Manipulation of Data Structure in a Numerical Analysis Problem Solving Systems

Transcription

1 Purdue University Purdue e-pubs Department of Computer Science Technical Reports Department of Computer Science 1969 Manipulation of Data Structure in a Numerical Analysis Problem Solving Systems Lawrence R. Symes Report Number: Symes, Lawrence R., "Manipulation of Data Structure in a Numerical Analysis Problem Solving Systems" (1969). Department of Computer Science Technical Reports. Paper This document has been made available through Purdue e-pubs, a service of the Purdue University Libraries. Please contact epubs@purdue.edu for additional information.

2 M A NIPULATION OF DATA STRUCTURES Iff A N U MERICAL A N ALYSIS PROBLEM SOLVING SYSTEM N APSS Lawrence R. Symes

3 L. R. Symes 1.

4 L. R. Symes 2. AIMS OF SYSTEM Dur i ng t he past severa l years cons i derab l e effort h as been expended des i gn i ng and i mp l e men t i ng sys t e ms wh i ch are i n t ended t o prov i de extended capab i l i t i es for p ersons wi th m a t h e m a t - i c a l prob l e ms to so l v e. Some of them i n add i t i on to N APSS are C ULLER F RIED, KLERER M A ", M P, R E C K O N E R, A M T R A N, and P OSE. These sys t e ms can b e c l ass i f i ed as prob l e m solving sys t e ms for appl i ed m a t h e m a t i cs. Before t he advent of t hese systems t h e research scient i st or eng i neer used a procedura l l anguage such as F ORTRAN or ALGOL when he employed t he compu t er t o aid him in solving a prob l e m. Both of t h ese l anguages, a l t hough t hey resemb l e m a t h e m a t i c a l no t a t i on more closely t han m a ch i n e l anguage, are somewhat ar t i f i c i a l and contain m any unnecessary, from the user ' s po i n t of v i e w, ru l e s. The ar t i f i c i a l appearance and t he rules must be mas t ered before t he l anguage can be u s ed. Therefore t he scient i st or engineer i s d i ver t ed from h i s ma i n purpose into becom i ng a programmer. Even af t er h e h as l earned the l anguage, i t s complexi ty i ncreases -he probab i l i t y of error, and t hus r educes h i s eff i c i ency. In add i t i on to t hose d i ff i cu l t i es, t h e user w i t h a m a t h e m a t i c a l prob l e m had to use program l i brar i es in order to obtain rou t i nes for solving common l y occurr i ng prob l e m s. These l i brar i es f r e ^ i o n t l y were i nadequa t e and almost a l ways confus i ng. Tho rou t i nes often w ere poor l y documen t ed and

5 L. R. Symes 3. performed l i t t l e or no mon i t or i ng of t he a c curacy of t h e r e su l t s. Thus t h e user of such a l i brary had to know enough numer i c a l ana l ys i s to select the bes t me t hod for solving prob l e ms and to de t erm i ne t h e accuracy of t h e r esu l t s. N APSS h as been des i gned to remove some or a l l of t hese prob l e ms and t o offer severa l o t her des i rab l e f e a t ures. I t, in some s ense, endeavors to have man do wha t he j s bes t equipped to do and to have t he computer do wha t i t i s bes t equipped t o d o. Six g enera l t echn i ques have been u t i l i zed to assist t he u s er in stat ing and solving h i s prob l e m. F i r s t, the source l anguage used to presen t a problem to t h e compu t er i s simi l ar t o norma l "text book" ' m a t h e m a t i c a l no t a t i on. Th i s perm i t s one to use t he system w i t hou t first hav i ng to i n t ens i ve l y study t h e input l anguage. It also reduced the probab i l i t y of user programming errors, because t h e user i s f a m i l i ar wi th the no t a t i on. S e cond, c l er i ca l s t a t e men t s used for d i mens i on i ng arrays and dec l ar i ng var i ab l es are removed from t h e source l anguage. These are t asks wh i ch t he compu t er can easi ly perform bu t wh i ch are a cons t an t source of errors if t he user does t h e m. Th i rd, N APSS perm i t s t h e direct man i pu l a t i on of quan t i t i es o t her t han s c a l ars. These i nc l ude numeric arrays, symbol i c express i ons, func t i ons, and arrays of func t i ons. This fur t her a l l o ws t he source l anguage to resemb l e more c l ose l y

6 L. R. Symes 4. "text book" form, and t h ereby l eads to f e w er statements; h ence f e w er oppor t un i t i es for programming errors. Four t h, solve s t a t e men t s are included in the source l anguage. These s t a t e men t s permi t the user t o state a problem h e w i shes to solve i n a conc i se, na t ura l form. The u s er may i nc l ude parame t ers such as i n i t i a l v a l u es, t he a c curacy d e s i r ed, t he method h e would l ike u s ed, or h e may o m i t any or a l l of the add i t i ona l p arame t ers. The solve statem en t s invoke rou t i n es, po l ya l gor i t h ms [41[5l j from a bu i l t - i n l i brary. They a t t e mp t to solve t he user ' s prob l e m automati c a l l y. They reques t add i t i ona l informat ion as needed and mon i t or the a c curacy of t he r esu l t s in order t o insure t ha t i t r e m a i ns w i t h i n t he specified l i m i t s. The i nc l us i on of t hese so l ve s t a t e men t s grea t l y reduces t he burden norma l l y i mposed on t he u s er. To solve common l y occurring prob l e ms wi th the a i d of t he solve s t a t e men t s, t he user i s on l y required to know how to d ef i n e the equa t i ons for the prob l e m ; he i s no t requ i red to know the numer i ca l ana l ys i s i nvo l ved or even t he m e t hod u s ed. The method i s selected by t he system and the a c curacy of t he resu l t s i s assured. F i f t h, on- l i ne commun i ca t i on between t he system and t he u s er i s prov i ded by ei ther a t e l e t ype or graph i c d i sp l ay d ev i c e. The use of t hese t erm i n a l s bring t he computer and t he u s er closer t oge t h er and consequen t l y i mprove t he user ' s eff i c i ency.

