Random-access machine

81: 1357: – by using a primitive recursive "operator" called "definition by cases" (defined in Kleene (1952) p. 229 and Boolos-Burgess-Jeffrey p. 74). Such a "bounded indirection" is a laborious, tedious affair. "Definition by cases" requires the machine to determine/distinguish the contents of the pointer register by attempting, time after time until success, to match this contents against a number/name that the case operator 2077:= { r0, r1, r2, ... } to an unbounded string of (very-) bounded-capacity pigeon-holes. These will do nothing but hold (very-) bounded numbers e.g. a lone bit with value { 0, 1 }. Likewise we shrink the accumulator to a single bit. We restrict any arithmetic to the registers { A, N }, use indirect operations to pull the contents of registers into the accumulator and write 0 or 1 from the accumulator to a register: 2138:

square" that "the head" is currently observing. The "head" can be thought of as being in the conditional jump – observe that it uses indirect addressing (cf Elgot-Robinson p. 398). As we decrement or increment "N" the (apparent) head will "move left" or "right" along the squares. We will move the contents of "E"=0 or "P"=1 to the "scanned square" as pointed to by N, using the indirect CPY.

1554:: Because any instruction acting on a single register can be augmented with its indirect "dual" (including conditional and unconditional jumps, cf the Elgot-Robinson model), the inclusion of indirect instructions will double the number of single parameter/register instructions (e.g. INC (d, r), INC (i, r)). Worse, every two parameter/register instruction will have 4 possible varieties, e.g.: 1157: – it has no indirection capability – cannot compute all "recursive sequential functions" (ones that have parameters of arbitrary length) if it does not have the capability of modifying its own instructions, but it can via Gödel numbers if it does (p. 395-397; in particular figure 2 and footnote p. 395). On the other hand their RASP model P' 40: 198: 143: 1443:

register – in some models every register can be a pointer register – is specified by the instruction). This "mutually exclusive but exhaustive choice" is yet another example of "definition by cases", and the arithmetic equivalent shown in the example below is derived from the definition in Kleene (1952) p. 229.

1310:

model – quite similar to Melzak's (1961) model – uses two and three-register adds and subtracts and two parameter copies; Cook and Reckhow's model reduce the number of parameters (registers called out in the program instructions) to one call-out by use of an accumulator "AC".

1771:

If we designate one register to be the "accumulator" (see below) and place strong restrictions on the various instructions allowed then we can greatly reduce the plethora of direct and indirect operations. However, one must be sure that the resulting reduced instruction-set is sufficient, and we must

1196:

finite both in the number of "states" (instructions) and the instructions' sizes (their capacity to hold symbols/signs). So how does a state machine move an arbitrarily large constant directly into a register, e.g. MOVE (k, r) (Move constant k to register r)? If huge constants are necessary they must

997:

of instructions created from the base set and given a mnemonic. In a formal sense, to use these blocks we need to either (i) "expand" them into their base-instruction equivalents – they will require the use of temporary or "auxiliary" registers so the model must take this into account,

600:

of a tape's "head" from its left end, measured in numbers of tape-squares, represents the natural number in "the register". To DECrement the count of squares the tape head moves left; INCrement it moves right. There is no need to print or erase marks on the tape; the only conditional instructions are

584:

Melzak (1961) provides an easy visualization of a counter machine: its "registers" are holes in the ground, and these holes hold pebbles. Per an instruction, into and out of these holes "the computer" (person or machine) adds (INCrements) or removes (DECrements) a single pebble. As needed, additional

2985:

We will build the indirect CPY ( i, q, d, φ ) with the CASE operator. The address of the target register will be specified by the contents of register "q"; once the CASE operator has determined what this number is, CPY will directly deposit the contents of the register with that number into register

2045:

For maximum flexibility, as we have done for the accumulator A – we will consider N just another register subject to increment, decrement, clear, test, direct copy, etc. Again we can shrink the instruction to a single-parameter that provides for direction and indirection, for example.

1270:-instructions (the ones stored in the registers) so that they contain more than one command. But this too can be exhausted unless an instruction is of (potentially) unbounded size. So why not use just one "über-instruction" – one really really big number – that contains 1205:

Sometimes the constant k will be created by use of CLR ( r ) followed by INC ( r ) repeated k times – e.g. to put the constant k=3 into register r, i.e. 3 → r, so at the end of the instruction =3: CLR (r), INC (r), INC (r), INC (r). This trick is mentioned by Kleene (1952) p. 223. The

2991:

So the following is actually a constructive demonstration or proof that we can indeed simulate the indirect CPY ( i, q, d, φ ) without a "hardware" design change to our counter machine/model. However, note that because this indirect CPY is "bounded" by the size/extent of the finite state machine, a

2137:

We give here a slightly more formal demonstration. Begin by designing our model with three reserved registers "E", "P", and "N", plus an unbounded set of registers 1, 2, ..., n to the right. The registers 1, 2, ..., n will be considered "the squares of the tape". Register "N" points to "the scanned

2041:

Another approach (Schönhage does this too) is to declare a specific register the "indirect address register" and confine indirection relative to this register (Schonhage's RAM0 model uses both A and N registers for indirect as well as direct instructions). Again our new register has no conventional

1967:

The typical accumulator-based model will have all its two-variable arithmetic and constant operations (e.g. ADD (A, r), SUB (A, r) ) use (i) the accumulator's contents, together with (ii) a specified register's contents. The one-variable operations (e.g. INC (A), DEC (A) and CLR (A) ) require only

1133:

In the following one must remember that these models are abstract models with two fundamental differences from anything physically real: unbounded numbers of registers each with unbounded capacities. The problem appears most dramatically when one tries to use a counter-machine model to build a RASP

964:

The three base sets 1, 2, or 3 above are equivalent in the sense that one can create the instructions of one set using the instructions of another set (an interesting exercise: a hint from Minsky (1967) – declare a reserved register e.g. call it "0" (or Z for "zero" or E for "erase")

2141:

The fact that our tape is left-ended presents us with a minor problem: Whenever LEFT occurs our instructions will have to test to determine whether or not the contents of "N" is zero; if so we should leave its count at "0" (this is our choice as designers – for example we might have

491:

Example: → 5; means "The contents of source register with address "3" is put into destination register with address "5". If =38, that is, the contents of register 3 is the number 38, then this number will be put into register 5. The contents of register 3 are not disturbed by this operation, so

3112:

We begin with a number in register q that represents the address of the target register. But what is this number? The "predicates" will test it to find out, one trial after another: JE (q, y, z) followed by INC (y). Once the number is identified explicitly, the CASE operator directly/explicitly

1792:

However, the accumulator comes at the expense of more instructions per arithmetic "operation", in particular with respect to what are called 'read-modify-write' instructions such as "Increment indirectly the contents of the register pointed to by register r2 ". "A" designates the "accumulator"

1442:

need to specify an additional parameter "i/d" – "indirect/direct". In a sense this new "i/d" parameter is a "switch" that flips one way to get the direct address as specified in the instruction or the other way to get the indirect address from the pointer register (which pointer

1147:

Melzak (1961) added indirection to his "hole-and-pebble" model so that his model could modify itself with a "computed goto" and provides two examples of its use ("Decimal representation in the scale of d" and "Sorting by magnitude", whether these are used in his proof that the model is Turing

1309:

From the references in Hartmanis (1971) it appears that Cook (in his lecture notes while at UC Berkeley, 1970) has firmed up the notion of indirect addressing. This becomes clearer in the paper of Cook and Reckhow (1973) – Cook is Reckhow's Master's thesis advisor. Hartmanis'

481:

Example: +1 → 3; means "The contents of source register with address "3", plus 1, is put into destination register with address "3" (here source and destination are the same place). If =37, that is, the contents of register 3 is the number "37", then 37+1 = 38 will be put into register

3643:: Schönhage's RAM0 machine has 6 instructions indicated by a single letter (the 6th "C xxx" seems to involve 'skip over next parameter'. Schönhage designated the accumulator with "z", "N" with "n", etc. Rather than Schönhage's mnemonics we will use the mnemonics developed above. 621:

capacity, called "registers". These registers hold only natural numbers (zero and positive integers). Per a list of sequential instructions in the finite state machine's TABLE, a few (e.g. 2) types of primitive operations operate on the contents of these "registers". Finally, a

1788:. It is quite conventional in principle in past and present computing machines of the most varied types, e.g. desk multipliers, standard IBM counters, more modern relay machines, the ENIAC" (boldface in original: Goldstine and von Neumann, 1946; p. 98 in Bell and Newell 1971). 1152:

encoding, the register model did not need indirection to be Turing equivalent; but it did need at least one unbounded register. As noted below, Minsky (1967) hints at the problem for a RASP but doesn't offer a solution. Elgot and Robinson (1964) proved that their RASP model

4165:(volume A). QA 76.H279 1990. van Emde Boas's treatment of SMMs appears on pp. 32–35. This treatment clarifies Schōnhage 1980 – it closely follows but expands slightly the Schōnhage treatment. Both references may be needed for effective understanding. 987:

will be discussed in context of indirect addressing). However, building the primitive recursive functions is difficult because the instruction sets are so ... primitive (tiny). One solution is to expand a particular set with "convenience instructions" from another set:

510:

is one that specifies a "pointer register", the contents of which is the address of a "target" register. The target register can be either a source or a destination (the various COPY instructions provide examples of this). A register can address itself indirectly.

1294:"In general a RPT operation could not be an instruction in the finite-state part of the machine ... this might exhaust any particular amount of storage allowed in the finite part of the computer . RPT operations require infinite registers of their own." (p. 214). 3005:

The CASE "operator" is described in Kleene (1952) (p. 229) and in Boolos-Burgess-Jeffrey (2002) (p. 74); the latter authors emphasize its utility. The following definition is per Kleene but modified to reflect the familiar "IF-THEN-ELSE" construction.

