System of automation of programming for analog machines (SAPAM)

CYBERNETICS 29

SYSTEM OF AUTOMATION OF PROGRAMMING FOR ANALOG MACHINES

(SAPAM)

A. Kh. Bereslavskii and G. S. Gol'denberg

Kibernetika, Vol. 4, No. 2, pp. 36-44, 1968

UDC 681.142.2:681.142.39

This paper gives a short description of an automated system enabling a digital computer to perform the syn-

thesis, analysis, and calculation of models.

w FUNDAMENTAL PROBLEMS

i.I. The time factor. One of the fundamental ad- vantages of an analog computer over a digital, dynamic response, is lost because of problem complexity and more rigid tolerances. The greatest time losses have to do with the fact that problem preparation for analog computer input is laborious and of unsatisfactory qual- ity.

1.2. Optimality of the model. The investigator con- structs the model by intuition and analogy, since a completely formalized approach is restricted (at least) by the volume of computation. In the general case there is little possibility of obtaining a satisfactory result without transformation of the mathematical description.

1.3. Consideration of the requirements. For var- ious reasons the investigator cannot take into account or compensate for all of the constraints which may be

imposed on the domain of possible values of the model parameters. In this case, as in w the formal work must be carried out by the consideration of a large amount of informat ion and i ts p rocess ing , which is feas ib le only with a digital computer . The poss ib i l i ty of using a digital computer for solving this problem has been considered in a number of papers [6].

1.4o Analys i s of these p rob lems shows that the following ma t t e r s mus t be considered:

a) the s t ruc tu re of SAPAM and the purpose of its par ts ;

b) the form and quanti ty of the input and output information;

e) the degree to which the cons t ra in t s and r e q u i r e - ments are allowed for;

d) opt imizat ion of the s t ruc tu ra l d iagram and methods of implementing it;

e) the possibility of the use of SAPAM.

w APPROACH AND SOLUTION

2.1. The SAPAM system. SAPAM is a unified system that can solve awhole complex of problems arising in analog computer work, while retaining the ability to solve individual problems. Figure 1 shows a functional diagram of SAPAM. The system consists of parts ensuring the synthesis of the model, its optimization, calculation, and analysis by a number of criteria, the correction of elements in accordance with the characteristics of specific analog computers, and so on.

2.2. The form of the data fed into and obtained from the digital computer is adapted to that generally ac- cepted by analog computer specialists, and does not require special knowledge or the use of specialized equipment. By its content and utilization the input information is divided into

Fig. 1

a) general data about the mathematical description; b) numerical values of the parameters of the de-

scription;

c) data about the analog computer and additional instructions (the variables to be registered, the re- quired accuracy, and so on).

2.3~ In the opinion of the authors the most important initial model requirements are the following:

a) versatiliy and reliability of circuits; b) circuits conforming to the possibilities of the

analog computer;

c) probable accuracy of the solution. 2o3ol. In the literature a number of methods for

calculating the productivity of analog computers are de scribed [7], based on the determination of the amount of information processed, and also methods for calculating the reliability of schemes [5] based on the definition of a failure as the occurrence of aprolonged and gross distortion of the solution.

2~ 3.2. Methods of determining the probable accuracy of an analog computer solution by means of a digital computer were set forth in reports by the authors at the scientific seminar on Methods of Mathematical Simulation and the Theory of Electrical Networks held at the Institute of Cybernetics AN UkrSSR, 24Decem- ber 1965.

2.4. Methods of constructing a general algorithm for synthesis of an optimal model are not described in the literature on analog computers. In practice this problem is solved by successive selection of parana-

30 KIBERNETIKA

eter values and switching with subsequent analysis of the variants obtained. It is obvious that the criteria are best satisfied by minimization of the model with respect to the number of operational units. (Known methods of constructing so-called reversible units also use irreversible operational units [3]. )

2.5. We consider the most important properties of the operational units used in constructing a structural

scheme of the model. 2.5.1. Unidirectional transmission of information. 2.5.2. Freedom of the form of the transfer function

of the unit f r om dependence on subsequent opera t iona l uni ts which may be connected to i t s output. Here uni ts of the f i r s t and second type a r e d is t inguished. In units of the f i r s t type the s ign of the output s ignal depends on the sign of the input s igna ls , while in those of the second type the s ign of the output s ignal is e s t ab l i shed by the inves t iga tor .

