mBMSm HN18ttmfiM RBDSDBHRffiMflUffi ImBIHSI B NMI hRm H H 13 ■■■■HI '»w } B HI H BB BHBfl t%® m m h ■ hOubHI HP I mam LIBRARY OF THE UNIVERSITY OF ILLINOIS AT URBANA-CHAMPAIGN 510. 84 no. Q>01- 612 cop. 2 CEMRAl CIKUIMION AMD BOOKSTACKS responsible or its J™™ a below . You ^X^»?TTi 0f $75 °° be causes for student '° r DEPARTMENT OF COMPUTER SCIENCE UNIVERSITY OF ILLINOIS AT URBANA-CHAMPAIGN URBANA, ILLINOIS Report No. UIUCDCS-R-73-611 calc2 - A PLATO IV Lesson on Differentiation by Axel T. Schreiner December 1973 Departments -of Computer Science and Mathematics University of Illinois at Urbana-Champaign Urbana, Illinois 61801 Digitized by the Internet Archive in 2013 http://archive.org/details/calc2platoivless611schr Ill ACKNOWLEDGMENT The PLATO IV consulting staff, especially Judy Sherwood and Jim Kraatz, helped me many times when TUTOR and I were on different wave lengths. After I was done programming, Rick Blomme and his staff further improved the PLATO system. Professor H. George Friedman, Jr. (as usual) read this paper and removed my hyphens. And Vivian Alsip read my handwriting and produced a perfect manuscript. Without all this help, I would still be reading in Knuth's book [1] on how to differentiate... IV TABLE OF CONTENTS Page 1. Introduction 1 2. Formula Management 2 3. Formula Representation 3 4. The Various Types 5 5. Input (readin, and postfix) 7 6. The Choice Page (options) 10 7. The Differentiation Algorithm (derive) 12 8. Interfacing with the Student (infix, and output) 18 9. Wrap up 20 10. Conclusions 21 REFERENCES 23 A Listing of calc2 24 LIST OF FIGURES Page 1. Storage Layout for the Formulas 4 2. Example of a Differentiation Situation 12 3. Example of the Differentiation Process 13 4. Layout of an Entry in the Driving Table 17 LIST OF TABLES Page 1. Types of a Node in the Formula 6 2. Codes in the Expanded Input String 8 3. The Input Parser Stack Mechanism 9 4. The Differentiation Situations 16 5. Displayed Information in a Differentiation Situation 18 6. Use of the Student Bank 22 calc2 - A PLATO IV Lesson on Differentiation 1. Introduction In the summer of 1971, I wrote a symbolic differentiator as a demonstration program for a course on computer applications in calculus [2]. This program was, in its central part, a direct and unmodified implementation of the algorithm described by Knuth [1], but it was able to differentiate all trigonometric, inverse trigonometric, and hyperbolic functions, as well as e x , and various logarithms. The program was written in BASIC. In fall 1972, I decided to recreate this program on the PLATO IV system [3], originally as a demonstration for the various stack-oriented algo- rithms used during input, differentiation, and output of a formula. BASIC had not been a good language for string- and tree-manipulation (neither is TUTOR, the language of the PLATO IV system), and the recreation was also intended to improve on the implementation of the algorithm. During the design of this PLATO IV lesson, calc2, however, I decided to make the student a participant rather than a spectator to the differentiation process. calc2 was to control a student's attempts at differentiation by guiding him through the steps and controlling each of his operations. To keep this guidance deterministic from the machine's point of view, the student must differentiate in postorder (i.e., for each operator, he must differentiate the operands before he is allowed to construct the derivative of the phrase involving the operator). An adapting control allows him to dynamically bypass some or all of the simpler steps in differentiation. Calc2 is table-driven; it only provides the routines to manipulate the data structures which represent the formula and the piece-wise derivatives; the actual differentiation is under control of a table of differentiation algorithms. It was thus possible during the design of this lesson to expand the set of differentiation situations—rules may be easily added or deleted. It is quite conceivable that a particular table might be introduced (then together with suitably chosen problems!) which only practices the chain rule, etc. Similar effects can be obtained by manipulating the dynamic bypass mechanism. In this report I intend to describe how calc2 operates. TUTOR, PLATO'S authoring language, is ever-expanding, and some coding in calc2 might now be inefficient; however, I believe the basic techniques still are acceptable. 2. Formula Management Knuth [1] presents a formula to his differentiator as a binary threaded tree. If one intends to perform a lot of (for the mathematician- user necessarily infixed) formula fragment display, and if the implementation language is not suited to tree manipulation, such a tree is not the best data structure. Instead, I decided to employ one queue and two stacks with variable sized entries. Initially, the queue is loaded with the given formula in post order (i.e., each operator follows its operands), each element of this queue containing one fragment (operator or operand) of the formula. Both stacks are empty. As the differentiation progresses, one stack is pushed with phrases of the formula, the other with the derivatives of these phrases; e.g., if a binary operator of the formula is processed, the top two phrases of each stack are combined to form a new top phrase of the derivative stack, and then the top two phrases of the formula stack and the binary operator are combined to form the new top phrase of the formula stack. For display and judging purposes, these processed phrases are maintained with binary operators infixed between their operands and parentheses are employed. Constructing an infixed formula 'bottom up' allows one to insert the minimal number of parentheses without any effort, provided the actual data structure underlying the formula is flexible enough to accommodate these parentheses without expanding in size. 3. Formula Representation PLATO IV operates on a Control Data Cyber 70 computer; the word size is 60 bits, and at the time calc2 was written, TUTOR did not provide for