1332:

that can, if necessary, hunt ad infinitum along the unbounded string of registers until it finds what it is looking for. The pointer register is exactly like any other register with one exception: under the circumstances called "indirect addressing" it provides

1281:

Elgot and Robinson (1964) come to a similar conclusion with respect to a RASP that is "finitely determined". Indeed it can access an unbounded number of registers (e.g. to fetch instructions from them) but only if the RASP allows "self modification" of its

4081:, Journal of the Association for Computing Machinery (JACM) 10:217-255, 1963. An extremely valuable reference paper. In their Appendix A the authors cite 4 others with reference to "Minimality of Instructions Used in 4.1: Comparison with Similar Systems". 3009:

The CASE operator "returns" a natural number into φ depending on which "case" is satisfied, starting with "case_0" and going successively through "case_last"; if no case is satisfied then the number called "default" (aka "woops") is returned into φ (here

573:

is where the instruction deposits its result. The source register's address can be specified either (i) directly by the instruction, or (ii) indirectly by the pointer register specified by the instruction. The source and destination registers can be one.

1784:"The first part of our arithmetic organ ... should be a parallel storage organ which can receive a number and add it to the one already in it, which is also able to clear its contents and which can store what it contains. We will call such an organ an 1384:

Suppose we had been able to continue on to number 65367, and in fact that register had what we were looking for. Then we could have completed our calculation successfully! But suppose 65367 didn't have what we needed. How far should we continue to

4013:, vol. 4, no. 3. September 1961 pages 279-293. Melzak offers no references but acknowledges "the benefit of conversations with Drs. R. Hamming, D. McIlroy and V. Vyssots of the Bell telephone Laborators and with Dr. H. Wang of Oxford University." 391:

varies depending on the author; common instructions include: increment, decrement, clear to zero, copy, conditional jump, halt; other instructions are unnecessary because they can be created by combinations of instructions from the instruction

2154:

The following table both defines the Post-Turing instructions in terms of their RAM equivalent instructions and gives an example of their functioning. The (apparent)location of the head along the tape of registers r0-r5 . . . is shown shaded:

1345:

If we eschew the Minsky approach of one monster number in one register, and specify that our machine model will be "like a computer" we have to confront this problem of indirection if we are to compute the recursive functions (also called the

301:

of a common computer, except for the additional ability of registers to store natural numbers of any size. Like the counter machine, the RA-machine contains the execution instructions in the finite-state portion of the machine (the so-called

1398:

method, or be augmented with an ability to explore the ends of its register string, ad infinitum if necessary. (A failure to find something "out there" defines what it means for an algorithm to fail to terminate; cf Kleene (1952) pp. 316ff

3853:, Cambridge University Press, Cambridge, England. The original Boolos-Jeffrey text has been extensively revised by Burgess: more advanced than an introductory textbook. "Abacus machine" model is extensively developed in Chapter 5 3713:, although it is "unbounded" in the sense that the model implies no upper limit to the number of registers necessary to do its job(s). For example, we do not require r < 83,617,563,821,029,283,746 nor r < 2^1,000,001, etc. 1306:, and he offers the unbounded RPT quoted above that as playing the role of μ operator; it together with CLR (r) and INC (r) can compute the mu recursive functions. But he does not discuss "indirection" or the RAM model per se. 423:

model. Two additions move it away from the counter machine, however. The first enhances the machine with the convenience of indirect addressing; the second moves the model toward the more conventional accumulator-based

3108:

that doing the testing are all mutually exclusive – "predicates" are functions that produce only { true, false } for output; Boolos-Burgess-Jeffrey add the requirement that the cases are "exhaustive".

3377:

The commonly encountered Cook and Rechkow model is a bit like the ternary-register Malzek model (written with Knuth mnemonics – the original instructions had no mnemonics excepting TRA, Read, Print).

1197:

either start out in the registers themselves or be created by the state machine using a finite number of instructions e.g. multiply and add subroutines using INC and DEC (but not a quasi-infinite number of these!).

1547:

with the COPY instructions, and Cook-Reckhow (1973) provide their accumulator-based model with only two indirect instructions – COPY to accumulator indirectly, COPY from accumulator indirectly.

4135:, Society for Industrial and Applied Mathematics, SIAM J. Comput. Vol. 9, No. 3, August 1980. Wherein Schōnhage shows the equivalence of his SMM with the "successor RAM" (Random Access Machine), etc. resp. 1289:

In the context of a more computer-like model using his RPT (repeat) instruction Minsky (1967) tantalizes us with a solution to the problem (cf p. 214, p. 259) but offers no firm resolution. He asserts:

467: – a single natural number. For precision we will use the quasi-formal symbolism from Boolos-Burgess-Jeffrey (2002) to specify a register, its contents, and an operation on a register: 1258:

state machine to address them, then registers outside the bounds will be unreachable. For example, if the finite state machine can only reach 65,536 = 2 registers then how can it reach the 65,537th?

1148:

equivalent is unclear since "the program itself is left to the reader as an exercise" (p. 292)). Minsky (1961, 1967) was able to demonstrate that, with suitable (but difficult-to-use)

544:

is used by the instruction. The source register's address can be specified either (i) directly by the instruction, or (ii) indirectly by the pointer register specified by the instruction.

3995:, Mathematical Bulletin, vol. 4, no. 3. September 1961 pages 295-302. In his Appendix II, Lambek proposes a "formal definition of 'program'. He references Melzak (1961) and Kleene (1952) 1366:

Is the number in register N equal to 0? If not then is it equal to 1? 2? 3? ... 65364? If not then we're at the last number 65365 and this had better be the one, else we have a problem!

3709:

The definitional fact that any sort of counter machine without an unbounded register-"address" register must specify a register "r" by name indicates that the model requires "r" to be

1075:

Most authors pick one or the other of the conditional jumps, e.g. Shepherdson-Sturgis (1963) use the above set minus JE (to be perfectly accurate they use JNZ – Jump if

617:

has, for a memory external to its finite-state machine – an unbounded (cf: footnote|countable and unbounded) collection of discrete and uniquely labelled locations with

3985:, Second Edition 1973, Addison-Wesley, Reading, Massachusetts. Cf pages 462-463 where he defines "a new kind of abstract machine or 'automaton' which deals with linked structures." 1328:

register requires an ability to be cleared and then incremented (and, possibly, decremented) by a potentially infinite loop. In this sense the solution represents the unbounded

3522:

Conditional jump if is positive; i.e. IF > 0 THEN jump to instruction z else continue in sequence (Cook and Reckhow call this: "TRAnsfer control to line m if Xj > 0")

4067:. In the former chapter he defines "Program machines" and in the later chapter he discusses "Universal Program machines with Two Registers" and "...with one register", etc. 1780:

Historical convention dedicates a register to the accumulator, an "arithmetic organ" that literally accumulates its number during a sequence of arithmetic operations:

1415:

indirection we require a "hardware" change in our machine model. Once we make this change the model is no longer a counter machine, but rather a random-access machine.

1233:) that uses its finite-state machine to interpret a "program of instructions" located in its registers – i.e. we are building what is nowadays called a 161: 3857:; it is one of three models extensively treated and compared – the Turing machine (still in Boolos' original 4-tuple form) and recursion the other two. 1990:

If we so choose, we can abbreviate the mnemonics because at least one source-register and the destination register is always the accumulator A. Thus we have :

1361:

declares. Thus the definition by cases starts from e.g. the lower bound address and continues ad nauseam toward the upper bound address attempting to make a match:

1161:

equipped with an "index register" (indirect addressing) can compute all the "partial recursive sequential functions" (the mu recursive functions) (p. 397-398).

596:

with as many left-ended tapes as "registers". Each tape's length is unbounded to the right, and every square is blank except for the left end, which is marked. The

585:

pebbles come from, and excess pebbles go back into, an infinite supply; if the hole is too small to accommodate the pebbles the "computer" digs the hole bigger.

1229:: This is more severe than the first problem. In particular, this problem arises when we attempt to build a so-called RASP, a "universal machine" (see more at 2109:

Rename the COPY instructions and call INC (N) = RIGHT, DEC (N) = LEFT and we have the same instructions as the Post-Turing machine, plus an extra CLRN :

847:: Used by Elgot-Robinson (1964) in their investigation of bounded and unbounded RASPs – the "successor" model with COPY in the place of CLEAR: 471:

means "the contents of register with address r". The label "r" here is a "variable" that can be filled with a natural number or a letter (e.g. "A") or a name.

3583:: Schönhage demonstrates how his construction can be used to form the more common, usable form of "successor"-like RAM (using this article's mnemonics): 515:

For want of a standard/convention this article will specify "direct/indirect", abbreviated as "d/i", as a parameter (or parameters) in the instruction:

2974:

which gives a sequence of operations for solving a specific type of problem, an algorithm has five important features " (italics added, Knuth p. 4-7).

3692:

Indirection comes (i) from CPYAN (copy/transfer contents A to N) working with store_A_via_N STAN, and from (ii) the peculiar indirection instruction

1468:

Assign a code to specify direct addressing as d="0" and indirect addressing as i="1". Then our machine can determine the source address as follows:

1002:

Example: Base set 1. To create CLR (r) use the block of instructions to count down register r to zero. Observe the use of the hint mentioned above:

534:

get the destination address from pointer-register N. Suppose =3, then register 3 is the destination and the instruction will do the following: → 3.

1913:

However, when we write the CPY instructions without the accumulator called out the instructions are ambiguous or they must have empty parameters:

1278:

he uses represents a great inconvenience to the model, and the result is nothing at all like our intuitive notion of a "stored program computer".

2980:

The difficulty arises because the registers have explicit "names" (numbers) and our machine must call each out by name in order to "access" it.

1337:

contents, rather than the address-operand in the state machine's TABLE, to be the address of the target register (including possibly itself!).

53: 630:

is available to test the contents of one or two registers and "branch/jump" the finite state machine out of the default instruction-sequence.