7 L. R. Symes 5. S i x t h, i ncremen t a l execut ion of a program i s a l l o w ed. Th i s, comb i ned w i t h t he use of on- l i ne t erm i n a l s, crea t es a closed loop between t h e user and the sys t e m. The user i s ab l e to mon i t or h i s program during execu t i on and the system i s ab l e to request i nforma t i on from the user and point out errors when t hey ar i s e. Th i s e l i m i na t es much of the t i m e t ha t i s was t ed in preparing and submi t t ing runs of a program wh i ch are unproduc t i v e b e c ause t he user tried several fru i t l ess c ases, h as an i ncorrec t program, or h as forgo t t en to ini t i ali ze a v ar i ab l e.

8 L. R. Symes 6. NAPSS LANGUAGE Rather t han present a detai l ed descript ion of the NAPSS l anguage [101, [12] w e describe a sampl ing of the al lowable assignment statements. The ari thmet i c expression in NAPSS permi t s the direct man i pu l a t i on of numeric scalars, vec t ors, arrays, symbol i c func t i ons and variables which denote symbol i c express i ons. The user need not worry about the type or mode of the operands; ra t her, al l that need concern him is whether or not the ari thmet i c expression i s mathemat i cal ly correct. Serval examples of ari thmet i c expressions and assignment statements appear below: i ) D - ( B + C ) I D * E I i i) ARRAY - ( [3,0:21 1, 2,..., 9 ) / 10 i i i) E = VI + V2 t 2 iv) F(X) - A X t 2 + B X + C, ( X < 0 ) ~ A X t 2 - B X + C v) G(X) = A X t 2 + B X + C, ( X < 0 ) = A X t 2 - B X + C v i ) X(X)[1, 1] * - X t 2 - B, ( X < = G ) - - ( X t 2 ) - B, ( X > C ) vi i) H{X, Y)[5, -21 = G ' (X) + / X t A ( I - 0 TO I ] vi i i) S - ir I A NAPPS STRING ' 1 i.x) Rri S!] '"ARRAY ELEMENT " The l eft arrow operator (-) indicates that the ari thmet i c expression on the right is to be evaluated and i ts value is to be assigned to the variable on the l ef t. The value assigned to D is ei ther a scalar or an array depending upon the operands

9 L. R. Symes 7- in the expression on the right ; wh i l e the value assigned to ARRAY i s a 3 by 3 array. The equals sign (=) has the more mathemat i cal m e an i ng. Statement three establ i shes that a future occurrence of E i s equivalent to the expression VI + V2 t 2. Values are only subst i tuted for the variables in the expression on the right of the = when a value of the variable on the l eft i s needed. Thus if the value of VI or V2 should change between the defini t ion of E and the use of E this is reflected in the value of E. Variables defined to the l eft of an = are referred to as equals variables) and variables defined to the l eft of an v- are cal l ed l eft arrow var i ab l es, or simply var i ab l es. Statements four and five i l lustrate that a symbol i c funct ions may be assigned different defini t ions on different doma i ns. The difference between statements four and five is simi l ar to the difference between statements t wo and t hree. In the defini t ion of F the variables A, B, and C have their current values subst i tuted for t h e m, whi l e in the defini t ion of G they do no t. Values are only subst i tuted for A, B, and C when a value of the funct ion G is n e eded. Funct ions defined to the l eft of an = sign are cal l ed equals funct ions and funct ions defined to the l eft of an are cal l ed l eft arrow funct i ons. Statements six and seven i l lustrate how arrays of funct ions are d ef i n ed. Al l t he elements in array of funct ions must have the same number of arguments and they al l must be ei ther l eft

10 L. R. Syme s 8. arrow or equals func t i ons. Statement eight ass i gns to S a s t r i ng, and state ment n i ne ass i gns a string to an element of an array. A l t hough NAPSS i s intended pr i mar i l y as a prob l e m statem en t l anguage, t h e f e a t ures of a procedura l l anguage have b e en i nc l uded to increase i t s power for the user who w i shes to crea t e a persona l l i brary of NAPSS rou t i n es. i n t erna l procedures may be wr i t t en in M APSS. Ex t erna l and The use of t h ese fac i l i t i es i s op t i ona l. The casual user need not b e concerned wi th the ru l es that procedures i n t roduce, for he can e mploy t he system on wha t i s cal l ed console l eve l. On console l eve l t he user does no t set up any prodecures. S t a t e men t s are entered w i t hou t hav i ng to go t hrough any i n i t i a l set u p, and are norma l l y executed as t hey are r e c e i v ed.

