■ ■fa mm Mm: BMOBBMHMflSVSj' HbH I BBBB BBS ■QHHH ■ mmm ■ JcW EKES #W&$£- iw<«. LtJ-vi I^H rtr.'l K>; +:*>:■:• m Wmm ■ i^h ,\v •:■>>:■ ^B mm wtiSfSSEl WWfSmi ■ I H iRii SHn Si *MflW CrBW cgBBiMK'yvffljyw ■nnffi HHHHHIffl wXmm LIBRARY OF THE UNIVERSITY OF ILLINOIS AT URBANA-CHAMPAIGN 510.84 vio.559-564 COJD. ^ Digitized by the Internet Archive in 2013 http://archive.org/details/algorithmmodels562hoff UIUCDCS-R-73-562 February 1973 ALGORITHM MODELS by DONALD JAY HOFFMAN DEPARTMENT OF COMPUTER SCIENCE UNIVERSITY OF ILLINOIS AT URBANA-CHAMPAIGN URBANA, ILLINOIS UIUCDCS-R-73-562 ALGORITHM MODELS* by DONALD JAY HOFFMAN February 1973 DEPARTMENT OF COMPUTER SCIENCE UNIVERSITY OF ILLINOIS AT URBANA- CHAMPAIGN URBANA, ILLINOIS 6l801 This work was supported in part by the Joint Services Electrical Program (US Army, US Navy, US Air Force) under contracts DAAB-07-67-C-0199 and DAAB-07-72-C-0259; advised by Prof. S. R. Ray, Department of Computer Science and submitted in partial fulfillment of the requirements of the Graduate College for the degree of Master of Science in Computer Science. Ill ACKNOWLEDGMENT The author would like to express his great appreciation to Dr. Sylvian Ray for his help and guidance in conceiving and working out this problem. My appreciation also goes to Mrs. Barbara Bunting for typing the manuscript, IV TABLE OF CONTENTS Page 1. INTRODUCTION 1 2. MODEL GENERATING AND IMPLEMENTATION CHECKING PHILOSOPHY . . 2 2.1 Approach 2 2.2 Method 2 2.3 Implementation Analysis 3 2.k The Algorithm Checker (AC) 3 3. EXAMPLE - METHOD OF FALSE POSITION k 3.1 Description of the Algorithm h 3.2 Create the Model k k. THE ALGORITHM CHECKER 10 k.l General Physical Description 10 k.2 Implementation Method of False Positions 10 5. CONCLUSION 16 APPENDICES 1. Implementation Restrictions - method of false position. IT 2. The Algorithm Checker Processing Some Test Implementations of the Method of False Position 19 3. The Algorithm Checker Program 32 1. INTRODUCTION A problem has always existed in basic computer science education in training students to take a problem and translate it into a working solution on a computer. This process can be broken down into three steps: 1. Create an algorithm 2. Make a flow chart of the algorithm 3. Code the algorithm A major area of intellectual difficulty is the process of conceptualizing an algorithm in a form that can be easily translated into a flow chart. There has not been and perhaps never will be a rigorously prescribed way to approach this problem. The human mind performs these processes in a way yet unknown. What I will try to do is treat the human mind as a black box and perform the same operations by creating a system of logical cognitive models of the required operations in taking an algorithm and translating it into a form suitable for coding. I will then demonstrate these cognitive models by demonstrating a method of checking the logical construction of implementations of some algorithms by using these models as part of a system for automatic fault detection. My specific approach will be: 1. Develop a generalized heuristic procedure for generating conceptual models of algorithmsj 2. Develop these conceptual models for some common algorithms. 3. Demonstrate step 1 and step 2 by using these models in a prepro- cessor written to perform automatic fault detection on implementa- tions of these algorithms* 2. MODEL GENERATING AND IMPLEMENTATION CHECKING PHILOSOPHY Decide which features are important and devise a technique for algorithm detection „ 2.1 Approach Separate the algorithm into a series of partitions that are each a logical operation and are easily physically separated. If the partitions are large and/or very complex then perform the separation operation again, and so on. 2.2 Method Represent the algorithm as a Type 3 language represented by a series of productions. Describe the language using several levels of non-terminals - the lower the numerical level assigned to the non-terminal the more complex is the sublanguage the non-terminal represents. 2. 3 Implementation Analysis Define a function, plausibility , whose range is between and 100, and whose initial value is 100. The plausibility will be an estimate of the likelihood that a specific implementation of an algorithm is an accurate mechanization of the algorithm in question. The plausibility is determined by two factors : 1. The existence of certain features of the algorithm in the implementation in question. 