the subdivision of a word. Bit operations (logical and, or, and circular shift) were available, as was a one-dimensional array facility. A student in the PLATO IV system has 150 "student variables" available, and it is the lesson author's responsibility to manage those variables; no dynamic variable allocation is provided in TUTOR. I decided to allocate the queue and both stacks in one array. The queue occupies the beginning of the array. As elements are disposed of from the beginning of the queue, the formula stack overlays these elements. If we accommodate the parenthesized representation at no increase in space require- ments, the queue will not have to be moved. The derivative stack follows the queue in the array. This method is destructive, but once differentiation is complete, formula and derivative exist infixed and either one could then again be converted to post-order to repeat the process. n75 (a TUTOR variable) i formula stack tformel * item queue deriv nl45 tderiv derivative stack this slack results from simplification tformel: pointer to top of formula stack tderiv: pointer to top of derivative stack deriv: pointer to bottom of derivative stack (one element beyond queue) item: pointer to current element in queue Figure 1. Storage Layout for the Formulas Elements in the queue are one word long, and these "nodes" have the following fields: M° type prin leng Ipar rpar wher info: bits 1...8 type: bits 9... 14 wher: bits 39... 60 (pointer to) auxiliary information type of this element begin position of this element in the master display line These elements are essentially copied from the queue into the stack, and then the following fields are also used: rpar: bits 34... 38: number of ) following this node lpar: bits 29... 33 number of ( preceding this node leng: bits 21... 28: length field poin: bits 15. ..20: principal type field The length and principal type field implement the variable sized stack entries and allow simplification: the last node of a phrase contains in its leng field the length of that phrase, i.e., the number of nodes constituting it, and the prin field indicates the principal type of the phrase. This principal type is defined to be the type of the root node of the formula tree into which the phrase could be expanded. If the phrase is a function evaluation, the node representing the function contains in its prin field the principal type of the function argument. It is thus possible to "collapse" the composition of a function and its inverse—like two successive unary minus operators. 4. The Various Types calc2 allows its user to specify the formula which he wishes to differentiate. The user may use 26 variables a...z, the binary operators +,-, *,/ and ** for exponentiation, the unary minus operator -, and the functions sin, cos, tan, cot, sec, esc (trigonometric), sinh, cosh, tanh, (hyperbolic), asin, acos, atan (inverse trigonometric), and In, exp, sqrt. Parentheses may be used. Following TUTOR conventions, multiplication takes precedence over division. At most points in calc2, the standard TUTOR formula evaluation mechanisms can be employed; these permit juxtaposition of variable names to denote multiplication, superscript to denote exponentiation, and omission of (most) parentheses in function application; however, in order not to overly complicate the input scanner for the given formula, it must be specified in a more rigorous, conventional syntax; we assign the unary minus the highest precedence so that the phrase a**-b would be legitimate. Upon input we have two types of terminal operands: numerical values which for simplification purposes are assigned the types 0, 1, and v, and variables which are assigned the type c. Only during differentiation do we make the further distinction between type c for constant variables and type x for the differentiation variable, type type name info 1 x(variable) poi 2 c(constant) poi 3 (zero) poi 4 1 (one) poi 5 v (value) poi 6 + 7 - 8 * 9 / 10 ** 11 -(unary) 12 In 13 exp 14 sin 15 as in 16 cos 17 acos 18 tan 19 atan 20 cot 21 sec 22 esc 23 sinh 24 cosh 25 tanh 26 sqrt nter to variable storage at random value for x nter to variable storage at random value nter to constant storage at value nter to constant storage at value 1 pointer to constant storage at value Table 1. Types of a Node in the Formula (to be constructed by the input-parser) In TUTOR, characters a through z have the internal codes 1 through 26; we allocate variable storage (used for randomization during answer judgment) into variables vl through v26 and thus have the info fields of types x and c also store the character associated with the node. Values pose a bigger problem. When judging phrase answers I randomize all variables (types c and x), evaluate the correct phrase known to the lesson and the answer phrase given by the student, and compare. In order to evaluate efficiently, I intended to use TUTOR'S "compute" command which takes a character string, compiles it (it must be a formula), and evaluates it. Such a formula may contain the usual operators, numbers, and "defined names". Names are defined with a "define" statement which assigns at compile time either a variable location or a constant as a string replacement for the name. Treating the variables a...z is simple: they are defined to be variable locations vl through v26, and before using compute, random numbers can be placed into those locations. How should I maintain values? There exists, of course, a table of values, and the info field of types 0, 1, and v, points to an entry in this table. But TUTOR does not provide a conversion from floating point to character string in memory. I could have stored values as character strings, but that would have severely limited the simplification techniques which may introduce new values. I, therefore, "defined" the names A, B, ... for the locations of the value table, and in phrases presented to the "compute" command, these names are used instead of values. This meant then that output of these phrases to the screen had to be accomplished interpretively: strings between values could be shown directly, values had to replace their names in the phrase strings. 