11 L. R, Symes O VER-ALL STRUCTURE OF THE SYSTEM The N APSS system curren t l y running on t h e Control Da t a at Purdue Un i vers i t y cons i s t s of four ma i n modu l es : t he superv i sor, t he comp i l er, t he i n t erpre t er and t he ed i t or. These modu l es are composed of 115 d i fferen t rou t i n es, wh i ch are combined into 28 over l ays. Almost a l l of the system i s wr i t t en in F O R T R A N, w i t h t h e except ion of a few mach i ne dependen t opera t i ons wh i ch are restricted to ' black-box ' 1 modu l es coded in assemb l y l anguage. Th i s i s done to aid t he goa l of mach i ne i ndependence for the sys t e m. The supervisor con t ro l s t he f l o w into each of the t hree o t her modu l e s. It d i s t i ngu i shes be t ween N APSS sources statem en t s, wh i ch are processed by the compi l er and edi t s t a t e m en t s, wh i ch are processed by t he ed i t or. The superv i sor i s also respons i b l e for invoking t he i n t erpre t er when a NAPSS statemen t i s to b e execu t ed. N APSS source state ments are t ransformed by the comp i l er into an i n t erna l t ext wh i ch the i n t erpre t er processes. Th i s scheme w a s adopted for severa l r e asons. F i r s t, the comp l ex i t y of the e l e men t s to be man i pu l a t ed and the absence of dec l arat i ons require execu t i on t i me decod i ng of operands. S e cond, i t easi ly a l l o ws for ex t ens i ons to the sys t e m. Th i rd, i t g i ves t he user i ncremen t a l execu t i on. Four t h, i t perm i t s extensive error d i agnos t i cs and perm i t s error correct ions w i t hou t hav i ng to recomp i l e the whole program. F i f t h,

12 L. R. Symes s t a t e men t s wh i ch are repea t ed l y execu t ed are on l y translated once into i n t erna l t ex t. The i n t erna l and source t ex t for each state ment i s stored in secondary s t orage. When a statement i s to be execu t ed, a copy of the i n t erna l t ext i s passed to the i n t erpre t er. Th i s reduces cons i derab l y the core storage requ i red for a u s er ' s programme. Since t he system i s intended for use in an i ncremen t a l l y execut ing mode, no reference to secondary storage i s norma l l y required to obtain t h e internal t ext of a statem en t. The system opera t es in one of t wo modes : suppress mode or execute mode. In t he suppress mode, each state ment i s comp i l ed into i n t erna l t ext and t h e internal and source t ext i s saved on secondary storage for l a t er execu t i on. Suppress mode i s entered by typing the statement. SUPPRESS. A block of s t a t e men t s wh i ch have b e en compi l ed in suppress mode may be execu t ed at any t i me by typing t he statement. GO. The norma l mode of execut ion i s execute mode. H ere, each s t a t e men t i s executed i m med i a t e l y af t er i t h as been compi l ed and a copy of i t s i n t erna l and source t ex t saved in secondary s t orage. The system au t o ma t i ca l l y enters suppress mode when t h e user starts a compound state ment (a FOR s t a t e men t ) or a procedure. Th i s i s necessary b e c ause a compound statement cannot be executed un t i l the who l e statement i s received and a procedure i s on l y executed when i nvoked. The system re-en t ers

13 L. R. Symes 11. execu t e mode au t o ma t i ca l l y as soon as the compound statement or procedure i s comp l e t ed. The m e mory of a NAPSS program is made up of a few pages of r e a l memory wh i ch res i de in core and a l arger number of v i r t ua l pages of virtual m e mory which reside in secondary s t orage and are brough t in and out of rea l m e mory. Two vec t ors (one deal ing w i t h v i r t ua l and the other w i t h rea l memory) and severa l po i n t ers are used to keep track of r e a l and virtual m e mory. Each element in the v i r t ua l memory vec t or i s subdivided into three t wen t y-b i t by t es. The first by t e contains a f l ag indicat ing wha t type of i nforma t i on i s stored in the p age. The second by t e i s a s w i t ch, used when a page i s in rea l m e mory t o indicate whe t h er or not a copy of the p age also resides in secondary s t orage. The third by t e contains the real page number t he v i r t ua l page is i n, when i t i s in rea l m e mory. The e l e men t s of the v i r t ua l memory vector wh i ch deno t e ava i l ab l e pages are l inked t oge t h er. In i t i a l l y, the element for virtual p age one points to t h e element for virtual page t wo and t he l ast element con t a i ns a zero. ' When a page of vi r t ua l memory i s returned to the system i t s element i s aga i n l inked to the top of the l i st of ava i l ab l e v i r t ua l p ages. The rea l m e mory vec t or e l e men t s contain one entry per r e a l p age. Th i s entry i s the number of the v i r t ua l page occupying i t (zero of i t is free). Th i s po i n t er from rea l memory to