2.5.3. Opera t iona l units a r e d e s c r i b e d by r e l a t i on equat ions and a r e made in the fol lowing types.

2.5.3.1~ G e n e r a l - p u r p o s e l i nea r units for p e r f o r m - ing summat ion (adder) , i nve r s ion ( inver t e r ) , i n t e g r a - t ion ( in tegra to r ) , gene ra t i ono f the independent va r i ab l e , gene ra t ion of constant vol tages .

2~ 5o3.2~ G e n e r a l - p u r p o s e nonl inear units r e a l i z i n g funct ional dependences by the method of p i e c e w i s e - l i n e a r app rox ima t ion (un ive rsa l funct ional conver t e r ) , mul t ip l i ca t ion (product block) and d iv is ion (divis ion block).

2.5~176 Spec ia l i zed units for r e a l i z i n g log ica l and s t anda rd m a t h e m a t i c a l dependences of the fo rm I xl, max(xl;x2) , s i n ~ t , coswt, " b a c k l a s h , " etc.

w TRANSLATOR PROGRAM

3.1. The t r a n s l a t o r p r o g r a m is intended for t r a n s - la t ing the d e s c r i p t i o n of the model f rom the input l an - guage of the t r a n s I a t o r SAPAM into the language of the s t r u c t u r a l scheme. The input language of the t r a n s - l a t o r SAPAM is based on ALGOL-60 [1]. The use of me ta l i ngn i s t i c f o r m u l a s is a l so s i m i l a r [1].

3.2. Fundamenta l symbols , i den t i f i e r s , and numbe r s ; fundamenta l concepts . The input language of the SAPAM t r a n s l a t o r is cons t ruc ted of the following b a s i c symbols :

<basic symbol> : := <letter) I <digit> i <delimiter> 3.2.1. Letters [i] 3.2.2. Digits [I] 3.2.3. Delimiters (delimiter} : : = <operator>l <separator> I (bracket>]

<declarator> <operator> : : = <arithmetic aperator) ] (differentia-

tion operator> I <relational operator} <arithmetic operator> : : = + I - I x I / I I <differentiation operator> : := p <relational operator) ::- -

<separator) :: = -I , I; <bracket> : : = ( I )[ begin I end <declarator> : := independent t cons tant I s cheme 3.2.4. I den t i f i e r s [1] 3.2.5. N u m b e r s [1] 3.2.6. Quant i t ies and c l a s s e s . The fol lowing c l a s s e s

of quant i t i es a r e d i s t inguished: the independent v a r i -

able, variables, and constants. The construction of the structural scheme, beginning with the operating characteristics of the analog computer, assumes that a) there is only one independent variable; b) some of the constant quantities are considered to be variables-- therefore the translator recognizes as constants only those quantities which are described as constants.

3.3~ E x p r e s s i o n s (express ion} : : = (complex express ion) t <equation)i

<system of equations} 3o3olo Variables <variable identifier} :: = (identifier> <variable) :: = (variable identifier)

303o2. Independent variables <independent variable identifier> :: = <identifier) <independent variable} :: = (independent variable

identifier> 3.3.3. Constants <constant identifier) :: = <identifier> <constant> :: = <constant identifier> 3.3o4. Function designators <function identifier> :: = <identifier) <parameter> :: = <complex expression> <parameter list} :: = (parameter>~{parameter list>,

(parameter} (parameter part> :: = (<parameter list}) <function designator> : := <function identifier)

<parameter part> The function designators determine the necessity

for realizing functional dependences by means of general-purpose nonlinearity blocks.

3.3.5. Standard scheme designators <standard scheme identifier> :: = <identifier> <actual parameter> :: = <complex expression} (actual parameter list) :: = <actual parameter)l

<actual parameter list>, <actual parameter> <actual parameter part> :: = <empty} I ((actual pa-

rameter part> :: = <empty}](<actual parameter list>) <standard scheme designator} :: = <standard scheme identifier> <actual parameter part>

The standard scheme designator shows that the structural scheme of some mathematical dependence stored in the memory of the digital computer, cor responding to a standard scheme identifier must con- form to the common structural scheme formed by the translator. Then the input quantities of the scheme (formal parameters) are replaced by the variables of the actual parameter list.