2- The physical location, precedence, of the various features in this implementation. The existence and precedence relations can both be discovered by analyzing the productions in the model. Each feature is assigned a step constant whose range is between 0.0 and 1.0. It is an estimate (heuristic) of the relative impor- tance of this feature in the algorithm model. The absence of this feature or its physical misplacement in the implementation shall suggest an error in the implementation. When checking the implementation, should such an error condition arise, the plausibility is reduced by multiplying it by the associated step constant. 2.1+ The Algorithm Checker (AC) The Algorithm Checker is a program written for the IBM 360/75 computer in SN0B0L language. It implements the above techniques to perform analysis of the implementations of some specific algorithms. The input to AC is a student program, coded as a FORTRAN subroutine intended to mechanize an algorithm which AC has been written to check. If AC is told which algorithm to check for AC will be able to: 1. Estimate the plausibility that the algorithm is being correctly implemented or not. 2. Recognize and comment on logical errors in the input program. 3. EXAMPLE - METHOD OF FALSE POSITION 3.1 Description of the Algorithm Method of False Position One approach to the solution of non-linear equations will be to use linear inverse interpolation to derive methods of the form: X... r \ X . X. n » ■ i • ■ X. _ J • i+l i i-l i-n+1 One such method is the method of false position . Suppose we can find two points x. and x_ such that f(x )*f(x )<0« The chord joining y and y inter- A rs A r> A r> sects the x axis at a point x„. Then chosing "X-. and x. (either x or x ) such that f(x^)*f(x. ) 0. We repeat the same procedure to obtain a new x^ and so on Terminate this process when f„ is sufficiently small. 3.2 Create the Model A. The method of false position algorithm can be broken down into steps: 1« Initialization 2 . Loop 3. Terminate These steps should occur in the above order both logically and physically with a high probability and the algorithm can be represented by the following flow chart : ( ^ START J INITIALIZE LOOP TERMINATE V STOP ) Or it can be described by this production: < false position> ::= < initialize:* < loop>< terminate> Assign level zero to the non-terminal on the left hand side and assign level one to the three non-terminals on the right hand side. From this production we can immediately derive a precedence relation- ship where the three steps should appear in this specific order with a very high liklihood. Precedence Table N. B AN. Initialize Loop Terminate Initialize -1 -1 Loop 1 -1 Terminate 1 1 i . , -1 A must precede B don't care +1 B must precede A B. The initialization step should perform these operations: 1. The definition of the non-linear function, F. If this operation is present, in FORTRAN, it would most likely be a function state- ment. 2. Make first two guesses at the root of F, X and X . 3. Calculate the values of F at X and X , f. and f g . The initialize step could be represented by the following flow chart: V START J \r DEFINE (A) FUNCTION V MAKE INITIAL GUESS X A (B) A " MAKE INITIAL GUESS X^ (C) J/ jf A = f(X A ) (D) i f B = f(X fi ) (E) \if GO TO LOOP However the possible orders of operation for the various operations in the initialization step are not unique; there could be several different flow charts for the initialization step. The initialization step can be represented by just one set of productions: < initialize> ::=< define function> < initialize X > (< initialize X > (< calculate f A >< calculate f > ) -D A _D ( < calculate f > < calculate f > ) ) (< calculate f > < initialize X > < calculate f > ) A rs rs This production is unique. The left-hand side of the production is level one, from before, and the non-terminals on the right-hand side are assigned level two. From this production we can derive the following precedence relations: PRECEDENCE TABLE X Y A B c D E A -1 -1 -1 -1 B 1* -1 C 1* -1 D 1 1 E 1 1 +1 Y must precede X don't care -1 X must precede Y * note that step A precedes steps B and C strictly due to FORTRAN syntax requirements. C An iterative process can be described by a generalized loop step, This generalized loop step can be represented by the following flow chart. Q ENTER J Step A: perform some operations Step B: are the termination conditions satisfied/ yes tKexit) no Step C: perform some operations Note that either step A or step C may be null. However for the method of false position both steps should exist. 