14 L. R. Symes 12. v i r t ua l memory i s used when a new virtual page i s placed in rea l m e mory. The virtual page current ly in t h e real page mus t b e copied.out into secondary storage if a copy of i t i s not already t h ere. The a mount of core ass i gned to real m e mory i s dynam i c. Pages are removed from the t op and bot tom of rea l memory in order to obtain cont iguous b l o cks of s t orage. Pages are removed from t he top of rea l memory for t wo purposes : f i r s t, to expand the name t ab l e, and s e cond, to obtain space for the work poo l. Pages are removed from the bo t t o m of rea l m e mory to ob t a i n space for l oca l name con t ro l blocks during the eva l uat i on of l eft arrow func t i ons. See f i gure 1. The work poo l i s used to hold arrays when perform i ng array ar i t h m e t i c. R eques t s for work poo l space are a l ways made in t erms of words. However, t h e a mount of rea l memory assigned t o the work poo l i s a l ways an integral number of p ages. -7hen a reques t i s m ade for work poo l space and the work poo l i s e mp t y, the space suppl i ed i s zeroed. When space i s reques t ed for the work poo l and t he work poo l is not e mp t y, one of t wo s i t ua t i ons ar i s e s. F i r s t, t he space requested i s l ess t han t he current size of the v.'ork poo l. If the d i fference b e t w e en t h e space reques t ed and the current si ze of the work poo l a moun t s to one or more p ages, a corresponding number of p ages i s returned to rea l memory from the bot tom of the work poo l. S e cond, the space requested exceeds the curren t size of t h e work

15 L. R. Symes I NTERPRETER ' S RECURSIVE V ARIABLES 13. TEMPORARY NAME CONTROL BLOCKS RESULT NAME CONTROL"BLOCK GLOBAL NAME TABLE N A ME T ABLE AENCBS REAL MEMORY PAGE 2 REAL M E M O R Y PAGE 3 REAL MEMORY REAL M E M O R Y PAGE N-4 REAL MEMORY PAGE N-3 LEFT ARROW FUNCTION LOCAL N A ME TABLE r e a l m e m o r y v e c t or VIRTUAL MEMORY VECTOR F i gure 1 N APSS M e mory Organ i za t i on

16 L. R. Symes 14. poo l. If add i t i ona l pages are ob t a i ned from rea l memory to sa t i sfy t he r eques t, t hey are z eroed. V i r t u a l p ages are assigned to rea l pages sequen t i a l l y. Thus a virtual page i s not removed un t i l a l l r e a l pages are ass i gned a v i r t ua l p age. Th i s sequen t i a l process may be broken whenever space i s ass i gned to the work poo l or to hold the local name con t ro l blocks for a l eft arrow func t i on, s i n c e, af t er t he space reques t i s s a t i sf i ed, the next r e a l page to receive a v i r t ua l page may no l onger belong to rea l m e mory. When t h i s occurs the po i n t er to t h e next rea l p age to receive a virtual page i s reset to the f i rs t page now in rea l m e mory. The algori thm for br i ng i ng v i r t ua l pages into rea l m e mory i s fur t h er mod i f i ed when ' t h e work poo l re t urns a page to real m e mory. Since the page re t urned i s e mp t y, a virtual page may be placed in it d i r e c t l y, avoiding t he poss i b i l i t y of hav i ng to save t he virtual p age current ly t here in secondary s t orage. Thus t he norma l sequen t i a l process i s interrupted un t i l a l l pages re t urned to rea l m e mory by t he work poo l are r e-used. The system does not assign al l of real memory to e i t her the work poo l or to space for l eft arrow func t i on ' s l o c a l name control b l o cks. A reques t for rea l memory space i s honored as long as t wo pages rema i n in real m e mory af t er t he reques t i s s a t i sf i ed. If more space i s reques t ed than can be supp l i ed, t he reques t i s mod i f i ed to correspond to the max i mum a mount of space ava i l ab l e. Th i s perm i t s the system to cont inue i f t h i s

17 L. R. Symes 15. i s adequa t e. Two pages are required in rea l memory to fac i l i t a t e the l i nk i ng of v i r t u a l p ages. Wi th t wo pages in rea l memory t he above algori th m guaran t ees t h a t t he prev i ous and current v i r t ua l p ages referenced rema i n in real m e mory. Thus t hey may be l inked t oge t her if n e c essary, w i t hou t having to save po i n t ers and re-read a v i r t ua l page to f i l l in l ink i nforma t i on. Assoc i a t ed w i t h each procedure is a name t able containing en t r i es for each v ar i ab l e, l abe l and constant in that procedure. The en t r i es, cal l ed name control b l o cks, are created during comp i l a t i on when t h e name or constant i s i n t roduced. At t h i s t i m e i t con t a i ns the name of t he v ar i ab l e, and some basic -. a t t r i bu t es descr i b i ng how the variable appears in the program. Dur i ng execu t i on t he name con t ro l block i s used to hold v a l u es, po i n t ers to va l ues and a complete set of a t t r i bu t es for t he vari ab l e. Th i s doub l e usage of t he n a m e control b l ock entries poses no problem if compi l at ion and execut ion are performed s epara t e l y. Bu t in NAPSS t h e norma l mode of opera t i on i s to execu t e each s t a t e men t as soon as i t i s comp i l ed. Thus, three s i t ua t i ons are poss i b l e when a variable i s entered in t he name t ab l e. F i r s t, the variable may never h ave been used before in t he program. S e cond, t he variable may have appeared before in t he program but have no value ass i gned to i t. Therefore, i t is just as i t w as when the compi l er l ast saw i t.

