[The following note and paper by Ershov appeared in the Communications
of the ACM 1, 8 (August 1958), pp. 3-6.

Ershov's paper clearly shows at least the following insights:
1.  Hash tables can provide O(n) v. O(n^2) behavior.
2.  Hash 'consing'.
3.  Hash techniques for 'economising', or 'compressing'.
4.  Labelling technique for minimizing the number of intermediate
result registers.

The figures referred to in the text are apparently in the original
Russian paper, but do not appear in the 1958 CACM translation.

-- Henry Baker.]

-----

TECHNIQUES DEPARTMENT

Editor's Note:

A revised summary of the material in the A.C.M. library is being
printed to encourage further contributions of missing material.

As the translation of the Russian paper appearing in this section does
not give any easy clues about its subject material or intent, a brief
description is attempted here.  It is nice to see that
English-speaking peoples are not the only experts at obfuscation.

This paper deals with the production of 3-address machine language
instructions (for the BESM computer) from algebraic statements of the
type found in Fortran, Unicode and other languages.  Assembly program
characteristics are included.  An algorithm is given for creating
rough machine language instructions in pseudo-form and then operating
upon these to alter them to the most efficient form.  For at least the
domain of a single formula, a check is made for duplicate strings of
any length.

The most pertinent point is that the algorithm itself operates more
efficiently than their previous methods.  Thus the processor takes an
amount of time to produce an efficient object program which is
linearly proportional to the number of instructions in the program,
NOT to the square of the number as previously.  Obviously, when the
interaction of instructions over the entire program is considered (as
in the Fortran processor) or when the formulae are exceptionally long,
this method becomes more valuable as the programs grow larger.

-----

ON PROGRAMMING OF ARITHMETIC OPERATIONS

A. P. Ershov

Doklady, AN USSR, vol. 118, No. 3, 1958, pp. 427-430

Translated by Morris D. Friedman, Lincoln Laboratory*

The concepts used without explanation are taken from [1].

#1.  Programming algorithms of arithmetic operations (AO) consist of
three parts.

The first part A1 successively generates the commands of the AO
program.

The second part A2 generates a conventional number (CN)* for each
command constructed, which denotes the result of the programmed
operation, and replaces it in the formula of the programmed
expression.  Identification of the entries of similar expressions in
the AO formula is made during the A2 operation so that similar
expressions will not be programmed repeatedly (economy of command).

The third part A3 replaces the CN in the constructed program, which
denotes the intermediate results, by a code of operating registers
(OR).  New principles for constructing the algorithms of A2 and A3 are
proposed herein.

#2.  Assumptions and Definitions.  Programming of arithmetic
operations is carried out on a three address computer.  The left side
of the AO formula is the superposition of binary and unary operations,
each of which is realized by a single command.  Each command has one
bit sigma, which neither enters into the operation code nor into the
address part of the command.  The A1 algorithm generates the AO
commands in the form

-----------------------------
| a | b | c | sigma | theta |
-----------------------------

where theta is the operation code, a and b are CN which denote
components and c is a CN which denotes the result (if theta is a unary
operation, then b=0; c/=0 only for a resultant command, i.e., the
concluding command of the calculation of the AO formula; the content
of the digit sigma is zero at first).  A _Block of Preparatory
Operations_ (BPO) is a group of n registers with the addresses
L+1,...,L+n in which are located the AO commands which are generated
by the algorithm A1.  A _Block of Resultant Commands_ (BRC) is a group
of registers in which are located all the resultant AO commands in
succession.  A conventional number of the first kind means a quantity
or constant in the formula.  A conventional number of the second kind
is an intermediate result in the calculation of the formula.  It is
generated by the algorithm A2 and equals the address of the
non-resultant command in the BPO for each such command.  A BPO scale
is a group of consecutively located registers of the memory with
continuous enumeration of the digits, with which the s-th digit of the
BPO scale corresponds to the L+s register of the BPO.  The scale of
the CN of the first kind has a similar apparatus.  A _Block of
Operating Registers_ (BOR) is a group of registers with the addresses
r+1,...,r+m, where r+1,...,r+m are the codes of the operating
registers.  A _Block of Preparatory Programs_ (BPP) is a group of
registers in which a preparatory AO program will be placed.  The
symbol (T) denotes the content of the register T.