3733:

We can escape this restriction by providing an unbounded register to provide the address of the register that specifies an indirect address.

1924:

Historically what has happened is these two CPY instructions have received distinctive names; however, no convention exists. Tradition (e.g.

1173:

The indirect instructions are necessary in order for a fixed program to access an unbounded number of registers as the inputs vary." (p. 73)

1108:(3) We go to location "under_Thatcher's_front_porch", jackhammer away the concrete, and discover "the treasure": a sack of rusty door-knobs. 318: 3356:. But it can't – its finite state machine's "state register" has reached its maximum count (e.g. 65365 = 11111111,11111111 1592:) = COPY contents of source-register indirectly into register using destination address to be found in the destination-pointer register r 448:'s TABLE, the machine derives a "target" register's address either (i) directly from the instruction itself, or (ii) indirectly from the 965:

to contain the number 0). The choice of model will depend on which an author finds easiest to use in a demonstration, or a proof, etc.

3935:(1971), "Computational Complexity of Random Access Stored Program Machines," Mathematical Systems Theory 5, 3 (1971) pp. 232–245. 3817:

Goldstine, Herman H., and von Neumann, John, "Planning and Coding of the Problems for an Electronic Computing Instrument", Rep. 1947,

1968:

the accumulator. Both instruction-types deposit the result (e.g. sum, difference, product, quotient or remainder) in the accumulator.

1324:"address" register that can potentially name (call out) any register no matter how many there are. For this to work, in general, the 4175:, JACM (Journal of the Association for Computing Machinery) 4; 63-92. Presented at the meeting of the Association, June 23–25, 1954. 4162: 3972: 3954: 3892: 3830: 3216:

Case_n (the induction step) looks like this; remember, each instance of "n", "n+1", ..., "last" must be an explicit natural number:

341: 252: 234: 216: 208: 179: 124: 102: 67: 1051:

For example: the most expanded set would include each unique instruction from the three sets, plus unconditional jump J (z) i.e.:

2035:

all the other registers? Not until we provide for at least one unbounded register from which we derive our indirect addresses.

1905:

If we stick with a specific name for the accumulator, e.g. "A", we can imply the accumulator in the instructions, for example,

1353:

Our simpler counter-machine model can do a "bounded" form of indirection – and thereby compute the sub-class of

4010: 349: 2993: 3908: 3818: 1354: 1303: 973: 506:

the address of the source or destination register whose contents will be the subject of the instruction. Definition: An

428:

with the addition of one or more auxiliary (dedicated) registers, the most common of which is called "the accumulator".

388: 297:

but with the added capability of 'indirect addressing' of its registers. The 'registers' are intuitively equivalent to

2093:

We push further and eliminate A altogether by the use of two "constant" registers called "ERASE" and "PRINT": =0, =1.

1371:"Bounded" indirection will not allow us to compute the partial recursive functions – for those we need 2134:. The Post–Turing machine is Turing equivalent, so we have shown that the RAM with indirection is Turing equivalent. 1479:

For example, suppose the contents of register 3 are "5" (i.e. =5 ) and the contents of register 4 are "2" (i.e. =2 ):

1099:

At location "Tom_&_Becky's_cave_in_pirate_chest" will be where we can find a map directing us to "the treasure":

1192:

state part of the machine is supposed to be – by the normal definition of algorithm –

4188: 2958:

Throughout this demonstration we have to keep in mind that the instructions in the finite state machine's TABLE is

59: 3717:

Thus our model can "expand" the number of registers, if necessary to perform a certain computation. However this

1577:) = COPY to destination-register indirectly using the source address to be found in the source-pointer register r 3790: 1539:

Probably the most useful of the added instructions is COPY. Indeed, Elgot-Robinson (1964) provide their models P

3560:

Schönhage (1980) describes a very primitive, atomized model chosen for his proof of the equivalence of his SMM

2073:

Posing as minimalists, we reduce all the registers excepting the accumulator A and indirection register N e.g.

1238: 1230: 601:

to check to see if the head is at the left end, by testing a left-end mark with a "Jump-if-marked instruction".

322: 310: 95: 89: 3957:. A difficult book centered around the issues of machine-interpretation of "languages", NP-Completeness, etc. 3281:

Case_last stops the induction and bounds the CASE operator (and thereby bounds the "indirect copy" operator):

2131: 1607:) = COPY indirectly the contents of the source register with address to be found in source-pointer register r 4020:(1961). "Recursive Unsolvability of Post's Problem of 'Tag' and Other Topics in Theory of Turing Machines". 2986:"φ". We will need an additional register that we will call "y" – it serves as an up-counter. 1347: 3747: 1929: 106: 4070: 2130:

In the section above we informally showed that a RAM with an unbounded indirection capability produces a

1266:

we address a register beyond the bounds of the finite state machine? One approach would be to modify the

2997: 1139: 1121:

to any other location (including itself): its contents (the treasure map) provides the "address" of the

984: 337: 4105:, (Russian) Dok. Akad. Nauk 122 (1958), 967-970. English translation, Automat. Express 1 (1959), 20-23. 3929:, Journal of the Association for Computing Machinery, Vol. 11, No. 4 (October, 1964), pp. 365–399. 1772:

be aware that the reduction will come at the expense of more instructions per "significant" operation.

445: 365: 303: 298: 282: 28: 4168: 4146: 4128: 4074: 1391: 1135: 4002: 3932: 3918: 1210:

state machine; there is always a bigger constant than the number of instructions available to the

1206:

problem arises when the number to be created exhausts the number of instructions available to the

1117:

specifies a location identified as the pirate chest in "Tom_&_Becky's_cave..." that acts as a

4037: 1048:

Again, all of this is for convenience only; none of this increases the model's intrinsic power.

976:( cf Minsky (1967), Boolos-Burgess-Jeffrey (2002) ). (How to cast the net wider to capture the 4158: 4108: 3968: 3950: 3888: 3826: 1611:, into the destination register with address to be found in the destination-pointer register r 1438:

is the address of interest. Whenever an instruction specifies a register address it now will

1418:

Now when e.g. INC is specified, the finite state machine's instruction will have to specify

4029: 3922: 3868: 3864: 3808: 614: 530:

get the source register's address (register "A") from the instruction itself but indirectly

403:

For a description of a similar concept, but humorous, see the esoteric programming language

314: 290: 286: 266: 3675:; contents of N points to register address; put contents of A into register pointed to by N 1105:(2) Inside the box is a map to the location of the treasure: "under_Thatcher's_front_porch" 1102:(1) We go to location "Tom_&_Becky's_cave..." and dig around until we find a wooden box 4150: 3846: 3842: 3561: 3014:

designates some selection of parameters, e.g. register q and the string r0, ... rlast )):

441: 420: 345: 294: 1619:

In a similar manner every three-register instruction that involves two source registers r

1395: 1275: 1274:

the program instructions encoded into it! This is how Minsky solves the problem, but the

1149: 1129:

Why the need for an indirect operation: Two major problems with the counter-machine model

1932:

computer) uses two names called LOAD and STORE. Here we are adding the "i/d" parameter:

592:

Minsky (1961) and Hopcroft-Ullman 1979 (p. 171) offer the visualization of a multi-tape

4051: 3988: 3960: 3942: 3821:, Princeton. Reprinted on pp. 92–119 in Bell, C. Gordon and Newell, Allen (1971), 608:

The following instruction "mnemonics" e.g. "CLR (r)" are arbitrary; no standard exists.

593: 399:" (IR); this register points to the instruction being executed in the instruction table 377: 373: 333: 3769: 3568:"In order to avoid any explicit addressing the RAM0 has the accumulator with contents 3486:, Indirectly copy the contents of the source-register pointed to by pointer-register r 4182: 4099: 4017: 3938: 3838: 1376: 1329: 463:(a unique, distinguishable designation/locator equivalent to a natural number) and a 416: 17: 2097:{ CPY (d, ERASE, i, N), CPY (d, PRINT, i, N), CLR (N), INC (N), DEC (N), JZ (i, N, I 444:

with the addition of indirect addressing. At the discretion of instruction from its

3978: 3898: 3880: 3860: 2042:

name – perhaps "N" from "iNdex", or "iNdirect" or "address Number".

1925: 737: 3873:

Preliminary discussion of the logical design of an electronic computing instrument

440:(RAM) is an abstract computational-machine model identical to a multiple-register 380:

or zero; each register can store exactly one natural number of any size, or a zero

3764: 1246:

Observe that the counter machine's finite state machine must call out a register

566: – the "target" may be either a source or a destination register. 474:→ means "copy/deposit into", or "replaces", but without destruction of the source 3876: 3353: 1114: 1394:

the counter machine needs to either use the unfortunate single-register Minsky

1250:(directly) by its name/number: INC (65,356) calls out register number "65,365" 3759: 4155:

Handbook of Theoretical Computer Science. Volume A: Algorithms and Complexity

452:(e.g. number, label) of the "pointer" register specified in the instruction. 2065:

Schönhage does this to produce his RAM0 instruction set. See section below.

1286:

instructions, and has encoded its "data" in a Gödel number (Fig. 2 p. 396).

1125:

location "under_Thatcher's_front_porch" where the real action is occurring.

404: 3927:

Random-Access Stored-Program Machines, an Approach to Programming Languages

3373:

2954:

Example: Bounded indirection yields a machine that is not Turing equivalent

2053:

STAN (i/d) = CPY (d, A, i/d, N). STore Accumulator via iNdirection register

636:: The model closest to Minsky's (1961) visualization and to Lambek (1961): 3669:; contents of A points to register address; put register's contents into A 2050:

LDAN (i/d) = CPY (i/d, N, d, A); LoaD Accumulator via iNdirection register

744:{ INCrement the contents of register r, CLeaR the contents of register r, 3742: 1234: 1092:

In our daily lives the notion of an "indirect operation" is not unusual.