8 2. 3. The loop step should perform these operations: 1. Calculate the next guess at the root of F, X^. Calculate the value of the function F, at L, f Check the value of f , (or some other function of L) if it is small enough , then exit the loop . h. Decide which old value of X to replace for use in the calcula- tions in the next pass through the loop. Then replace either X or X and the associated f or f with X and f respectively. A £5 A d N N Substituting into the generalized loop step above, the loop step of the method of false position can be represented by the following flow chart: From initialize step Perform inverse interpolation to find value of new *■ Using value of X^ Calculate new value of f N Are the termination conditions satisfied? (if abs(f ) small enough? ) t no Replace old value of X and f with X and f . yes To Termin- ate Step Each operation in this flow chart can be expected to be found in the logical position corresponding to its position in the flow chart with a high probability, The physical position os these operations should also be similar to the flow chart with a somewhat lower probability. The operations that must be performed to satisfy the requirements of the box are physically unique; there is more than one way to perform any operation. The loop step of the method of false position algorithm can be represented by the following production: < loop> ::= < calculate new X , calculate f from X > < branch if f small enough> < test signed f , replace X and f> From the above production we can derive a precedence relationship for all elements of the loop step. In this case the flow chart above is the only probable way to order the steps. D. The termination step of the algorithm is not well defined. The only requirements on it are exclusively syntactic. The termination step must be ended by a RETURN statement followed by an END statement. 10 k. THE ALGORITHM CHECKER U.l General Physical Description The Algorithm Checker program will be coded in IBM/360 SN0B0L IV. It consists of a supervisor program which calls various subprograms written to check for specific features. AC will search for the specific features of an algorithm in a sequence of operations derived from the productions in the algorithm model. AC will first separate the major program sections (level 1 non-terminals) and then search these major program sections for subsections specified by- level 2 non-terminals, and so on. While this is being performed AC will also be continuously updating the plausibility of this implementation each time an error condition occurs. k.2 Implementation Method of False Positions A. Level 1 The first thing AC must do is separate the input program into the three distinct steps specified by the production; < false position> ::= < initialize> < loop> < terminate> This step breakdown operation can most easily be performed by first looking for distinctive physical code structures in the various steps. The most dis- tinctive and easiest-to-detect feature is the structure of an iterative opera- tion. In the method of false position there is only one iterative operation expected, the loop step. The loop step is found by scanning the input program for one of these patterns : 11 Pattern 1 * IF(A < relational operator> B)GOTOn Pattern 2 IF(A < relational operator> B)GOTOm GOTOn m There are other more complex ways of coding iterative operations but for now I will assume that only one of these more simple patterns has been used. Should a pattern match be successfully performed, it will be assumed that all code physically preceding the loop step is the initialize step and that all code physically following the loop step is the terminate step. If the loop step cannot be successfully isolated it becomes extremely difficult for AC to proceed beyond this point and obtain any meaningful results, So, if the loop step cannot be detected and isolated, AC will reduce its plausibility estimate to a very low value and will terminate processing this implement at i on . * < relational operator> ::= -LT. , .LE. , .GT. , .GE 12 B. Level 2 - Initialize The initialize step was defined by the productions: < initialize> : := < define function> < initialize X > A ( < initialize X > ( < calculate f > < calculate f > ) r> A is ( < calculate f_> < calculate f > ) ) J a A ( < calculate f > < initialize X_> < calculate f > ) The left-hand side of the production is a level 2 non-terminal and the right-hand side is composed of level 3 non-terminals. The level 3 non- terminals, in this case, will all reduce to terminals. The terminals for this language are all equivalent to one or more FORTRAN statements. < define function> : := { var (var 2 ) = exp(var ? )} is one such level 3 production, var, is the name of the function of var~ is the name of the functions dummy variable. Similarly < initialize X > ::= {var... = exp} ' {var = const} < initialize X > ::= (var, = exp} | { var, = const} < calculate f > ::={var c; = var n (var )} A j A. j < calculate f > ::={var/ = var (var, )} Whe re var._ is X , var, is X^, var,- is f , var^- is X , exp is any valid FORTRAN arithmetic expression, exp(var ) is any valid FORTRAN arithmetic expression which contains var , and const is any valid FORTRAN constant. All complete FORTRAN statements are enclosed in braces. We scan the code searching for the patterns specified by the above productions. If we have pattern matches we assign the names of the variables 13 in the code to the variables X. , X^, f. f_, respectively. If any of the pat- A a A , is tern matches fail, or the operations are out of correct order, the plausibility is reduced by multiplying it by the appropriate step constant. If DELTA, the loop termination control value (see next section), is initialized in the initialize step. There are some additional complications caused by certain ambiguities. We must diverge from a completely systematic approach and use some additional heuristics and first determine the additional value of DELTA. C. Level 2 - Loop The logical expression in the IF-GOTO , {IF (logical expression) GOTOn}, statement which controls loop termination should reduce to the form: (A. LT. DELTA) where A is a positive valued expression and DELTA is a positive constant such that when the expression is true the loop step is exited. However, there are several equivalent forms of this expression which are not immediately identifiable. For the expression A.GT.B. assuming that A and B are variables of date type REAL. Here is a list of some equivalent expressions: A.GT.B A.GE.B B.LT.A B.LE.A .NOT.(B.GT.A) .NOT. (B.GE.A) .NOT. (A.LT. B) .NOT.(A.LE.B) A-B.GT.O A-B.GE.O B-A.LT.O B-A.LE.O I will consider the relational operations .LT. equivalent to .LE., and .GT. equivalent to .GE. because for real numbers of finite length due to computer Ik truncation error, the relational operators . LT. and .LE., and .GT. and .GE. , are indistinguishable. The loop step can be represented by the following production: < }oop> ::=< calculate new X , calculate f from X> < branch on condition satisfied> < replace old X and associated f> The level 3 non-terminals on the right-hand side of the above reduce to: < calculate new X^, calculate f from X> :: = < calculate X > < calculate f > , -, i v t X A f A " X B f B . < calculate X> : := { var = — } ' A " B where var„ is X,. T and X. , X_. , f . , and f^ were defined in the initialization I a A £5 A £5 step. < branch on condition satisfied> ::={IF (logical expression) GOTOn} < calculate f R > ::={ f^ = f(Xjj)} < replace old X and associated f> ::= < test> < replace a> { GOTOm} < replace B> < test> ::=< IF (logical expression) GOTOn} < replace a> ::= ({X A = X N > { f A = f }) ({f A = V {X A =X N }) < replace b> : := ({X fi = X N ) { f fi = f^}) C{f B = f N HX B = X N }) If the loop control IF - GOTO statement contains a logical expression that does not reduce to the form of A < relational operator> B it is not accept- able and is rejected. 15 Within the loop the termination value DELTA should be a constant. We select it from the positional and structural relationships defined in the productions. If logical expression is of the form { X.LT. DELTA} where DELTA is a constant, then we assume we have found the value of DELTA. If DELTA is a variable we must check the initialize step for an expression of the form { var = const.