18 L. R. Symes 16. Here a l i mi t ed compa t ab i l i t y check i s made b e t w e en the t wo uses of the var i ab l e in t he program. For examp l e, the use of a name as a l abe l and as a variable in an ari th met i c expression i s i l l ega l. Th i rd, t he variable h a s appeared before i n t he program and h a s been ass i gned a value and a complete set of a t t r i bu t es. This enab l es more checking to be p erformed. However, the name t ab l e rou t i ne mus t not d i srup t any of t he a t t r i bu t e f l ags, for if any of them are changed t he a t t r i bu t e m ay no l onger correspond to t he value assoc i a t ed w i t h t he name control b l o ck. The name t ab l e i s constructed s equen t i a l l y. This me t hod requ i res a m i n i mum amount of space, and p erm i t s the.name t ab l e to grow dynam i c a l l y. The name t ab l e i s expanded by removing p ages permanen t l y from rea l m e mory. Th i s me t hod of name t ab l e cons t ruc t i on does requ i re t ha t the name t ab l e be searched s equen t i a l l y. bo t t o m to t op. The search goes through the n a m e t ab l e from Th i s i s done b e c ause frequen t l y the grea t es t percen t age of references to a var i ab l e occur in the i m med i a t e v i c i n i t y of i t s d ef i n i t i on. A var i ab l e wh i ch i s dec l ared to be g l oba l in N d i fferen t procedures has N+ l n a m e con t ro l blocks assoc i a t ed wi th i t. There i s a name control block for t he variable in t he name t ab l e of each of the procedures in which i t appears. On l y comp i l e t i me i nforma t i on and a po i n t er to t he N+lst copy i s con t a i ned in t hese name control b l o cks. The N+ l s t copy i s in

19 L. R. Symes 17. t h e g l oba l variable n a m e t able and con t a i ns a complete set of a t t r i bu t es for t he var i ab l e and i t s value or a po i n t er to i t s v a l u e. The N+lst copy of a g l oba l var i ab l e ' s name control b l ock i s placed in the g l oba l n a m e t able when t he first procedure i s invoked in which t he g l oba l variable appears, or when t he var i ab l e i s declared g l oba l on the console l eve l {the por t ion of the program not contained in a procedure). When a g l oba l var i ab l e i s added to t he g l oba l name t ab l e and i t a l r e ady appears t h ere, a check i s made on t he compa t ab i l i t y of the a t t r i bu t es. An error r esu l t s when t hey conf l i c t. O t herw i se a po i n t er to the N+lst copy i s placed in t he procedure ' s copy of t he var i ab l e ' s name con t ro l b l o ck. A count i s kept in t he g l oba l n a m e control b l ock of t he number of procedures referenc i ng the g l oba l v ar i ab l e, "/hen a g l oba l variable i s no l onger r eferenced, then i t s name control b l ock i s removed from t he g l oba l name t ab l e and the storage assoc i a t ed w i t h i t i s re t urned to the sys t e m. A procedure i s compi l ed when i t i s d ef i n ed. To perm i t it t o be l inked into the program, t he t ex t genera t ed u s e s on l y re l a t i ve po i n t ers to n a m e t able en t r i es, and al l l inking be t ween en t r i es in a procedure ' s name t ab l e i s done w i t h relat i ve po i n t ers. This a l l o ws procedure A, for examp l e, to be compi l ed as an external procedure and to be invoked e i t her d i r e c t l y from t he conso l e l eve l or from ano t her procedure

20 L. R. Symes 18. wh i ch i t self i s invoked from the console l eve l. The name t ab l e for procedure A i s placed in the name t ab l e af t er the l ast en t ry presen t l y there when i t is invoked and a base address i s set u p. Var i ab l es wh i ch are not dec l ared to be e i t her l oca l or g l oba l in an i n t erna l procedure are assumed to be known in t he containing block"' '. Af t er t he procedure i s compi l ed and a copy of i t s n a m e t able s aved, a pass i s m ade t hrough the procedure ' s n a m e t ab l e. Th i s pass goes t hrough the name t ab l e from t op to bot tom and p l a c es a copy of the name con t ro l b l o ck for each variable not dec l ared to be e i t her l oca l or g l oba l, in the name t ab l e of t he containing b l o ck. If the variable h as appeared in the containing b l o ck, a compatabi l i. ty check i s made b e t w e en the a t t r i bu t es. During execu t i on only one name control b l o ck i s used for t he value and a t t r i bu t es of a variable wh i ch i s not dec l ared t o be local or g l oba l. in t he ou t ermos t b l o ck. Th i s i s t h e name con t ro l block entry The name control b l o ck in the i n t erna l procedure i s l inked to this when the i n t erva l procedure i s i nvoked. The l i nkage i s constructed so t ha t on l y one step i s requ i red to ob t a i n t he value of the variable regard l ess of the depth of the procedure. There are three types of name control b l o cks in d i fferen t A b l ock i s e i t her a procedure or t he console l eve l rou t i n e.

21 L. R. Symes 19. m e mory areas : ord i n ary, local for l eft arrow func t i ons, and t e mporary. See f i gure 1. Temporary name control b l o cks are used to hold t e mporary resu l t s dur i ng the eva l ua t i on of an ari th met i c express i on. A central rout ine is used to decode variable name control b l o cks during execu t i on. Th i s rout ine de t erm i nes t he t ype of t he n a m e control b l ock and h and l es the l inkage b e t w e en g l oba l, and non- l o c a l, non-g l oba l name con t ro l b l o cks. Three t h i ngs are re t urned when a name control b l ock is decoded : t h e a t t r i bu t e number, t he da t a po i n t er f i e l d and the index of the array AENCBS of first word of the da t a po i n t er por t i on of the name con t ro l b l o ck. See f i gures 1 and 2.