3.3.6o Derivative designator (differentiation order) :: = <unsigned integer) <differentiating operator) : : = p I P (<differentiation order})

<derivative designator) : :-'= (differentiating operator> (<variable>)

3o3.7. Complex expression (adding operator) : := +l- <multiplying operator> : : = x I / <primary expression> :: = <variable>l <function des-

ignator>l <standard scheme designator)l(<complex expression>) I <derivative designator)

<factor> : : = <pr imary express ion) [ <unsigned num-

CYBERNETICS 31

ber} (mult iplying operator} (p r imary expres - sion}l (constant} (mult iplying operator> (p r imary expression>t (unsigned integer)

(complex} : : = (factor)l (complex> (mult iplying operator) (factor)

(complex expression) : := (complex>l (adding oper - ator> (complex)] (complex expression> (adding operator} (complex>

3o3.7.1. In the cons t ruc t ion of a s t ruc tu ra l scheme of a model the fo rmat ion of the operat ional uni ts within the l imi t s of one express ion is ca r r i ed out f rom left to r ight, taking into account the following additional rules:

a) Syntactically the following priority is retained: first: (derivative designator>, (function designator) ,

(standard scheme designator); second: I ; third: x;

fourth: +, - b) operational units for expressions between a left

bracket and the corresponding right bracket are formed

independently and are used in the constructions that

follow.

3.3.8. Equation

(right-hand side of equation> :: : (complex expres-

sion)

(equation) : : = (variable} = ( r ight-hand side of equation}l (der ivat ive designator} = ( r ight-hand side of equation)

3.3.9. System of equat ions (beginning of system) : := begin (descript ion)t(end

of system); (descript ion) (body of system) :: = (equation}l (body of system);

(equation) (end of system} :: = end ( sys tem of equations) : : = begin (body of system}

(end of system)] (beginning of system}; (body of system) (end of system)

3.4. Descr ip t ions 3.4.1. Description of constants

(constant list} :: = (constant)l(constant list), (constant}

(descr ip t ion of constants} : : =constant (constant l ist) 3.4.2. Descr ip t ion of independent var iab le

(descr ip t ion of independent variable} : := independent ( independent var iable)

3~176176 Descr ip t ion of s tandard schemes (scheme list) :: = (s tandard scheme identifier)l

(scheme list}, ( s tandard scheme identifier} (scheme descr ipt ion) :: = scheme (scheme list} 3.5. Let the complex A be given. We a r r ange in it

by the ru l e s of w 7.1. the min imal number of brackets , without changing the o rder of the operat ions , suehthat within each pa i r of brackets there occur only opera- t ion des ignators and operat ional uni ts of the same p r i - ority. The express ion in brackets is a complex (or group of complexes separated by the separa to r ", "), and we say that it f o rms a par t of the complex A. The complexes (variable} and (independent variable) will be cal led z e r o - r a n k complexes.

A s tandard scheme des ignator with a l is t (empty) is a complex of the f i r s t rank.

If at leas t one complex of the (n - 1)-th rank forms a par t of a complex, it is called a complex of the n- th rank.

If A is a complex r e p r e s e n t i n g ((complex expres - sion)), where (complex expression} is the complex B, then the ranks of A and B are equal.

3.5.1. Equations to be t r ans la ted must be wr i t ten in the form

p " % = F ( x , , p x , . . . , x2, px2 , . . . , x m, pxm . . . . , t),

where x i is a variable to be determined, n i is the

largest exponent of p for the x i in the system, and m

is the number of variables in the system.

EXAMPLE

As an example we cons ider the notation in the input language of Mathieu 's equation:

p2x 1 ---- 2 q x , cos o~t - - a x l

begin

constant q, a; scheme cos wt;

p(2) (xl) = 2 • x c o s m t - - a ~ x l

end 3.6. Each operat ional unit of the s t ruc tu ra l scheme

cor responds ei ther to a complex occur r ing on the r igh t - hand side of the equation, or to a var iab le definable by the equation or its der ivat ive . In crea t ing the device rea l iz ing the complex the number of inputs is checked. If this exceeds the p e r m i s s i b l e number the complex is rea l ized by severa l optimal units~ If this separa t ion is not poss ible , a device with the neces - sa ry number of inputs is organized provis ional ly , and a signal about this is t r ansmi t t ed to the invest igator .

3.6.1o Input va r i ab les occur in gene ra l -pu rpose l inear uni ts with the signs that are ascr ibed to them in the equation. In gene ra l -pu rpose nonl inear and special ized units the input and output var iab les a re posi- tive.