}. Where var is DELTA then the con- stant is the value of DELTA. If we cannot find the value of DELTA we assume DELTA undefined, and we reduce the plausibility by an appropriate amount. Next we will check the calculation of )L by performing a match of statements to see if any are mathematically equivalent to the production usini the proper variable names. Then check to see if its followed by the calcula- tion of f . Next check to see if the replacement of old X and f is properly per- formed by checking for pattern matches with the productions. Again, if there are failures in the pattern matches or order of operations we again reduce the plausibility by an appropriate step constant. 16 5. CONCLUSION The Algorithm Checker is operational for checking implementations of the method of false position algorithm implementations. It has proven to be capable of parsing the implementation into specific steps, isolating specific variables, and making a reasonable guess as to whether a given implementation will in fact perform a false position root finding function. IT APPENDIX 1 Implementation Restrictions - method of false position 1. Algorithm Restrictions A. Assume termination within a reasonable number of iterations; therefore there is no need to count the number of steps. B. Assume that the first guess of X and X^ will give values of f A d A and f_ such that (f. * f D ) < 0. C. Starting values of X. and X^ will not yield zero values of f. & A B J A of f B . D. The termination conditions are reasonable. E. The termination value, DELTA, is a constant defined within the input program. 2. AC Input Restrictions Student programs should observe these constraints; A. The input program must be coded in syntacically legal IBM 360 FORTRAN IV. B. The overall structure of program is that of a FORTRAN subroutine. C. All functions will be defined by function statements. D. All arrays are dimensioned. E. The arithmetic IF is not allowed. 3. Output from AC The output from AC will be a printed copy of the input program (above) and comments for AC as to: A. Program variable names and their relation to the algorithm. B. Comments as to coding accuracy. 18 C. AC's estimate of the plausibility that the correctly implemented. D. Recommended corrections for the input program. U. Notation Conventions X , X_^ The two values of X used to calculate the next estimate of the root. X^ The new estimate of the root. f The function whose root we're interested in. f(X) Calculate the function at X. f . , f_ The values of the function at X and X . 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X CO » » C U' co • • a «i a Uj • • -» CC UJ w — II >• X jt _l _J • I a. a. co <_> ►- I — 3 co »» » • OUUi X 3- X | i_j • • « II MHO O I 3 — — I— »- I- CO l-Ht-3-33 • z z o. nan UJXUiI-Mt-1-C O IC3 •33« — O^-OUCOO- co 3 o c CO — - JL Uj o ./£ -» M O or — »- CO CO c_ o. n — C fvi a a. — i a. a T jr. UJ UJ X I- I- J- — I "Ml I a. JL ♦ I. UJ V. U. »- — M 5* O «J r\j Z UJ4- u. I a 00 UJ I— CO u u U u < < < it -J -J mi UJ o_ a a a KJ LU UJ uj Z U. CL at ♦ at uj z z o o ♦ U (J ♦ • r *o h~ cd a O <^ M i^ »t ui «o h- co nrompi n ^ * *-• <-* IBLIOGRAPHIC DATA HEET 1. Report No. UIUCDCS-R-73-562 3. Recipient's Accession No. 5. Report Date February 1973 Title and Subtitle ALGORITHM MODELS 6. . Author(s) DONALD JAY HOFFMAN 8. Performing Organization Rept. No - UIUCDCS-R-73-562 Performing Organization Name and Address Department of Computer Science University of Illinois at Urbana-Champaign Urbana, Illinois 6l801 10. Project/Task/Work Unit No. 11. Contract /Grant No. 2. Sponsoring Organization Name and Address 13. Type of Report & Period Covered Thesis Research 14. i5. Supplementary Notes 16. Abstracts The human mind performs in a way yet unknown. What I will try to do is treat the human mind as a black box and perform the same operations by creating a system of logical cognitive models of the required operations in taking an algorithm and translating it into a form suitable for coding. I will then demonstrate these cognitive models by demonstrating a method of checking the logical construction of implementations of some algorithms by using these models as part of a system for automatic fault detection. 17. Key Words and Document Analysis. 17a. Descriptors algorithm automatic fault detection heuristic procedure 17b. Identifiers/Open-Ended Terms 17c. COSATI Fie Id /Group 18. Availability Statement unlimited distribution 19. Security Class (This Report) UNCLASSIFIED 20. Security Class (This Page UNCLASSIFIED 21. No. of Pages 6o 22. Price FORM NTIS-35 ( 10-70) USCOMM-DC 40329-P7I 1CT 29 1973 ModuOr mlcrop.ogr.mm.d oomp _^Himi. = ■ '""a 0112 088400566 PL. i!^^^ B8900HH HMDe^HhqSBIB HMflHOBH BM WClgw H IBB MJMMSObHH