#3.  In existing command economy methods, the total time of operation
of the A2 algorithm is proportional to the square of the number of
commands in the AO program.

Shown on figure 1 is a diagram of the A2 algorithm which permits the
realization of command economy within a time proportional to the
number of commands in the AO program.  The basis of the algorithm
proposed is the assumption that there exists a certain integer
function F=F(theta,a,b) (L+1<=F<=L+n) defined for any AO command

-----------------------------
| a | b | c | sigma | theta |
-----------------------------

Operation of algorithm A2 is started after construction of the next AO
command K by the A1 algorithm (for simplicity, an A2 algorithm is
describe which does not produce economy of the resultant commands).

Operation 1 investigates whether the command K is the resultant (if
not, do operation 2).

Operation 2 calculates F(theta,a,b) for the command K and directs the
result into the register S.  It is evident that L+1<=(S)<=L+n.  Let
(S)=L+p.

Operation 3 verifies whether (L+p) equals zero (if not, do operation
4).

Operation 4 verifies whether the operation codes, the first two
addresses and the digit sigma agree for the K and (L+p) commands (if
yes, output I).

Operation 5 increases p by one if p<n and puts L+1 into the register S
if p=n.

Operation 6-8 perform if the command K is not economized.

Operation 6 investigates the CN from the address part of the command K.
If a CN of the first kind is among them, ones are put in the
appropriate digits of the scale of the CN of the first kind and in the
p-th digit of the BPO scale (zeroes are in all the digits of both
scales before the start of the AO programming).

Operation 7 calculates certain quantities needed for the operation of
the A3 algorithm for the command K (see #5).

Operation 8 directs the command K into the L+p register.

Operation 9 performs if K is a resultant command (c/=0).  If a one is
in the digit of the scale of the CN of the first kind corresponding to
the CN c, then BPO commands containing a CN of the first kind are
scanned by using the BPO scale.  Commands containing the CN c are
marked by ones in the sigma digit.  Consequently, none of the commands
containing the CN c in the address part and having been constructed
after K will coincide with any of the commands constructed before K
during the execution of operation 4.  Then K is transmitted to the
next free BRC register.

The A2 diagram has two outputs I and II.  A CN of the second kind
which denotes the result of a constructed nonresultant command is
obtained in the register S at the output I.  The output II corresponds
to a resultant command.

#4.  The duration of the execution of A2 is determined by the number
of repetitions of operations 3-5.  This number depends on the
distribution of the values of F(theta,a,b) in the strip [L+1,L+n].
[Let us note that the usual command economy algorithms correspond to
F(theta,a,b)=L+1.]  Evidently, the most favorable case is the uniform
distribution of the F(theta,a,b) values in [L+1,L+n] for a random
composition of the AO formula.  In this case, the mathematical
expectation Phi of the number of repetitions of the operations 3-5 can
be calculated as a function of the number of commands k making up the
program and of the quantity of registers n in the BPO [Phi =
Phi_n(k)].  A derivation of analytic estimates appears to be difficult
an the values of Phi_n(k) were computed by the Monte Carlo method.
Presented on figure 2 are the curves of log_10(k) obtained for n=150
(50) 450.  Curves of log_10(k) and log_10(k^2/2) are given for
comparison.  It follows from an analysis of the results obtained that,
in practice, not more than one execution of the 3-5 operations will
occur in each AO command if the BPO exceeds the number of commands in
the AO by not less than one and one-half times for all n.

The simplicity of the calculation and the sufficiently uniform
distribution of the values is the unique criterion limiting the choice
of F(theta,a,b).  It is expedient to use the methods of producing
uniformly distributed pseudo-random numbers for the actual
construction of F(theta,a,b).  An investigation of the statistical
structure of the formulas of the AO to be programmed is of great value
in the successful choice of F(theta,a,b).

