954 1955 8 wang 1957 12 hints hints péter 1958 9 davis 1958 30 ...