22 L. R. Symes DATA STRUCTURES A name con t ro l block i s t he bas i c un i t of a l l data s t ruc t ures in t he sys t e m. In some cases i t ho l ds the ac t ua l v a l u es of the v ar i ab l e, and in o t hers i t con t a i ns a po i n t er t o t he ac t ua l va l ues and descr i p t i ve i nforma t i on. A name con t ro l block is made up of seven sixty-bi t words of t wentyone t wen t y b i t b y t e s. See f i gure 2. ITERATION POINTER DATA : ATTRIBUTE! POINTER I FLAGS "! NAME DATA PORTION The Layout of a Name Control B l ock F i gure 2 A name con t ro l block wh i ch deno t es a numeric scalar contai t he value of the scalar in i t s d a t a por t i on. One or t wo words of t he da t a por t ion are used depend i ng upon whe t h er t he value i s single prec i s i on r e a l, doub l e prec i s i on rea l or single prec i s i on comp l ex.

23 L. R. Symes When a name control b l ock deno t es a numer i c array, t he da t a por t ion of t he name con t ro l block con t a i ns t he ac t ua l bounds for t he array, t he dec l ared bounds for t he array (these may or m ay not have b e en specified by t he programmer), and t he number of d i mens i ons in t he array. The data po i n t er by t e of t he n a m e control block po i n t s to where the ac t ua l array i s s t ored, by rows, as a con t i guous b l o ck. The array i s stored as a con t i guous b l o ck to speed up array opera t i ons. If the da t a po i n t er by t e of the.name con t ro l block i s non zero, a copy of the array exists in secondary storage in t h e array f i l e. The data po i n t er i s then t he number of the record used to store t he array and an index in t he vector A E P A R. The v e c t or AEPAR con t a i ns add i t i ona l i nforma t i on about t h e array. Each word in AEPAR i s subdivided into three by t es. The first by t e con t a i ns the reference count for t he array. Th i s i s i ncremen t ed by one each t i me the array appears in a l ef t arrow func t i on d ef i n i t i on. The va l ues of a l l non-parame t er var i ab l es are fixed when a l eft arrow func t i on i s d ef i n ed. The use of a reference count for arrays p erm i t s on l y one copy of the ac t ua l array to be k ep t, and if the non-parame t er array variable i s ass i gned a new value the value of the func t i on w i l l not change. The second by t e contains t h e number of d i mens i ons in t h e array. And t he third by t e contains t he number of words in the array. The number of words i s equa l

24 L. R. Symes 22. t o t he number of e l e men t s in t he array t i mes the number of words in each e l e m en t. Th i s f a c t or i s one for a single prec i s i on rea l array and t wo for a doub l e prec i s i on real array or single prec i s i on complex array. If the da t a po i n t er byte of t he array ' s name control b l ock i s zero, the on l y copy of t he array ex i s t s in t he work poo l, and the array i s t he resul t of t he l ast array operat ion p erformed. The work poo l can contain anywhere from zero to three arrays. A coun t er i s kep t of the number of arrays i n t he work poo l. In add i t i on, for each array in t he work poo l t he index of t h e first word of the array, the index of the first word of the data por t i on of the array ' s name con t ro l b l o ck, and t he informat ion contained in t h e array ' s AEPAR entry i s k ep t. When an array operat ion i s to be performed a check is m ade t o see if any of t he arrays i nvo l ved a l ready exist in t he work poo l. If t hey do, -no reference to secondary storage needs to be made to ob t a i n the operands. A check i s also made to de t erm i ne if t he resul t of the prev i ous array operat ion i s an operand of the curren t array opera t i on. If i t i s not t he prev i ous resul t array must be stored t e mporar i l y in secondary s t orage. A name control b l ock wh i ch d eno t es an equals variable con t a i ns the virtual page number of the first page used to store t he internal t ex t for the express i on in i ts da t a po i n t er

25 L. R. Symes 23. by t e. The first word of each v i r t ua l page ' i s used for l inkage. The l i nk con t a i ns t h e virtual page number of t he next page used to hold t he t ext of the express i on or zero if t he page i s the l as t. When an equals variable is an operand of an ar i t h me t i c express i on t h i s i n t erna l t ext i s evaluated to ob t a i n a value for t h e equals v ar i ab l e. If a name control b l ock deno t es a scalar symbol i c l eft arrow func t i on, t he da t a pointer con t a i ns t he page number of t he f i rs t v i r t ua l page used to store t h e i n t erna l t ex t of t he ar i t h me t i c expression for the first doma i n of d ef i n i t i on. The f i rs t by t e of t he fourth word of the data por t ion con t a i ns t h e number of argumen t s of the func t i on. The first four words of the f i rs t vi tual page used to store t he i n t erna l ar i t h me t i c express i on t ext for each domain con t a i ns a set of po i n t ers. The f i rs t word i s used to l ink t oge t h er t he pages requ i red to store t he i n t erna l t ex t for t he ar i t h me t i c express i on for the doma i n. It con t a i ns the v i r t ua l page number of the next v i r t ua l page u s ed. A zero l i nk d eno t es the l ast p age. The nex t t hree words are subdivided into n i ne by t es. The first by t e con t a i ns the number of words of internal t ex t in the boo l ean express i on for the doma i n. Th i s i s used when t he boolean express i on i s being moved pr i or t o i t s eva l u a t i on. The second by t e con t a i ns t he reference count for the func t i on. If t he func t i on appears in t he def i n i t i on of ano t her l eft arrow func t i on this i s increased by