3.7. After informat ion en te rs the memory of a digital computer , l i s t s of constants and s tandard schemes a r e t r a n s f e r r e d into special fi les. Then the in fo rma- tion about each equation and the format ion of operat ional uni ts is considered. Operat ional uni ts a re formed only for complexes of the f i r s t rank. The contents of a complex of the f i r s t rank are t r ansmi t t ed to a working file, after which it is decided whether the unit ini t ia ted has been considered previously. If it has been considered, then into its place in the equation is t r a n s f e r r e d the ident i f ier of the output 'value of the complex identical with the given one. If the complex is being considered for the f i r s t t ime, the constants and numbe r s belonging to it a re withdrawn and t r a n s - fe r red to the file cor responding to this complex: the contents of the f i l e - - in fo rmat ion about the p r og ram- - a re sent to the operator .

Then a uni t is formed rea l iz ing the given complex (apart f rom the complex (der ivat ive designator)) , and to the output var iable of the unit an ident i f ie r is ass igned which is t r a n s f e r r e d to the equation in place of the complex. To the output var iab le of the complex (de- r iva t ive designator) there is ass igned the ident i f ier t r a n s f e r r e d to the equation in place of the complex, and a fi le is formed in which the o rder of d i f ferent ia-

32 KIBERNETIKA

tion, the variable identifier for the operation of dif- fe rent ia t ion , and the complex ident i f ier a r e located.

After these opera t ions the rank of the complex is reduced by unity. As a r e su l t of s u c c e s s i v e reduct ion of rank, the r igh t -hand side of the equation will be- come the a lgebra ic sum of z e r o - r a n k complexes connected by the equal i ty sign to a de t e rminab le equation of the va r i ab le or its de r iva t i ve (in the gene ra l ca se of o rder n) and r ea l i z ab l e e i the r by an adder , i n t eg ra - ing adder , or in tegra t ing adder with subsequent i n t eg ra - tion. If the va r i ab le to be de te rmined co r re sponds to complex ident ical to one cons idered p rev ious ly , the output va lues of the ident ical complexes a r e identified by this var iable . A compar i son is made of the o rde r of d i f fe ren t ia t ion and of the va r i ab le in the ease of the opera t ion of d i f ferent ia t ion with the content of f i les fo rmed for complexes of the type (der iva t ive des ig - nator}.

In the ease of coincidence , the output va r iab le of the in tegra tor a s s u m e s the ident i f ie r of the output value of the complex for which the given fi le was formed. If there is no compar i son the output value is a ssumed by a new ident i f ier .

3.8. Var i ab le s not dependent on o thers , but not de sc r ibed as independent, a r e r ea l i z ed by a constant- vol tage genera to r .

3.9. If t he re is a des e r ip t ion an independent va r iab le is r e a l i z ed by an independent va r iab le genera to r .

3.10. The s t ruc tu ra l scheme is comple ted by the c rea t ion of an i nve r t e r for each operat ional unit.

3.11. Because of lack of space arid inexper ience in the use of the SAPAM t r ans l a to r , the authors do not give all r e c o m m e n d a t i o n s on the t r ans fo rma t ion of the ma thema t i ca l desc r ip t ion such that it would co r re spond to the f o r m indicated in w and ensure that a sa t i s f ac to ry s t ruc tu ra l scheme is obtained (however, it can be shown that quest ions propounded above can to a cons ide rab le extent be decided by the in t roduct ion of additional equations into the descr ipt ion) . F o r the same r easons we have been compel led to omi t a d i scuss ion of the l i s t of syntact ic e r r o r s which may be detected by the t r ans la to r . It is obvious that the r e s t r i c t i o n of the input language and the -smal l va r i e ty of p rob lems make it poss ib le to compi le a f a i r ly c o m -

plete l i s t of e r r o r s . 3.12. For r e m o v a l of units f r o m the s t ruc tu ra l

scheme in connection with i ts min imiza t ion it is nec- e s s a r y to bear in mind the following:

a) It is forbidden to r e m o v e some units because of the r e q u i r e m e n t s imposed by the nature of the inves - t igat ions (the output va r i ab le s a re then labeled by the d i spa tcher p r o g r a m on t r ans f e r to the op t imize r p ro -

gram). b) We do not d i scuss r emova l of units r ea l i z ing

v a r i a b l e s definable by equations (apart f r o m ease s of

identity). c) ~r of s c h e m e s with r e s p e c t to the

number of i n v e r t e r s is explained below (w The in- t roduct ion in accordance with w of a l l poss ib le i n v e r t e r s not only s impl i f i e s the a lgor i thm of the t r ans l a to r , but en su re s comple te minimiza t ion .

d) The authors have not found genera l ru les for the r emova l of units r ea l i z ing complexes ; however the known methods of the d i r ec t ed graph, the in terchange of adders and nodal points, the use of s tandard schemes , and a lso a number of p rac t i ca l methods es tabl ished by the authors ensure , in the opinion of the authors, that sa t i s fac to ry r e su l t s will be obtained.

w OPTIMIZER PROGRAM

4. i. The optimizer program is intended to minimize the structural scheme with respect to the number of

inverters.