425: 326: 4041: 1566:) = COPY directly from source-register directly to destination-register 851:{ INCrement the contents of register r, COPY the contents of register r 3142:

case_last: IF INC (y), = ="last" THEN CPY ( rlast, φ ), J (exit) ELSE

640:{ INCrement contents of register r, DECrement contents of register r, 2146:

Instruction set 1 (augmented): { INC (N), DEC (N), CLR (N), CPY (d, r

1857:

Contents of r2 points to r378,426 with contents "17": copy this to A

1486:

Example: CPY ( 1, 3, 0, 4 ) = CPY ( indirect, reg 3, direct, reg 4 )

4092:, Zeitschrift fur mathematische Logik und Grundlagen der Mathematik: 4033: 1899:

Contents of r2 points to r378,426: copy contents of A into r378,426

1403:, in particular p. 323-325.) See more on this in the example below. 1186:

of registers versus bounded capacities of state-machine instructions

1079:

Zero in place of JZ; yet another possible convenience instruction).

998:

or (ii) design our machines/models with the instructions 'built in'.

317:

in the registers as well as its data – is called the

1426:

can be either (i) the state machine's instruction that provides an

993:

These will not be subroutines in the conventional sense but rather

736:: The "successor" model (named after the successor function of the 419:

machine (RAM) starts with the simplest model of all, the so-called

4007:

An informal Arithmetical Approach to Computability and Computation

3508:

into the destination-register pointed to by the pointer-register r

2081:{ LDA (i, N), STA (i, N), CLR (A/N), INC (A/N), DEC(N), JZ (A/N, I 3807:

With a few exceptions, these references are the same as those at

2057:

Why is this such an interesting approach? At least two reasons:

2038:

The minimalist approach is to use itself (Schönhage does this).

387:, or just "table", containing execution instructions; the exact 3151:

Case_0 ( the base step of the recursion on y) looks like this:

3725: – it must be indexable with a natural number: 191: 136: 74: 33: 1422:

the address of the register of interest will come from. This

4057:(1st ed.). Englewood Cliffs, N. J.: Prentice-Hall, Inc. 3967:, North-Holland Publishing Company, Amsterdam, Netherlands. 3136:

case_n: IF INC (y), = =n THEN CPY ( rn, φ ), J (exit) ELSE

3130:

case_1: IF INC (y), = =1 THEN CPY ( r1, φ ), J (exit) ELSE

3127:

case_0: IF CLR (y), - =0 THEN CPY ( r0, φ ), J (exit) ELSE

3068:

cases_2 through case_next_to_last: etc. . . . . . . . . ELSE

1302:

RPT that together with CLR (r) and INC (r) can compute any

1254:. If the number of registers exceeds the capability of the 344:. Van Emde Boas (1990) calls these three together with the 3947:

Introduction to Automata Theory, Languages and Computation

897:

Action on finite state machine's Instruction Register, IR

782:

Action on finite state machine's Instruction Register, IR

671:

Action on finite state machine's Instruction Register, IR

4028:(3). The Annals of Mathematics, Vol. 74, No. 3: 437–455. 3572:

and an additional address register with current contents

2613: 1407:

Unbounded indirection and the partial recursive functions

1341:

Bounded indirection and the primitive recursive functions

348:, "sequential machine" models, to distinguish them from " 4121:

Math.-Phys. Semsterberichte (Göttingen) 4 (1954), 42-53.

3352:

If the CASE could continue ad infinitum it would be the

3721:

mean that whatever number the model expands to must be

2206:

Action on finite state machine Instruction Register IR

2142:

the machine/model "trigger an event" of our choosing).

157: 4149:, "Machine Models and Simulations" pp. 3–66, in: 4119:

Die Universalität programmgesteuerter Rechenmaschinen.

1631:

will result in 8 varieties, for example the addition:

1227:

of registers versus bounded state-machine instructions

960:

Creating "convenience instructions" from the base sets

340:, the RA-machine and RASP-machine models are used for 1350:) – both total and partial varieties. 3360:) or its table has run out of instructions; it is a 2992:

RASP using this indirect CPY can only calculate the

3905:, Journal of Computer Systems Science 7(4):354-375. 321:or RASP-machine. It is an example of the so-called 152:

may be too technical for most readers to understand

4173:A Variant to Turing's Theory of Computing Machines 4050: 3594:, k is a constant, an explicit number such as "47" 968:Moreover, from base sets 1, 2, or 3 we can create 4090:Eine Abstrakte programmgesteuerte Rechenmaschine' 3139:case_n+1 to case_last: IF . . . THEN . . . ELSE 3133:case_2 through case n: IF . . . THEN . . . ELSE 1168:Cook and Reckhow (1973) say it most succinctly: 2126:Turing equivalence of the RAM with indirection 2113:{ ERASE, PRINT, CLRN, RIGHT, LEFT, JZ (i, N, I 3791:"From Register Machines to Brainfuck, part 1" 3458:, the registers can be the same or different: 3426:, the registers can be the same or different; 1509:0* + 1*4 = 4 = destination-register address 4 1492:0* + 1*4 = 4 = destination-register address 4 8: 3532:copy "the input" into destination register r 1526:1* + 0*4 = = destination-register address 2 3113:copies the contents of this register to φ: 2023:The notion of indirect address register "N" 68:Learn how and when to remove these messages 3915:, McGraw-Hill Book Company, Inc. New York. 3885:Computer Structures: Readings and Examples 3823:Computer Structures: Readings and Examples 3340:how do we handle an out-of-bounds attempt? 2823:IF N =r3] =0 THEN "end" → IR else +1 → IR 2741:IF N =r3] =0 THEN "end" → IR else +1 → IR 2157: 2069:(2) Reduce a RAM to a Post-Turing machine: 2061:(1) An instruction set with no parameters: 4053:Computation: Finite and Infinite Machines 3949:, 1st ed., Reading Mass: Addison-Wesley. 360:An RA-machine consists of the following: 325:and is closest to the common notion of a 253:Learn how and when to remove this message 235:Learn how and when to remove this message 180:Learn how and when to remove this message 164:, without removing the technical details. 125:Learn how and when to remove this message 3504:. Copy the contents of source register r 1795: 1523:0* + 1*3 = 3 = source-register address 3 1506:0* + 1*3 = 3 = source-register address 3 1316:Design our machine/model with unbounded 884: 769: 658: 293:. The RA-machine is very similar to the 88:This article includes a list of general 3851:Computability and Logic: Fourth Edition 3781: 3435:will double the contents of register A. 1489:1* + 0*3 = = source-register address 5 1401:Chapter XII Partial Recursive Functions 4112:Graphschemata und rekursive Funktionen 3887:, mcGraw-Hill Book Company, New York. 3825:, McGraw-Hill Book Company, New York. 3545:copy the contents of source register r 3104:Kleene require that the "predicates" Q 1920:CPY ( d, A, d, r2 ) = CPY ( , , d, r2) 2616:=r4] =0 THEN "end" → IR else +1 → IR 1917:CPY ( d, r2, d, A ) = CPY (d, r2, , ) 492:continues to be 38, now the same as . 162:make it understandable to non-experts 7: 4079:Computability of Recursive Functions 3555: 1055:{ CLR (r), DEC (r), INC (r), CPY ( r 713:IF = 0 THEN z → IR ELSE + 1 → IR 578:Refresher: The counter-machine model 319:random-access stored-program machine 4065:Very Simple Bases for Computability 4061:Models Similar to Digital Computers 3903:Time-bounded random access machines 3167:CLR ( y ) ; set register y = 0 939:IF = THEN z → IR ELSE + 1 → IR 824:IF = THEN z → IR ELSE + 1 → IR 309:The RA-machine's equivalent of the 502:instruction is one that specifies 207:tone or style may not reflect the 94:it lacks sufficient corresponding 25: 3993:How to Program an Infinite Abacus 3913:Computability & Unsolvability 3679:(C), JAZ ( z ): = 0 then go to I 3622:JEA ( r, z ) ; IF = then I 867:Equals the contents of register r 752:Equals the contents of register r 395:one special register called the " 342:computational complexity analysis 49:This article has multiple issues. 4157:, The MIT PRESS/Elsevier, 1990. 3789:Érdi, Gergő (6 September 2010). 3604:LDA ( i, r ) ; ] → A ; 3556:Schönhage's RAM0 and RAM1 (1980) 558:Definition: The contents of the 547:Definition: The contents of the 217:guide to writing better articles 196: 141: 79: 38: 3997:Introduction to Metamathematics 3991:(1961, received 15 June 1961), 3983:The Art of Computer Programming 3965:Introduction to Metamathematics 3610:STA ( d, r ) ; → r ; 3598:LDA ( d, r ) ; → A ; 655:continue to next instruction }: 644:contents of register r is Zero 57:or discuss these issues on the 4077:(1961) received December 1961 4059:In particular see chapter 11: 4011:Canadian Mathematical Bulletin 4005:(1961, received 15 May 1961), 3901:and Robert A. Reckhow (1973), 3770:Random Access Machine Emulator 3765:Random Access Machine Emulator 3760:Random Access Machine Emulator 3616:STA ( i, r ) ; → ; 3490:into the destination register. 2150:,i, N), JZ ( i, r, z ), HALT } 1088:Example of indirect addressing 350:parallel random-access machine 1: 4137:Storage Modification Machines 4133:Storage Modification Machines 3875:, reprinted pp. 92ff in 2994:primitive recursive functions 1776:The notion of "accumulator A" 1535:The indirect COPY instruction 1355:primitive recursive functions 1320: – provide an 1138:and thus compute any partial 974:primitive recursive functions 4141:Theoretical Computer Science 4114:, Dialectica 12 (1958), 373. 3819:Institute for Advanced Study 3684:; ambiguous in his treatment 2002:), CLRA, INCA, DECA, ADDA (r 1627:and a destination register r 1520:Example: CPY ( 0, 3, 1, 4 ) 1503:Example: CPY ( 0, 3, 0, 4 ) 1304:primitive recursive function 540:Definition: The contents of 3124:φ (q, r0, ..., rlast, y) = 1314:The solution in a nutshell: 459:is a location with both an 313: – with its 4205: 3631:INCA ; + 1 --> A 2970:"Besides a merely being a 1552:A plethora of instructions 894:Action on register(s) "r" 881:goto to next instruction } 779:Action on register(s) "r" 766:goto to next instruction } 668:Action on register(s) "r" 555:of the "target" register. 26: 1096:Example: A treasure hunt. 504:in the instruction itself 411:Introduction to the model 3592:LDA k ; k --> A 2996:, not the full suite of 1239:von Neumann architecture 1231:Universal Turing machine 1083:The "indirect" operation 372:"; each register has an 323:von Neumann architecture 311:universal Turing machine 289:in the general class of 27:Not to be confused with 3576:(initially 0)" (p. 494) 1972:Example: INCA = +1 → A 1878:Incement contents of A 1447:Example: CPY ( indirect 109:more precise citations. 4143:(1979), pp. 36–37 4103:On operator algorithms 4049:Marvin Minsky (1967). 3748:Transdichotomous model 3467:will clear register 3. 3403:will clear register 5. 2998:mu recursive functions 985:mu recursive functions 863:contents of register r 748:contents of register r 624:conditional-expression 364:an infinite number of 338:counter-machine models 4022:Annals of Mathematics 2029:unbounded accumulator 1348:μ-recursive functions 1140:mu recursive function 874:Jump to instruction I 759:Jump to instruction I 648:Jump to instruction I 526:, N ) means directly 438:random-access machine 271:random-access machine 18:Random Access Machine 3855:Abacus Computability 3364:machine, after all. 2203:Action on registers 2027:If our model has an 1928:'s (1973) imaginary 1885:CPY ( d, A, i, r2 ) 1843:CPY ( i, r2, d, A ) 1375:indirection aka the 1063:), JZ (r, z), JE ( r 571:destination register 508:indirect instruction 446:finite state machine 397:instruction register 356:Informal description 304:Harvard architecture 283:model of computation 29:Random-access memory 4147:Peter van Emde Boas 4071:John C. Shepherdson 3705:Finite vs unbounded 3657:(A), INCA: +1 → A 3118:Definition by cases 3079:, y) is true THEN φ 3057:, y) is true THEN φ 3038:, y) is true THEN φ 3018:Definition by cases 2972:finite set of rules 2495:CPY ( d, E, i, N ) 2413:CPY ( d, P, i, N ) 2132:Post–Turing machine 4088:Kaphengst, Heinz, 3727:ω is not an option 3618:indirectly store A 3394:, C is any integer 3368:Examples of models 3321:: CPY ( rlast, φ ) 3120:CPY (i, q, d, φ) = 2788:JZ ( i, N, halt ) 1959:CPY ( d, A, d/i, r 626:in the form of an 332:Together with the 285:that describes an 4189:Register machines 3667:(A), LDAA: ] → A 3662:(N), CPYAN: → N 3652:(Z), CLRA: 0 → A 3606:indirectly load A 2951: 2950: 2706:JZ ( i, N, end ) 2577:JZ ( i, N, end ) 1903: 1902: 1392:Turing equivalent 1136:Turing equivalent 957: 956: 842: 841: 731: 730: 455:By definition: A 432:Formal definition 415:The concept of a 385:instruction table 291:register machines 263: 262: 255: 245: 244: 237: 211:used on Knowledge 209:encyclopedic tone 190: 189: 182: 135: 134: 127: 72: 16:(Redirected from 4196: 4129:Arnold Schönhage 4096:(1959), 366-379. 