26 L. R. Symes 24. one so t ha t only one copy of this func t i on needs to be k ep t. The t h i rd by t e i s t he v i r t ua l page number of t he f i rs t page used to hold t he t ext for the boo l ean expression for t he doma i n. Th i s by t e i s zero if the doma i n has no boo l ean express i on. The fourth by t e con t a i ns the number of virtual pages that are requ i red to hold t he l o c a l name t able for t he doma i n. The l oca l name t ab l e con t a i ns a name control block for each non-parame t er variable appear i ng in the boo l ean and ar i t h me t i c expression for t he doma i n. Th i s i s n e c essary so t h a t t h e va l ue of these var i ab l es can be fixed when the funct ion i s d ef i n ed. Byte f i ve is unused. Byte six con t a i ns the v i r t ua l page number of t he first page used to hold the l oca l name t ab l e for the doma i n. Byte seven contains the number of words of internal t ext in t he ar i t h me t i c express i on for t he doma i n. Byte eight i s unused and by t e n i ne con t a i ns the v i r t ua l page number of the first page of i n t erna l, ari th met i c express i on, t ext for the next doma i n. If this by t e i s zero, there i s not ano t her domain defined for the func t i on. The v i r t ua l pages used to store the t ext for a boo l ean express i on or a local name t able are l inked t oge t her by t he first word of each p age. A zero l ink spec i f i es the l ast p age. The name control block for a scalar symbol i c equals func t i on contains t he same i nforma t i on as a scalar symbol i c l eft arrow func t i on. The t ext for t he func t i on i s also stored in a s i m i l ar fashion except that in t he first v i r t ua l page

27 L. R. Symes 25. used to store the i n t erna l t ex t for a doma i n ' s ar i t h me t i c expression by t es t wo, four and six are not u s ed. There i s ho l oca l n a m e t ab l e requ i red for an equa l s func t i on since al l non-parame t er var i ab l es appear i ng in t he func t i on def i n i t i on assume their current va l ues when the func t i on is eva l u a t ed. There i s no reference count because if an equa l s func t i on appears in the def i n i t i on of a l eft arrow func t i on, a copy of t he equa l s func t i on mus t be crea t ed. Wh i l e t he copy i s be i ng m ade t he equals func t i on i s t ransformed into a l eft arrow func t i on to insure that t he va l ues of a l l non-parame t er var i ab l es are f i x ed. If a name control b l o ck deno t es an array of symbol i c func t i ons, i t contains t he same informat ion as a numeric array name control b l o ck. In add i t i on the first by t e of the four t h word of the data por t i on con t a i ns t he number of argumen t s in each of the func t i ons. The array is treated as if i t i s an array of rea l single prec i s i on numbers^ Each element con t a i ns t he virtual page number of -the first page used to store t he ari th met i c express i on t ext for t he first doma i n of the e l e men t ' s d ef i n i t i on. If an e l e men t I s no t d ef i n ed, i t s value i s zero. The t ex t for t he def i n i t i on of each e l e men t i.s l inked t oge t her in t he same m anner as a scalar symbol i c func t i on. N APSS i s not des i gned for string process i ng but i t does al low t he user to create s t r i ngs, conca t ena t e them and assign

28 L. R. Symes 26. t hem to v ar i ab l es. to l abe l h i s ou t pu t. Th i s i s done to permi t t he programmer The d a t a po i n t er by t e of a string valued var i ab l e ' s name con t ro l block con t a i ns t h e number of t h e s t r i ng. The string number i s the index of an entry in t he string re l oca t i on v e c t or. Each entry i s subd i v i ded into t hree by t e s. Byte one con t a i ns t h e i ndex of t he start of t he a c t u a l string descr i p t i on in the string picture t ab l e. by t e contains t h e reference count for the s t r i ng. The second The r eference count des i gna t es t he number of t i mes t he string var i ab l e has been concatenated to form ano t her s t r i ng, p l us one. The t h i rd by t e con t a i ns t he i ndex of the f i rs t word of the da t a por t i on of t he name control b l ock for t he string v ar i ab l e. The string picture t ab l e con t a i ns a descr i p t i on of each s t r i ng. Severa l en t r i es compose the descr i p t i on of a s t r i ng. E a ch entry d eno t es ei ther a l i t era l s t r i ng, a reference to a prev i ous l y def i ned string v ar i ab l e, or t h e end of a string p i c t ure. An entry in t he string picture t ab l e in subdivided into three by t e s. If byte one is not aero t he entry d escr i b es a l i t era l. Byte one i s t he number of charac t ers in t he l i t era l, byte t hree i s the number of the virtual page in wh i ch t he l i t eral i s s t ored, and by t e t wo i s the d i sp l acemen t on that page t o where t h e l i t era l b eg i ns. Each word in a virtual p age used to ho l d string l i t era l s is subdivided into three by t e s. A l i t eral i s divided into