4.2. After the formation of inverters by w

may be variables in the structural scheme with the same identifiers but different signs. We call a struc-

tural scheme having at least one variable with given

identifiers, a scheme with a complete set. The re-

moval of inverters indicates, first, an equivalent transformation of methematical description, connected

with the change in sign of the variables, and secondly,

the removal from the scheme of variables which are

outputs of the inverters to be removed. Labeled vari-

ables cannot be removed (w nor variables whose

removal would lead to the destruction of the complete- ness of a set. We call these variables forbidden for

r e m o v a l or s imple forbidden va r i ab les . It is obvious that the number of forbidden va r i ab l e s

n is not g r e a t e r than N/2, where N is the total number of va r i ab les in the scheme.

4.3. The number of blocks and the na ture of the switching between them de t e rmine the s ta te of the s t ruc tu ra l scheme. The r e m o v a l of an i n v e r t e r leads to a change of the s ta te of the scheme. We es tabl ish a co r respondence between each state of the s t ruc tu ra l scheme and the set of va r i ab les forbidden in the given state.

L e m m a 1. F o r a given state of the s e h e m e l e t t h e r e ex is t the va r i ab l e s x m and -Xm, and also the groups of va r i ab le s Xr and -Xr , where

X m ~ - ~ r n r X r ,

- - . , V m ~ U X m , (1)

and E and 1I a re r e spec t i ve ly symbols of units of the

f i r st f o r m (apart f rom inver te r s ) and i n v e r t e r s ( w 2.5.2}. Then the forbidding of x m does not lead to the forb id-

ding of x r. Indeed, if x r is forbidden, then x m can be r ea l i z ed

by the relations

- - xm = Z ~ , ~ ( - - x,) ,

x m = l i ( - - x . , ) . (2)

L e m m a 2. If the conditions of L e m m a 1 a re sa t - isf ied, the forbidding of Xr does not lead to the f o r -

bidding of Xm. The forbidding of x m does notdes t : 'oy the comple t e -

ness of the set, s ince - x m can be rea l i zed by the r e -

la t ion

- - x ,~ ---= s162 r (-- xr ) .

CYBERNETICS 33

T h e o r e m of the c o m p l e t e n e s s of p r o h i b i t i o n s . Le t x i be a v a r i a b I e w h o s e e x c l u s i o n l e a d s to a new s t a t e of the s t r u c t u r a l s c h e m e . F o r the se t of f o r b i d d e n

v a r i a b l e s c o r r e s p o n d i n g to th is s t a t e to be c o m p l e t e , i t is n e c e s s a r y and su f f i c i en t to s u p p l e m e n t the s e t of fo rb idden v a r i a b l e s c o r r e s p o n d i n g to the p r e c e d i n g s t a t e of the s c h e m e by the v a r i a b l e - x i , and by the v a r i a b l e s xj , Xl, and Xk, if they a r e r e l a t e d to - x i a s fo l lows :

- - x ~ ~ ~otox !,

~ = 2%ix ~ - - %,x~. (3)

N e c e s s i t y is obv ious . Suf f i c i ency . 1. If - x i is connec t ed wi th un i t s of the

s econd type, a change in the s igns of the v a r i a b l e s c o n n e c t e d wi th t h e s e b locks does not l e a d to a change in the s ign of - x i.

2. L e t - x i be c o n n e c t e d with uni ts of the f i r s t type. By L e m m a s 1 and 2, i f x i is e x e l u d e d f r o m the o r ig inM s t a t e of the s c h e m e , the t h e o r e m is vMid. Le t the e x - c lu s ion lead to an (n + l ) - t h s t a t e of the d i a g r a m , w h e r e the t h e o r e m was va l id f o r the p r e c e d i n g s t a t e s . Then two c a s e s a r e p o s s i b l e :

a) In the (n + 1)- th s t a t e of the s c h e m e t h e r e e x i s t both +Xk, + X l , +xj and -Xk , -Xl , - x j , w h e r e by Xk, Xl, xj we m e a n the v a r i a b l e s de f ines in (3); then by the l e m m a s the e x c l u s i o n of any of the r e m a i n i n g p e r - m i t t e d v a r i a b l e s does not l ead to v i o l a t i o n of the c o m - p l e t e n e s s of the se t and the t h e o r e m i s vMid.

b) Any of the -Xk, - X l , - x j has been exc luded p r e - v ious ly : in th is c a s e , by h y p o t h e s i s , a l l the p r o h i b i t i o n s c o n n e c t e d wi th the p r e s e r v a t i o n of c o m p l e t e n e s s of the s e t w e r e m a d e at the p r e v i o u s e x c l u s i o n s .