4063:and chapter 14: 4058: 4056: 4045: 3923:Abraham Robinson 3869:John von Neumann 3865:Herman Goldstine 3809:Register machine 3801: 3800: 3798: 3797: 3786: 3695: 3683: 3674: 3668: 3663: 3658: 3653: 3632: 3627: 3617: 3612:directly store A 3611: 3605: 3599: 3593: 3549:to "the output." 3544: 3531: 3521: 3503: 3485: 3481:) ; ] → r 3466: 3457: 3453:) ; - → r 3434: 3425: 3421:) ; + → r 3402: 3393: 2158: 1982:Example: MULA (r 1975:Example: ADDA (r 1909:INC ( A ) = INCA 1796: 1432:pointer-register 1188:: The so-called 885: 770: 659: 615:register machine 569:Definition: The 560:pointer register 549:pointer register 518:Example: COPY ( 366:memory locations 287:abstract machine 267:computer science 258: 251: 240: 233: 229: 226: 220: 219:for suggestions. 215:See Knowledge's 200: 199: 192: 185: 178: 174: 171: 165: 145: 144: 137: 130: 123: 119: 116: 110: 105:this article by 96:inline citations 83: 82: 75: 64: 42: 41: 34: 21: 4204: 4203: 4199: 4198: 4197: 4195: 4194: 4193: 4179: 4178: 4151:Jan van Leeuwen 4048: 4034:10.2307/1970290 4016: 3899:Stephen A. Cook 3847:Richard Jeffrey 3843:John P. Burgess 3805: 3804: 3795: 3793: 3788: 3787: 3783: 3778: 3756: 3739: 3707: 3702: 3693: 3682: 3678: 3673:(S), STAN: → 3672: 3666: 3661: 3656: 3651: 3630: 3625: 3621: 3615: 3609: 3603: 3600:directly load A 3597: 3591: 3562:pointer machine 3558: 3548: 3542: 3538: 3535: 3529: 3525: 3519: 3515: 3511: 3507: 3501: 3497: 3493: 3489: 3484: 3480: 3476: 3472: 3465:SUB ( 3, 3, 3 ) 3464: 3456: 3452: 3448: 3444: 3440: 3433:ADD ( A, A, A ) 3432: 3424: 3420: 3416: 3412: 3408: 3400: 3392: 3388: 3384: 3375: 3370: 3359: 3123: 3107: 3093: 3082: 3074: 3071:case_last: IF Q 3060: 3052: 3041: 3033: 2956: 2703:jump_if_blank: 2149: 2128: 2120: 2116: 2104: 2100: 2088: 2084: 2025: 2017: 2013: 2009: 2005: 2001: 1997: 1985: 1978: 1962: 1958: 1954: 1947: 1943: 1939: 1778: 1766: 1762: 1758: 1751: 1747: 1743: 1736: 1732: 1728: 1721: 1717: 1713: 1706: 1702: 1698: 1691: 1687: 1683: 1676: 1672: 1668: 1661: 1657: 1653: 1640: 1630: 1626: 1622: 1614: 1610: 1606: 1602: 1595: 1591: 1587: 1580: 1576: 1572: 1565: 1561: 1546: 1542: 1537: 1474: 1462: 1458: 1454: 1450: 1409: 1343: 1298:He offers us a 1276:Gödel numbering 1160: 1156: 1131: 1090: 1085: 1070: 1066: 1062: 1058: 1008: 962: 933:JE (r1, r2, z) 877: 870: 866: 858: 854: 818:JE (r1, r2, z) 762: 755: 751: 651: 580: 564:target register 542:source register 442:counter machine 434: 421:counter machine 413: 389:instruction set 358: 346:pointer machine 295:counter machine 259: 248: 247: 246: 241: 230: 224: 221: 214: 205:This article's 201: 197: 186: 175: 169: 166: 158:help improve it 155: 146: 142: 131: 120: 114: 111: 101:Please help to 100: 84: 80: 43: 39: 32: 23: 22: 15: 12: 11: 5: 4202: 4200: 4192: 4191: 4181: 4180: 4177: 4176: 4166: 4144: 4125: 4124: 4123: 4122: 4115: 4106: 4097: 4083: 4082: 4068: 4046: 4014: 4000: 3989:Joachim Lambek 3986: 3976: 3961:Stephen Kleene 3958: 3943:Jeffrey Ullman 3936: 3930: 3916: 3906: 3896: 3858: 3836: 3835: 3834: 3803: 3802: 3780: 3779: 3777: 3774: 3773: 3772: 3767: 3762: 3755: 3754:External links 3752: 3751: 3750: 3745: 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2595: 2593: 2591: 2589: 2586: 2583: 2580: 2578: 2575: 2572: 2568: 2567: 2565: 2563: 2561: 2559: 2557: 2555: 2553: 2551: 2549: 2547: 2545: 2543: 2541: 2539: 2537: 2535: 2532: 2531: 2528: 2525: 2523: 2521: 2518: 2515: 2513: 2511: 2509: 2507: 2504: 2501: 2498: 2496: 2493: 2490: 2486: 2485: 2483: 2481: 2479: 2477: 2475: 2473: 2471: 2469: 2467: 2465: 2463: 2461: 2459: 2457: 2455: 2453: 2450: 2449: 2446: 2443: 2441: 2439: 2436: 2433: 2431: 2429: 2427: 2425: 2422: 2419: 2416: 2414: 2411: 2408: 2404: 2403: 2401: 2399: 2397: 2395: 2393: 2391: 2389: 2387: 2385: 2383: 2381: 2379: 2377: 2375: 2373: 2371: 2368: 2367: 2364: 2361: 2359: 2357: 2354: 2351: 2349: 2347: 2345: 2343: 2340: 2337: 2334: 2332: 2329: 2326: 2322: 2321: 2319: 2317: 2315: 2313: 2311: 2309: 2307: 2305: 2303: 2301: 2299: 2297: 2295: 2293: 2291: 2289: 2286: 2285: 2283: 2281: 2279: 2277: 2274: 2271: 2269: 2267: 2265: 2263: 2260: 2257: 2254: 2252: 2250: 2247: 2244: 2243: 2241: 2239: 2237: 2235: 2233: 2231: 2229: 2227: 2225: 2223: 2221: 2219: 2217: 2215: 2213: 2211: 2208: 2207: 2204: 2201: 2198: 2195: 2192: 2189: 2186: 2183: 2180: 2178: 2175: 2172: 2169: 2167: 2165: 2162: 2152: 2151: 2147: 2127: 2124: 2123: 2122: 2118: 2114: 2107: 2106: 2102: 2098: 2091: 2090: 2086: 2082: 2055: 2054: 2051: 2024: 2021: 2020: 2019: 2015: 2011: 2007: 2003: 1999: 1998:), STA (i/d, r 1995: 1988: 1987: 1983: 1980: 1976: 1973: 1965: 1964: 1960: 1956: 1952: 1949: 1945: 1941: 1937: 1922: 1921: 1918: 1911: 1910: 1901: 1900: 1897: 1894: 1891: 1888: 1886: 1883: 1880: 1879: 1876: 1873: 1870: 1867: 1865: 1862: 1859: 1858: 1855: 1852: 1849: 1846: 1844: 1841: 1837: 1836: 1834: 1831: 1828: 1825: 1823: 1821: 1818: 1817: 1814: 1811: 1808: 1805: 1803: 1800: 1790: 1789: 1777: 1774: 1769: 1768: 1764: 1760: 1756: 1753: 1749: 1745: 1741: 1738: 1734: 1730: 1726: 1723: 1719: 1715: 1711: 1708: 1704: 1700: 1696: 1693: 1689: 1685: 1681: 1678: 1674: 1670: 1666: 1663: 1659: 1655: 1651: 1644: 1643: 1642: 1641: 1638: 1628: 1624: 1620: 1617: 1616: 1612: 1608: 1604: 1600: 1597: 1593: 1589: 1585: 1582: 1578: 1574: 1570: 1567: 1563: 1559: 1544: 1540: 1536: 1533: 1532: 1531: 1530: 1529: 1528: 1527: 1524: 1515: 1514: 1513: 1512: 1511: 1510: 1507: 1498: 1497: 1496: 1495: 1494: 1493: 1490: 1481: 1480: 1477: 1476: 1475: 1472: 1465: 1464: 1460: 1456: 1452: 1448: 1430:, or (ii) the 1428:explicit label 1408: 1405: 1388: 1387: 1369: 1368: 1342: 1339: 1296: 1295: 1260: 1259: 1243: 1242: 1219: 1218: 1217: 1216: 1214:state machine. 1199: 1198: 1178: 1177: 1176: 1175: 1164: 1163: 1158: 1154: 1130: 1127: 1112: 1111: 1110: 1109: 1106: 1103: 1097: 1089: 1086: 1084: 1081: 1073: 1072: 1068: 1064: 1060: 1056: 1046: 1045: 1044: 1043: 1036: 1035: 1034: 1033: 1026: 1020: 1019: 1009: 1006: 1000: 961: 958: 955: 954: 951: 948: 945: 941: 940: 937: 934: 931: 930:Jump if Equal 927: 926: 923: 920: 917: 913: 912: 909: 906: 905:COPY (r1, r2) 903: 899: 898: 895: 892: 889: 883: 882: 875: 868: 864: 856: 852: 840: 839: 836: 833: 830: 826: 825: 822: 819: 816: 815:Jump if Equal 812: 811: 808: 805: 802: 798: 797: 794: 791: 788: 784: 783: 780: 777: 774: 768: 767: 760: 753: 749: 729: 728: 725: 722: 719: 715: 714: 711: 708: 705: 701: 700: 697: 694: 691: 687: 686: 683: 680: 677: 673: 672: 669: 666: 663: 657: 656: 649: 611: 610: 604: 603: 594:Turing machine 588: 587: 579: 576: 562:points to the 538: 537: 536: 535: 498:Definition: A 496: 495: 494: 493: 486: 485: 484: 483: 476: 475: 472: 433: 430: 412: 409: 401: 400: 393: 381: 378:natural number 357: 354: 334:Turing machine 261: 260: 243: 242: 204: 202: 195: 188: 187: 149: 147: 140: 133: 132: 87: 85: 78: 73: 47: 46: 44: 37: 24: 14: 13: 10: 9: 6: 4: 3: 2: 4201: 4190: 4187: 4186: 4184: 4174: 4170: 4167: 4164: 4163:0-444-88071-2 4160: 4156: 4152: 4148: 4145: 4142: 4138: 4134: 4130: 4127: 4126: 4120: 4117:Hermes, Hans 4116: 4113: 4110: 4107: 4104: 4101: 4100:Ershov, A. P. 4098: 4095: 4091: 4087: 4086: 4085: 4084: 4080: 4076: 4075:H. E. Sturgis 4072: 4069: 4066: 4062: 4055: 