29 L. R. Symes 776. segmen t s of three charac t ers. Each segment i s stored in a by t e. If a string l i t era l w i l l not fi t in t he curren t string p age, t he l i t era l is broken. As m any segments of t he l i t eral as poss i b l e are placed in t he current page and t he rema i nder are p l aced in a new string p age. When t h i s occurs t wo entries are placed in the string picture t ab l e. This avo i ds t he prob l e m of l inking p ages used to hold string l i t era l s. The m ax i mum l ength of one string l i t era l i s 576 charac t ers. If by t e one of a string picture t able entry i s 1313, t hen the entry d eno t es t he nu l l s t r i ng. It has no. l ength and does not requ i re any s t orage, so by t e t wo and t hree are unused. If by t e one i s zero and byte t hree i s not 501, t he en t ry d eno t es a reference to a prev i ous l y def i ned string v ar i ab l e. So that a new copy of t he prev i ous l y defined var i ab l e ' s string i s not crea t ed, by t e three con t a i ns t he index of i t s entry in t he string re l oca t i on t ab l e. When t h i s occurs the reference count in t he re l oca t i on t ab l e for the variable i s increased by one. If by t e one i s zero and by t e t hree i s 501, t he entry d eno t es the end of a string p i c t ure. When a name con t ro l block d eno t es an array of s t r i ngs, i t con t a i ns t he same informat ion as a numeric array. array i s treated as a single prec i s i on rea l array. The The e l e m en t s of t he array contain t he i nd i ces of t he en t r i es in

30 L. R. Symes 28. t h e string re l oca t i on t ab l e for t he string d escr i p t i ons. If an e l e men t i s undef i n ed, i t s value i s z ero. A s can be seen from t he descr i p t i ons of t he var i ous da t a s t ruc t ures, t he pr i mary concerns in t he i r design has been to fac i l i t a t e t he i r use as operands whi l e at the same t i me reduc i ng t he a moun t of phys i ca l storage r equ i r ed.

31 R. Symes 29- ACKNOWLEDGEMENT The work was suppor t ed in par t by NSF Con t rac t GP REFERENCES 1. Cu l l er, G. J. (1968), " Mathemat i cal Laboratories: A New Power for the Physical Sciences, " Interact ive Systems for Experimental Appl i ed Ma t hema t i cs, (Klerer, M. and Re i nfe l ds, J., eds), 1968, pp K l erer, M. and M ay, J- "An Experiment in a User- Oriented Computer System, " Communicat ions ACM 7. No. 5, 1964, pp K l erer, M. and M ay, J., "A User Oriented Programming Language, " Computer Journal. 8, Ho. 2, 1965, pp R i c e, J. R., "On The Construct ion of Polyalgori thms for Automat i c Numerical Analysis, " Interact] ' ve Systems for Experimental Appl i ed Ma t hema t i cs, (Klerer, M. and Re i nfe l ds, J., eds), 1968, pp R i ce, J. R., ''A Polyalgori thm for the Automat i c Solut ion of Non-l inear Equat ions, " Proc. ACM 24th Nat ional Conference, San Franc i sco, 1969, ACM Publ i cat ion P-69, pp

32 L. R. Symes Ru l y e, A., Bracke t t, J. W., and K ap l o w, R., "The Status of Systems for On-Line M a t h e m a t i c a l Ass i s t ance, " Proceed i ngs - ACM Na t i ona l Mee t i ng 1967, p p S ch l es i nger, S. and S ashk i n, L., "POSE: A Language for Posing Prob l e ms to the Compu t er, " Commun i ca t i ons A C M, 10, No. 5, S e i t z, R. N., Wood, L. H., and E l y, C. A. "AMTRAN: Au t o ma t i c M a t h e m a t i c a l Trans l a t i on, " Interact ive. Sys t e ms for Exper i m en t a l Appl i ed M a t h e m a t i cs, ( K l erer, M. and R e i nfe l ds, J., eds. ), 1968, pp S t o w e, A. N., Wei s en, R. A., Yn t e m a, D. B., and Forg i e, J. W., "The Lincoln Reckoner : An Opera t i on- Or i en t ed On-Line F a c i l i t y wi th D i s t r i bu t ed Con t ro l, " Proceed i ngs - F a l l Joint Compu t er Conference 1966, pp Symes, L. R. and Roman, R. V., "Structure of a Language for a Numer i ca l Ana l ys i s Problem Solving Syste m, " In t erac t i ve Sys t e ms for Exper i men t a l Appl i ed M a t h e m a t i cs, (Klerer, M. and R e i nfe l ds, J., eds), , p p

33 L. R. Symes Symes, L. R., Eva l ua t i on of NAPSS Express i ons Involving Po l ya l gor i t h ms, Func t i ons, Recurs i on and Untyped V ar i ab l es. Purdue Un i vers i t y Techn i ca l R epor t, CSD TR 33, (1967). 12. Symes, L. R-. and Roman, R. V., Syntat i c and Semant i c Descr i p t i on of the Numer i c a l Ana l ys i s Programming Language (NAPSS). Purdue Un i vers i t y Techn i ca l R epor t, CSD TR 11 R ev i s ed, (1969).