The t h e o r e m is p roved .

C o r o l l a r y 1. If when x i i s r e m o v e d , x i is f o rb idden , then xi is a f o r b i d d e n v a r i a b l e fo r a l l s t a t e s of the s y s t e m .

C o r o l l a r y 2. I f w h e n x i is r e m o v e d , x s is f o rb idden , then when x s i s r e m o v e d x i i s p roh ib i t ed .

If x s is one of the xj , then by i t s e x c l u s i o n

x~ = 2 % (- - xi) - - % x , (4)

that i s , x i is f o r b i d d e n by the second equa t ion of (3). If x s i s one of the x k and xl , then by the r e m o v a l o f x s

- - x, , = 52% (-- x~) - - % x , + % % ( 5 )

o r

- - x s = Eo:~l ( - - xl) @ asixe, (6)

that is , xj is f o r b i d d e n by (3).

C o r o l l a r y 3. The se t of f o r b i d d e n v a r i a b l e s ob ta ined by r e m o v a l of the - x i has the s a m e p o w e r and con t a in s v a r i a b l e s wi th the s a m e i d e n t i f i e r s , but wi th oppos i t e s i gns , a s the set of f o r b i d d e n v a r i a b l e s ob ta ined by r e m o v a l of the x i.

The p roo f fo l lows f r o m (4) - (6) . C o r o l l a r y 4. L e t {M} be the se t of v a r i a b l e s f o r -

b idden by r e m o v a l of the c o m b i n a t i o n of the p e r m i t t e d v a r i a b l e s xl, x 2 . . . . . x m (m _< N/2) ; {X1} , {X2} . . . .

� 9 , {Xm} a r e the c o r r e s p o n d i n g se t s of v a r i a b l e s f o r b i d - den by r e m o v a l of one of the v a r i a b l e s e n u m e r a t e d .

Then , s i nce the p r o h i b i t i o n s a r e independen t of the

s t a t e of the d i a g r a m {M} = ~ {X~}. I

4.4. We can d r a w up a s c h e m e m i n i m a l wi th re-- s p e c t to the n u m b e r of i n v e r t e r s if i n f o r m a t i o n is a v a i l a b l e about the p r o p e r t i e s of the o p e r a t i o n a l un i t s p r e s e n t e d in w167 and 2.5.3 and the swi tch ing b e - tween t hem, g e n e r a l i z i n g f r o m the spec i f i c v a l u e s of the v a r i a b l e s and cons tan t s . We r e p r e s e n t a s t a t e of the s t r u c t u r a l d i a g r a m as a s q u a r e m a t r i x whose r o w s c o r r e s p o n d to o p e r a t i o n a l un i t s , and the c o l u m n s to v a r i a b l e s . H e r e , fo r c o n v e n i e n c e , the o p e r a t i o n a l un i t s a r e denoted by the i d e n t i f i e r of the output va lue . The i n t e r s e c t i o n of the i - t h row and j - t h co lumn has the va lue one if the j - t h v a r i a b l e e n t e r s the uni t hav ing the i - t h v a r i a b l e at the output. The n u m b e r of r o w s and c o l u m n s and the a r r a n g e m e n t of the uni t s d e t e r - m i n e the s t a t e of the m a t r i x , which is un ique ly c o n - n e c t e d wi th the c o r r e s p o n d i n g s t a t e of the s t r u c t u r a l s c h e m e . V a r i a t i o n of the s ta te of the s c h e m e l e a d s to a un ique v a r i a t i o n of the s ta te of the m a t r i x . F o r un i t s of the second type and i n v e r t e r s , it m a k e s s e n s e to c o n s i d e r the inputs only in the f ina l s t a te . T h e r e - f o r e the m a t r i x can be s i m p l i f i e d by put t ing the e l e - m e n t s of r o w s of the second type and i n v e r t e r s equal to ze ro . We ca l l th is m a t r i x the s t a t e m a t r i x of the s c h e m e .