#5.  There are definite relations in the order of the performance of
the operations entering into the AO formula.  These relations are
given by a rule that the components are calculated, at the beginning,
for the components of the formula and then the operation itself is
calculated.  Consequently, the formula can be considered as a
semi-ordered set of the operations therein.  Ordering of the
operations, caused by the successive location of the command in the
program, occurs in the construction of the program to calculate the
formula, consequently, the problem of programming the formula can be
formulated as a problem in ordering the operations of the formula by
retaining a given semi-order.  It is evident that the quantity of OR
required to calculate the formula depends upon the method of ordering
its operations.  For example, in order to calculate the formula

ab + (cd - ef (gh + ij (kl - mn))) -> y

seven OR are needed to perform the action from left to right while
only 2 OR are needed if the calculation is started with the innermost
parenthesis.  In this connection, the problem arises of finding such
an admissible ordering of the operations of the formula for which the
minimum quantity of OR would be required for its calculation.

The problem posed is solved partially by using the algorithm A3 of the
ordering of the operations of the formula whose diagram is presented
on figure 3.  Calculation of the operation 7 of the A2 algorithm for
each nonresultant command K of two integer functions whose values are
put in the third address of the command before it is transmitted to
the BPO is preparatory to the operation of A3.  The first function, a
function of the order P(K), is given by an inductive definition:

A)  If the command K does not contain a CN of the second kind, then
P(K)=1.

B1)  If a CN of the second kind, denoting the result of the command
K_1 is an address of the command K, then P(K)=P(K_1).

B2)  If a CN of the second kind, denoting the result of the commands
K_1 and K_2, are in the first and second addresses of the command K,
then

P(K) = / max{P(K_1) P(K_2)} if P(K_1) /= P(K_2)
       \ P(K_1)+1           if P(K_1) =  P(K_2)

The second function, the entry counter, is calculated as follows.
When a command K is transmitted to the BPO, its entry counter equals
0.  If a command K', containing a CN which denotes the result of the
command K, is then transmitted to the BPO, then 1 will be added to the
entry counter of the command K.

The algorithm A3 starts to perform after the termination of the
operation of A1 and A2.

The operation 1 transmits the next AO resultant command into the
register R, starting with the last register of the BRC.  Let the
command K be in R.

Operation 2 replaces the CN of the second kind in K by a code of OR.
If a CN of the second kind L+s enters into K, the content of the L+s
register is investigated.  The command K' from L+s is transmitted to
the first free register of the BOR.  The command K' in L+s is replaced
by the address r+i, which indicates where K' was transmitted to.  If
K' has already been replaced by the r+i address during the processing
of one of the preceding AO commands, 1 is subtracted from the entry
counter of the command K' in r+i.  The CN L+s in K is replaced by the
r+i OR code.  If two CN of the second kind L+s_1 and L+s_2 enter into
K, where the commands K_1 and K_2 are in the L+s_1 and L+s_2 registers,
that one of the commands K_1, K_2 is transmitted first into the BOR for
which the value of the order function is larger.

Operation 3 transmits K to the next register of the BPP, starting with
the last register.

Operation 4, scanning from the end of the BOR, finds the first command
with entry counter equal to 1.  If such a command is not found in the
BOR or if no commands are in the BOR, control is transferred to
operation 6.

Operation 5 transmits the command found from the r+j register into the
R register, puts r+j into the third address of this command and then
clears the r+j register.

Operation 6 transfers control to operation 1 if not all the commands
are transmitted from the BRC.

The algorithm described solves completely the problem of the most
favorable ordering for an AO for which the entry count of each command
is 1.  This follows from the following two statements which are valid
under the above-mentioned limitations:

1.  In the interests of the minimum expenditure of operating registers
for any binary operation, it is first necessary to calculate those of
its components for which the minimum number of OR required for its
calculation is larger.

2.  The order function for each command equals the minimum quantity of
OR required to calculate the expression in which the last operation is
realized by the given command.

Moscow University      June 27, 1957

* Translator's note: Apparently, the author means a number given
meaning by certain agreed upon conditions.  The translator is grateful
to Sheldon Best of MIT for having read the translation and making
corrections.

[1] A.P.Ershov.  Programming programs for the BESM, Moscow.  1958.

-----