4054: 4047: 4043: 4039: 4035: 4031: 4027: 4023: 4019: 4018:Marvin Minsky 4015: 4012: 4008: 4004: 4001: 3998: 3994: 3990: 3987: 3984: 3980: 3977: 3974: 3973:0-7204-2103-9 3970: 3966: 3962: 3959: 3956: 3955:0-201-02988-X 3952: 3948: 3944: 3940: 3939:John Hopcroft 3937: 3934: 3931: 3928: 3924: 3920: 3917: 3914: 3910: 3907: 3904: 3900: 3897: 3894: 3893:0-07-004357-4 3890: 3886: 3882: 3878: 3874: 3870: 3866: 3862: 3859: 3856: 3852: 3848: 3844: 3840: 3839:George Boolos 3837: 3832: 3831:0-07-004357-4 3828: 3824: 3820: 3816: 3815: 3814: 3813: 3812: 3810: 3792: 3785: 3782: 3775: 3771: 3768: 3766: 3763: 3761: 3758: 3757: 3753: 3749: 3746: 3744: 3741: 3740: 3736: 3734: 3728: 3724: 3720: 3716: 3715: 3714: 3712: 3704: 3699: 3697: 3694:LDAA ( ] → ) 3677: 3671: 3665: 3660: 3655: 3650: 3649: 3648: 3647: 3646: 3645: 3644: 3642: 3629: 3626:else continue 3620: 3614: 3608: 3602: 3596: 3590: 3589: 3588: 3587: 3586: 3585: 3584: 3582: 3575: 3571: 3567: 3566: 3565: 3563: 3537: 3524: 3514: 3492: 3471: 3470: 3462: 3461: 3439: 3438: 3430: 3429: 3407: 3406: 3401:LOAD ( 0, 5 ) 3398: 3397: 3383: 3382: 3381: 3380: 3379: 3372: 3367: 3365: 3363: 3355: 3345: 3342: 3339: 3336: 3335: 3327: 3323: 3320: 3317: 3316: 3315: 3314: 3310: 3306: 3303: 3299: 3296: 3295: 3294: 3293: 3289: 3286: 3285: 3284: 3283: 3282: 3274: 3271: 3270: 3262: 3258: 3256:CPY ( rn, φ ) 3255: 3252: 3251: 3250: 3249: 3245: 3241: 3238: 3234: 3231: 3230: 3229: 3228: 3224: 3221: 3220: 3219: 3218: 3217: 3209: 3206: 3205: 3197: 3193: 3191:CPY ( r0, φ ) 3190: 3187: 3186: 3185: 3184: 3180: 3176: 3173: 3169: 3166: 3165: 3164: 3163: 3159: 3156: 3155: 3154: 3153: 3152: 3144: 3141: 3138: 3135: 3132: 3129: 3126: 3125: 3119: 3116: 3115: 3114: 3110: 3097: 3090:default: do φ 3089: 3086: 3078: 3070: 3067: 3064: 3056: 3048: 3045: 3037: 3029: 3028: 3027: 3026: 3025: 3023: 3019: 3015: 3013: 3007: 3001: 2999: 2995: 2989: 2988: 2987: 2981: 2978: 2977: 2973: 2969: 2968: 2967: 2965: 2961: 2953: 2946: 2943: 2941: 2939: 2936: 2933: 2931: 2929: 2927: 2925: 2922: 2919: 2916: 2914: 2911: 2908: 2906: 2905: 2902: 2900: 2898: 2896: 2894: 2892: 2890: 2888: 2886: 2884: 2882: 2880: 2878: 2876: 2874: 2872: 2870: 2869: 2866: 2864: 2862: 2860: 2857: 2854: 2852: 2850: 2848: 2846: 2843: 2840: 2837: 2835: 2832: 2829: 2827: 2826: 2822: 2819: 2817: 2815: 2812: 2809: 2807: 2805: 2803: 2801: 2798: 2795: 2792: 2790: 2787: 2784: 2781: 2780: 2777: 2775: 2773: 2771: 2769: 2767: 2765: 2763: 2761: 2759: 2757: 2755: 2753: 2751: 2749: 2747: 2745: 2744: 2740: 2737: 2735: 2733: 2730: 2727: 2725: 2723: 2721: 2719: 2716: 2713: 2710: 2708: 2705: 2702: 2699: 2698: 2695: 2693: 2691: 2689: 2687: 2685: 2683: 2681: 2679: 2677: 2675: 2673: 2671: 2669: 2667: 2665: 2663: 2662: 2659: 2656: 2654: 2652: 2649: 2646: 2644: 2642: 2640: 2638: 2635: 2632: 2629: 2627: 2624: 2622: 2620: 2619: 2615: 2611: 2608: 2606: 2604: 2601: 2598: 2596: 2594: 2592: 2590: 2587: 2584: 2581: 2579: 2576: 2573: 2570: 2569: 2566: 2564: 2562: 2560: 2558: 2556: 2554: 2552: 2550: 2548: 2546: 2544: 2542: 2540: 2538: 2536: 2534: 2533: 2529: 2526: 2524: 2522: 2519: 2516: 2514: 2512: 2510: 2508: 2505: 2502: 2499: 2497: 2494: 2491: 2488: 2487: 2484: 2482: 2480: 2478: 2476: 2474: 2472: 2470: 2468: 2466: 2464: 2462: 2460: 2458: 2456: 2454: 2452: 2451: 2447: 2444: 2442: 2440: 2437: 2434: 2432: 2430: 2428: 2426: 2423: 2420: 2417: 2415: 2412: 2409: 2406: 2405: 2402: 2400: 2398: 2396: 2394: 2392: 2390: 2388: 2386: 2384: 2382: 2380: 2378: 2376: 2374: 2372: 2370: 2369: 2365: 2362: 2360: 2358: 2355: 2352: 2350: 2348: 2346: 2344: 2341: 2338: 2335: 2333: 2330: 2327: 2324: 2323: 2320: 2318: 2316: 2314: 2312: 2310: 2308: 2306: 2304: 2302: 2300: 2298: 2296: 2294: 2292: 2290: 2288: 2287: 2284: 2282: 2280: 2278: 2275: 2272: 2270: 2268: 2266: 2264: 2261: 2258: 2255: 2253: 2251: 2248: 2246: 2245: 2242: 2240: 2238: 2236: 2234: 2232: 2230: 2228: 2226: 2224: 2222: 2220: 2218: 2216: 2214: 2212: 2210: 2209: 2205: 2202: 2199: 2196: 2193: 2190: 2187: 2184: 2181: 2179: 2176: 2173: 2170: 2168: 2166: 2163: 2160: 2159: 2156: 2145: 2144: 2143: 2139: 2135: 2133: 2125: 2112: 2111: 2110: 2096: 2095: 2094: 2080: 2079: 2078: 2076: 2071: 2070: 2066: 2063: 2062: 2058: 2052: 2049: 2048: 2047: 2043: 2039: 2036: 2034: 2030: 2022: 1994:{ LDA (i/d, r 1993: 1992: 1991: 1981: 1974: 1971: 1970: 1969: 1950: 1935: 1934: 1933: 1931: 1927: 1919: 1916: 1915: 1914: 1908: 1907: 1906: 1898: 1895: 1892: 1889: 1887: 1884: 1882: 1881: 1877: 1874: 1871: 1868: 1866: 1863: 1861: 1860: 1856: 1853: 1850: 1847: 1845: 1842: 1840:INCi ( r2 ): 1839: 1838: 1835: 1832: 1829: 1826: 1824: 1822: 1820: 1819: 1815: 1812: 1809: 1806: 1804: 1801: 1798: 1797: 1794: 1787: 1783: 1782: 1781: 1775: 1773: 1754: 1739: 1724: 1709: 1694: 1679: 1664: 1649: 1648: 1647: 1636: 1635: 1634: 1633: 1632: 1598: 1583: 1568: 1557: 1556: 1555: 1553: 1549: 1534: 1525: 1522: 1521: 1519: 1518: 1517: 1516: 1508: 1505: 1504: 1502: 1501: 1500: 1499: 1491: 1488: 1487: 1485: 1484: 1483: 1482: 1478: 1470: 1469: 1467: 1466: 1446: 1445: 1444: 1441: 1437: 1433: 1429: 1425: 1421: 1416: 1414: 1406: 1404: 1402: 1397: 1393: 1386: 1382: 1381: 1380: 1378: 1374: 1367: 1364: 1363: 1362: 1360: 1356: 1351: 1349: 1340: 1338: 1336: 1331: 1327: 1323: 1319: 1315: 1311: 1307: 1305: 1301: 1293: 1292: 1291: 1287: 1285: 1279: 1277: 1273: 1269: 1265: 1257: 1253: 1249: 1245: 1244: 1240: 1236: 1232: 1228: 1226: 1221: 1220: 1215: 1211: 1207: 1203: 1202: 1201: 1200: 1195: 1191: 1187: 1185: 1180: 1179: 1174: 1171: 1170: 1169: 1166: 1165: 1162: 1151: 1145: 1144: 1143: 1141: 1137: 1128: 1126: 1124: 1120: 1116: 1107: 1104: 1101: 1100: 1098: 1095: 1094: 1093: 1087: 1082: 1080: 1078: 1071:, z ), J(z) } 1054: 1053: 1052: 1049: 1041: 1038: 1037: 1031: 1027: 1024: 1023: 1022: 1021: 1017: 1013: 1010: 1004: 1003: 1001: 999: 994: 991: 990: 989: 986: 983: 979: 975: 971: 966: 959: 952: 949: 946: 943: 942: 938: 935: 932: 929: 928: 924: 921: 918: 915: 914: 910: 907: 904: 901: 900: 896: 893: 890: 887: 886: 880: 873: 862: 855:to register r 850: 849: 848: 846: 837: 834: 831: 828: 827: 823: 820: 817: 814: 813: 809: 806: 803: 800: 799: 795: 792: 789: 786: 785: 781: 778: 775: 772: 771: 765: 758: 747: 743: 742: 741: 739: 735: 726: 723: 720: 717: 716: 712: 709: 706: 704:Jump if Zero 703: 702: 698: 695: 692: 689: 688: 684: 681: 678: 675: 674: 670: 667: 664: 661: 660: 654: 647: 643: 639: 638: 637: 635: 631: 629: 625: 620: 616: 609: 606: 605: 602: 597: 595: 590: 589: 586: 582: 581: 577: 575: 572: 567: 565: 561: 556: 554: 550: 545: 543: 533: 529: 525: 521: 517: 516: 514: 513: 512: 509: 505: 501: 490: 489: 488: 487: 480: 479: 478: 477: 473: 470: 469: 468: 466: 462: 458: 453: 451: 447: 443: 439: 431: 429: 427: 422: 418: 417:random-access 410: 408: 406: 398: 394: 390: 386: 382: 379: 375: 371: 367: 363: 362: 361: 355: 353: 351: 347: 343: 339: 335: 330: 328: 324: 320: 316: 312: 307: 305: 300: 296: 292: 288: 284: 280: 276: 272: 268: 257: 254: 239: 236: 228: 225:December 2017 218: 212: 210: 203: 194: 193: 184: 181: 173: 170:December 2017 163: 159: 153: 150:This article 148: 139: 138: 129: 126: 118: 115:December 2017 108: 104: 98: 97: 91: 86: 77: 76: 71: 69: 62: 61: 56: 55: 50: 45: 36: 35: 30: 19: 4172: 4154: 4140: 4136: 4132: 4118: 4111: 4109:Péter, Rózsa 4102: 4093: 4089: 4078: 4064: 4060: 4052: 4025: 4021: 4006: 4003:Z. A. Melzak 3996: 3992: 3982: 3979:Donald Knuth 3964: 3946: 3933:J. Hartmanis 3926: 3919:Calvin Elgot 3912: 3909:Martin Davis 3902: 3884: 3881:Allen Newell 3872: 3861:Arthur Burks 3854: 3850: 3822: 3806: 3794:. Retrieved 3784: 3732: 3726: 3722: 3718: 3710: 3708: 3691: 3640: 3639: 3580: 3579: 3573: 3569: 3559: 3502:) ; → 3376: 3361: 3351: 3343: 3337: 3325: 3318: 3308: 3301: 3287: 3280: 3272: 3260: 3253: 3243: 3236: 3222: 3215: 3207: 3195: 3188: 3178: 3171: 3157: 3150: 3117: 3111: 3103: 3095: 3084: 3076: 3062: 3054: 3049:case_1: IF Q 3043: 3035: 3030:case_0: IF Q 3021: 3017: 3016: 3011: 3008: 3004: 2990: 2984: 2979: 2971: 2963: 2959: 2957: 2782:J1 ( halt ) 2700:J0 ( halt ) 2153: 2140: 2136: 2129: 2108: 2092: 2074: 2072: 2068: 2067: 2064: 2060: 2059: 2056: 2044: 2040: 2037: 2032: 2028: 2026: 1989: 1966: 1951:STA ( d/i, r 1944:CPY ( d/i, r 1936:LDA ( d/i, r 1923: 1912: 1904: 1816:Description 1802:Instruction 1793:register A: 1791: 1785: 1779: 1770: 1646:will yield: 1645: 1618: 1551: 1550: 1538: 1471:i* + (1-i)*r 1439: 1435: 1431: 1427: 1423: 1419: 1417: 1412: 1410: 1400: 1396:Gödel number 1389: 1383: 1372: 1370: 1365: 1358: 1352: 1344: 1334: 1325: 1321: 1317: 1313: 1312: 1308: 1299: 1297: 1288: 1283: 1280: 1271: 1267: 1263: 1261: 1255: 1251: 1247: 1224: 1222: 1213: 1209: 1204: 1193: 1189: 1183: 1181: 1172: 1167: 1150:Gödel number 1146: 1132: 1122: 1118: 1113: 1091: 1076: 1074: 1050: 1047: 1039: 1029: 1015: 1011: 996: 992: 981: 977: 969: 967: 963: 888:Instruction 878: 871: 860: 845:Base model 3 844: 843: 773:Instruction 763: 756: 745: 738:Peano axioms 734:Base model 2 733: 732: 707:JZ ( r, z ) 662:Instruction 652: 645: 641: 634:Base model 1 633: 632: 628:IF-THEN-ELSE 627: 623: 618: 612: 607: 599: 591: 583: 570: 568: 563: 559: 557: 552: 548: 546: 541: 539: 531: 527: 523: 519: 507: 503: 499: 497: 464: 460: 456: 454: 449: 437: 435: 414: 402: 396: 384: 369: 359: 331: 308: 278: 274: 270: 264: 249: 231: 222: 206: 176: 167: 151: 121: 112: 93: 65: 58: 52: 51:Please help 48: 3877:Gordon Bell 3494:COPY ( d, r 3473:COPY ( i, r 3385:LOAD ( C, r 3354:mu operator 3300:JE ( q, y, 3235:JE ( q, y, 3170:JE ( q, y, 2833:. . . etc. 2820:N =r3] → A 1986:) = * → A 1979:) = + → A 1786:Accumulator 1461:destination 1457:destination 1318:indirection 1115:Indirection 376:which is a 299:main memory 107:introducing 3796:2024-02-07 3776:References 3641:RAM0 model 3581:RAM1 model 3516:JNZ ( r, I 3273:case__n+1: 2625:DEC ( N ) 2331:INC ( N ) 2014:), DIVA (r 2010:), MULA (r 2006:), SUBA (r 1864:INC ( A ) 1755:ADD ( i, r 1740:ADD ( d, r 1725:ADD ( i, r 1710:ADD ( d, r 1695:ADD ( i, r 1680:ADD ( d, r 1665:ADD ( i, r 1650:ADD ( d, r 1377:μ operator 1359:explicitly 1330:μ operator 1252:explicitly 1248:explicitly 1223:Unbounded 1184:capacities 1182:Unbounded 919:INC ( r ) 916:INCrement 804:INC ( r ) 801:INCrement 790:CLR ( r ) 693:DEC ( r ) 690:DECrement 679:INC ( r ) 676:INCrement 352:" models. 279:RA-machine 90:references 54:improve it 3700:Footnotes 3539:PRINT ( r 3463:Example: 3431:Example: 3399:Example: 3297:INC ( y ) 3288:case_last 3232:INC ( y ) 3087:, y) ELSE 3065:, y) ELSE 3046:, y) ELSE 2527:=0 → =r4 2445:=1 → =r4 2161:Mnemonic 1813:r378,426 1599:CPY (i, r 1584:CPY (d, r 1569:CPY (i, r 1558:CPY (d, r 1413:unbounded 1373:unbounded 1326:unbounded 1322:unbounded 1237:with the 1014:: JZ (r, 1005:CLR (r) = 925:+ 1 → IR 911:+ 1 → IR 891:Mnemonic 810:+ 1 → IR 796:+ 1 → IR 776:Mnemonic 699:+ 1 → IR 685:+ 1 → IR 665:Mnemonic 619:unbounded 405:Brainfuck 370:registers 60:talk page 4183:Category 4171:(1957), 4169:Hao Wang 4131:(1980), 3981:(1968), 3963:(1952), 3945:(1979). 3925:(1964), 3911:(1958), 3883:(1971), 3871:(1946), 3849:(2002), 3743:Real RAM 3737:See also 3543:) ; 3530:) ; 3526:READ ( r 3520:) ; 3244:case_n+1 2947:+1 → IR 2530:+1 → IR 2448:+1 → IR 2366:+1 → IR 2117:), JZ (I 2101:), JZ (I 2085:), JZ (I 2018:), etc.) 1948:, d, A ) 1893:378,426 1872:378,426 1851:378,426 1830:378,426 1455:, direct 1436:contents 1235:computer 1134:that is 922:+ 1 → r 807:+ 1 → r 696:- 1 → r 682:+ 1 → r 598:distance 457:register 450:contents 426:computer 368:called " 327:computer 4042:1970290 3564:model: 3441:SUB ( r 3409:ADD ( r 3208:case_1: 3092:default 2962:, i.e. 2960:bounded 2657:-1 → N 2492:erase: 2410:print: 2363:+1 → N 2328:right: 2249:start: 2164:label: 2031:can we 1300:bounded 1284:program 1268:program 1262:So how 1225:numbers 1119:pointer 1028:JZ (0, 1025:DEC (r) 982:partial 972:of the 553:address 551:is the 465:content 461:address 374:address 315:program 281:) is a 156:Please 103:improve 4161: 4153:, ed. 4040: 3971: 3953: 3891: 3829: 3723:finite 3711:finite 3498:, i, r 3477:, d, r 3362:finite 3338:woops: 3319:_φlast 3302:_φlast 3223:case_n 3179:case_1 3158:case_0 3024:, y): 2964:finite 2909:halt: 2574:left: 2121:), H } 2105:), H } 2089:), H } 1827:. . . 1799:Label 1763:, i, r 1759:, i, r 1748:, i, r 1744:, i, r 1733:, i, r 1729:, d, r 1718:, i, r 1714:, d, r 1703:, d, r 1699:, i, r 1688:, d, r 1684:, i, r 1673:, d, r 1669:, d, r 1658:, d, r 1654:, d, r 1637:+ → r 1603:, i, r 1588:, i, r 1573:, d, r 1562:, d, r 1543:and P' 1453:source 1449:source 1434:whose 1390:To be 1256:finite 1212:finite 1208:finite 1190:finite 1123:target 1042:: etc. 995:blocks 793:0 → r 787:CLeaR 500:direct 92:, but 4139:, in 4038:JSTOR 3344:exit: 3309:woops 2944:none 2738:none 2609:none 2200:etc. 2033:bound 1926:Knuth 1424:where 1420:where 1007:equiv 978:total 953:→ IR 950:none 944:Halt 936:none 908:→ r2 902:COPY 838:→ IR 835:none 829:Halt 821:none 727:→ IR 724:none 718:Halt 710:none 522:, A, 4159:ISBN 4073:and 3969:ISBN 3951:ISBN 3921:and 3889:ISBN 3879:and 3827:ISBN 3719:does 3346:etc. 3326:exit 3324:J ( 3307:J ( 3275:etc. 3261:exit 3259:J ( 3254:_φn: 3242:J ( 3210:etc. 3196:exit 3194:J ( 3189:_φ0: 3177:J ( 3098:, y) 3081:last 3073:last 2830:end 1440:also 1411:For 1194:very 1040:exit 1030:loop 1016:exit 1012:loop 980:and 879:ELSE 872:then 764:ELSE 757:THEN 653:ELSE 646:THEN 613:The 383:the 336:and 4030:doi 3449:, r 3445:, r 3417:, r 3413:, r 3237:_φn 3172:_φ0 3122:def 3020:φ ( 2612:IF 2197:r5 2194:r4 2191:r3 2188:r2 2185:r1 2182:r0 1957:def 1955:) = 1942:def 1940:) = 1930:MIX 1896:18 1890:18 1875:17 1869:18 1854:17 1848:17 1833:17 1810:r2 1761:sp2 1757:sp1 1746:sp2 1727:sp1 1701:sp2 1697:sp1 1686:sp2 1667:sp1 1459:, r 1451:, r 1385:go? 1335:its 1272:all 1077:Not 1067:, r 1059:, r 970:any 740:): 392:set 306:). 277:or 275:RAM 265:In 160:to 4185:: 4036:. 4026:74 4024:. 4009:, 3941:, 3867:, 3863:, 3845:, 3841:, 3833:}. 3811:. 3696:. 3447:s2 3443:s1 3415:s2 3411:s1 2966:: 2937:0 2934:1 2923:3 2920:1 2917:0 2912:H 2858:0 2855:1 2844:3 2841:1 2838:0 2813:0 2810:1 2799:3 2796:1 2793:0 2731:0 2728:1 2717:3 2714:1 2711:0 2650:0 2647:1 2636:3 2633:1 2630:0 2602:0 2599:1 2588:4 2585:1 2582:0 2571:L 2520:0 2517:1 2506:4 2503:1 2500:0 2489:E 2438:1 2435:1 2424:4 2421:1 2418:0 2407:P 2356:0 2353:1 2342:4 2339:1 2336:0 2325:R 2276:0 2273:1 2262:3 2259:1 2256:0 2177:N 2174:P 2171:E 1807:A 1765:dp 1750:dp 1742:s1 1735:dp 1731:s2 1720:dp 1716:s2 1712:s1 1682:s1 1671:s2 1656:s2 1652:s1 1625:s2 1621:s1 1613:dp 1609:sp 1605:dp 1601:sp 1594:dp 1590:dp 1579:sp 1571:sp 1379:. 1264:do 1142:: 947:H 861:IF 859:, 832:H 746:IF 721:H 642:IF 482:3. 436:A 407:. 329:. 269:, 63:. 4094:5 4044:. 4032:: 3999:. 3975:. 3895:. 3799:. 3729:. 3681:z 3624:z 3574:n 3570:z 3547:s 3541:s 3534:d 3528:d 3518:z 3512:. 3510:p 3506:s 3500:p 3496:s 3488:p 3483:d 3479:d 3475:p 3455:d 3451:d 3423:d 3419:d 3391:d 3387:d 3358:2 3328:) 3311:) 3304:) 3290:: 3263:) 3246:) 3239:) 3225:: 3198:) 3181:) 3174:) 3160:: 3106:n 3096:x 3094:( 3085:x 3083:( 3077:x 3075:( 3063:x 3061:( 3059:1 3055:x 3053:( 3051:1 3044:x 3042:( 3040:0 3036:x 3034:( 3032:0 3022:x 3012:x 3000:. 2614:N 2148:s 2119:z 2115:z 2103:z 2099:z 2087:z 2083:z 2075:r 2016:s 2012:s 2008:s 2004:s 2000:d 1996:s 1984:s 1977:s 1963:) 1961:d 1953:d 1946:s 1938:s 1767:) 1752:) 1737:) 1722:) 1707:) 1705:d 1692:) 1690:d 1677:) 1675:d 1662:) 1660:d 1639:d 1629:d 1623:r 1615:) 1596:. 1586:s 1581:. 1575:d 1564:d 1560:s 1545:0 1541:0 1473:s 1463:) 1241:. 1159:0 1155:0 1153:P 1069:k 1065:j 1061:d 1057:s 1032:) 1018:) 876:z 869:k 865:j 857:k 853:j 761:z 754:k 750:j 650:z 532:i 528:d 524:i 520:d 273:( 256:) 250:( 238:) 232:( 227:) 223:( 213:. 183:) 177:( 172:) 168:( 154:. 128:) 122:( 117:) 113:( 99:. 70:) 66:( 31:. 20:)

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