4.5~ The m i n i m i z a t i o n of a s c h e m e deno t e s the r e m o v a l of the r o w and c o l u m n of a m a t r i x cor re - - sponding to a r e m o v e d v a r i a b l e . A change of the s ta te of a m a t r i x i s c a r r i e d out in a c c o r d a n c e With the f o l - lowing r u l e s .

4 .5.1. The r e m o v a l f r o m a m a t r i x of a row and c o l u m n iden t i f i ed by one v a r i a b l e is p e r m i t t e d if t h e r e a r e z e r o s in a l l t h e i r e l e m e n t s .

4~ In o r d e r to r e d u c e a row to the f o r m of w the ones in i t s c e i l s a r e c a n c e l e d , and ones a r e e n t e r e d a t the i n t e r s e c t i o n of the row c o r r e s p o n d - ing to the oppos i t e v a r i a b l e , wi th the c o l u m n s c o r r e - sponding to the v a r i a b l e oppos i t e to that us ed tO iden t i fy the c o l u m n s hav ing ones in the e l e m e n t s at the i n t e r - s e c t i o n s wi th the r o w to be r e m o v e d .

4.5,3. To r e d u c e c o l u m n s to the f o r m of w ones at the i n t e r s e c t i o n of t h e s e m a t r i x c o l m n n s and r o w s a r e c a n c e l e d , and the r o w s ind ica t ed a r e t r e a t e d a s in w Obv ious ly , the t r a n s f o r m a t i o n of the s t a t e m a t r i x can be c o n s i d e r e d in s t ead of the t r a n s - f o r m a t i o n of the s c h e m e .

4.6. We c o n s i d e r a m a t r i x the r o w s and c o l u m n s of which a r e iden t i f i ed in the s a m e way as the s t a t e m a t r i x . If ceij is the con ten t of a ce l l of the newly f o r m e d m a t r i x , then

l c~ij = I 1, if the i - t h v a r i a b l e p r o h i b i t s the j - th ;

[ 0, o t h e r w i s e .

We ca t l the m a t r i x thus ob ta ined the c o m p a t i b i l i t y m a t r i x . C a s e s a r e p o s s i b l e w h e r e in the j - t h c o l u m n (~ij = 1 fo r a l l ceij; that i s , the v a r i a b l e whose i den - t i f i e r is a s s o c i a t e d with the j - t h c o l u m n is fo rb idden to e a c h of the v a r i a b l e s ( l abe led v a r i a b l e s and o thers ) . )

34 KIBERNETIKA

dx-

_d.~x#r.,. | dt f ~ acz ~ �9

Standard scheme

I s , / o ~ s~t ,

I ~ I r c , , , , t , , t ~ i I L.:,/ ~ I

, ~-~,~

~ COS ld~

-;CCz~t

dx,

dt

cos~at

b

sg -~: Co3fdt-Co3fi)t

r, cas~

dt

r

xa~gt dx, z , ~ t -d-~

1

I 1

I

1 l

I /

I /

d2c~ d~ z~ -x: co~Qt-cc~Jt

1 I

1

1 1 1 / /

1

1

11 I L

O -

0 i 0 E n d ,

~c~Wt

- ~ ,

d x~

Cos~)t

-c#~ Q t

Standard scheme

i" ~ r

I

I cos:3t V(V~/-cosut I

L 2 ~ -.~: COS 5)t

-N cos ~ t r,... d__xt

dL Xr

f

Fig. 2

CYBERNETICS 35

4.7. We call an i-th combination of mutually per-

mitted variables complete, if

N ~ / 7 / i ~ 12i == 0,

where m i is the number of variables in the i-th combination, and n i is the number of variables forbidden to the i-th combination.*

It is obvious that

N m . = T - - Pi , (7)

where Pi is the number of identifiers belonging only to the set of identifiers of the forbidden variables. For incomplete combinations,

N m~. = ~-- p~ > m~.

The problem of minimization consists of the discovery of a complete combination containing the greatest number of variables. In this form the problem yields to solution by the method of sequential analysis of variants. The essence of the algorithm is as follows. As- sume that the i-th complete combination with the corresponding value of m i has been found. We consider the j-tb combination (not necessarily complete), for which we determine mj by formula (7). If mj - mi, the combination is discarded as unpromising, and none of the combinations including it are considered.

4.8. As an example we consider the minimization of the structural scheme formed by the translator in w The scheme obtained as a result of the transla- tion, the state matrix, and the compatibility matrix are shown in Fig. 2a-c; the minimization scheme is shown in Fig. 2d (the symbol "--" denotes unprom-

ising variants). It does not make sense to consider combinations for opposite variables and combinations

which are no better than those considered. The state

matrix is transformed in accordance with the maximal combination obtained, and then correspondingly the structural scheme, the final forms of which are shown in Fig. 2e,f.

w OTHER SAPAM PROGRAMS

5. i. Programs which form a part of SAPAM (apart from translators and optimizers) are constructed by algorithms, some of which have been published [4-7], or have been realized in practice, and the development of the remainder presents no theoretical difficulties. This enables us to restrict ourselves to the question of the purpose of each program.

5,2. The dispatcher program controls the working of the system and the process of transmitting information from one program to another and to the external memory of the digital eomputer.

*The value of n i is determined by Corollary 4. In practice this is easy to do if the combinations of ones and zeros in the rows of the compatibility matrix corresponding to these variables are regarded as binary numbers and are added logically.

5.3~ The calculator program calculates the ele-

ments of the structural scheme. The fundamental al-

gorithms are explained in [4, 6]. 5.4. The debugger program is intended to correct

the structural scheme--taking into account the requirements imposed by tile analog computer to be used-- to read out additional information, to help carry out debugging, and to search for defects in setting up the problem. The fundamental part of the algorithm of the program is explained in [6].

5.5. The compiler program is intended for dynamic introduction of changes in the information stored in the external memory of the digital computer at any stage of programming of the problem.

5.6, The investigator program investigates the structural scheme for setting-up accuracy, reliability, and complexity, The algorithm for calculating reliability and complexity can be based on the methods described in [5, 7]. Analysis of the probable accuracy of the performance of the analog computer can be done by means of a program described at a scientific seminar given at the Institute of Cybernetics, AN UkrSSR, 24 December ]965.

5.7, The decipherer program is intended for the

decipherment of information and generation of the printed form.

w CONCLUSIONS

Turning to the question of the possibilities of

SAPAN[, we should first mention that application of the system assumes the use of a digital computer with

a sufficiently powerful external memory. Application of SAPAM must be expected to be most successful in the investigation of objects of the same kind. But since

SAPAM has been developed with fairly general as- sumptions, it should possess universality. In the opinion of the authors, extension of the basic ideas of SAPAM will also make it possible to solve the following im - portant problems.

6.1~ The creation of a single cycle for the input of information into analog computers.

6.2. Automation of the programming of digital- analog complexes.

6.3. Design of new types of hybrid computers. 6.4o Centralization of analog computing, aeeumula-.

tion of experience of working with analog computers. 6.5. Providing the software package for series-

produced analog computers.

REFERENCES

i. J. W. Baekus, F. L. Bauer, et alo, ~'Commu- nication on the algorithmic language ALGOL-60, ~r Zhurnal vychislitel'noi matematiki i matematicheskoi fiziki, i, no. 2, Moscow, 1964.

2. V. G. Boltyanskii, "Sufficient optimality conditions and the basis of the method of dynamic program- ruing," Izvestiya AN SSSR, Ser iya matematicheskaya, 28, no. 3, ~oseow, 1964.

3. G. E. Pukhov, Selected Problems in the Theory of Mathematical l~[achines [in Russian], izd-vo Naukova dumka, lqiev, 1964.

36 KIBERNETIKA

4. A. Kh. Bereslavskii, G. S. Gol'denberg, Yu. A. Red'kov, and E. P. Yakovenko, "Some aspects of the methods of preparation of problems for solution on continuous-operation digital computers," Izvestiya AN BSSR, Fiziko-teehnical series, 1965.

5. A. D. Epifanov, Reliability of Automatic Sys- tems [in Russian], izd-vo Mashinostroenie, Moscow, 1964.

6. A. Kh. Bereslavskii, G. S. Gol'denberg, Yu. A. Red'kov, and E. P. Yakovenko, "Aspects of the

automation of the preparation of problems for solution on continuous-operation digital computers," in: Mathematical Simulation and Electrical Networks [in Russian], no. 4, izd-vo Naukova dumka, Kiev, 1966.

7. G. M. Petrov, "Aspects of the construction of the structural circuits of analog-digital devices," in: Analog and Analog-Digital Computing Techniques [in Russian], i~zd-vo Mashinostroenie, Moscow, 1965.

2 January 1967 Nikolaev

Documents

System of automation of programming for analog machines (SAPAM)