5. Input (readin, and postfix) The input parser accepts a formula and constructs the queue from it; 8 the formula is in infix notation and in a conventional syntax, the queue is in postorder, parentheses-free, and encoded into nodes. The value table must also be constructed. The input parser has two phases: first the formula string is converted from a character string into a sequence of codes, one per computer word. Blanks are eliminated and illegal characters are rejected; function names including left parentheses are verified. The codes are outlined in Table 2. They overlay the area which is then occupied by the (more compact) queue. Code Meaning 1 ... 26 variable a . . . z 27 + 28 - 29 * 30 / 31 ** 32 the next compute 33 the next compute (this always indicates a left parenthesis) 34 1 subcode meaning 1 ( 2 ... 16 functions of type = subcode + 10 Table 2. Codes in the Expanded Input String The second phase of the input parser is a syntax analyzer, driven by an operator precedence grammar. A stack is used and the coded formula is converted into the postorder nodes in the queue. This part of the parser is straightforward and I therefore just present it as Table 3. state next symbol code action next state expect operand variable 1 ... 26 emit expect operator value 32 emit and enter into value table expect operator function or left paren 33, si bcode stack, increase # of parens expect operand + 27 ignore unary plus expect operand - 28 stack or collapse unary minus expect operand other SYNTAX error retrieve new formula expect operator T J > 5 / > 27 ...31 emit from stack and push by precedence expect operand ) 34 emit from stack decrease # of parens expect operator end of formula if no parens open exit other SYNTAX ERR0R retrieve new formula Table 3. The Input Parser Stack Mechanism TUTOR is not overly cooperative in constructing either phase of the input scanner: the formula is read as a student response to a TUTOR arrow, and at least conversion to the first internal representation should have been possible by using TUTOR'S rather eye-pleasing array of answer-judging techniques. Un- fortunately, no coding could be devised to overcome certain time slicing problems 10 which still exist: answer judging may not take more than 25ms of computer time or the user will be swapped and some of the judging material is lost. I was unable to determine whether my code was troubled by bugs in TUTOR or whether I was violating a principle of the language. Consequently, the first phase is accomplished by "brute force", the characters of the given formula are spread out, one to a computer word, and then analyzed. The coding is certainly in- efficient, FORTRAN-style, and TUTOR'S limits on the size of a program segment ("unit") were felt to be a severe restriction. The second phase of the scanner charged with creating postorder is simple, except for the treatment of the wher field in each output node: even- tually our lesson will present on each page a differentiation situation taken from the given formula. This formula is also shown on each page, and an arrow indicates the location of the presented situation in context. We thus must maintain with each node an indication where it is displayed in the formula. This information is maintained in the wher field, and it must be stacked for operators during the postorder conversion. The information is obtained by writing the element onto the screen (with a standard-length format for constants) and observing how much space is needed for writing. (Theoretically, TUTOR'S screen location variable "where" could be used, but "where" had not yet been completely debugged.) In order to generate the given formula display efficiently on each new page (since a selective screen erase conserving the formula turned out to be too slow), it is generated and saved in infixed form in memory, with values re- placed by capital letter-defined names, while the endorder queue is constructed. 6. The Choice Page (options) Having successfully presented a formula to the system, the user is 11 given four choices: he may read a new formula (implemented as a return to the reinitialized input scanner), replace a variable by a value, evaluate or dif- ferentiate the given formula. The replacement is implemented as a simple substitution: an entry in the value table is made and in all nodes referencing the variable, the type and info fields are changed to indicate a value and to point to the entry in the value-table. All wher fields to the right of the value must be updated to allow for the display of the value, and in the infix representation of the formula, the letter variable must be replaced by a capital letter value reference. (Since in TUTOR such a capital letter takes two six-bit codes, the first of which indicates capitalization, the replacement would be rather cumbersome. Instead, each letter variable in infix is preceded by a blank character, and this blank-code then can be overwritten with a shift-code during replacement.) Evaluation is implemented by presenting the infix string to the compute command. This, of course, must be preceded by assigning user-supplied values to the variables in the given formulas; these values are stored into those locations in the student bank which the defined names for the variables reference. The differentiation driver requests the name of the differentiation variable and changes the type field of all nodes referencing this variable to indicate type "x". Control then passes to the differentiation driver. A special, non-published choice is also available; differentiation can be requested to be automatic, bypassing any questioning of the user. This is accomplished by setting the request control for the questioning to a special value as will be outlined below. 7. The Differentiation Algorithm (derive) Consider the differentiation situation in Figure 2: 12 leng ^_ prin ** ** 1 * tformel item tderiv V DU DV Figure 2. Example of a Differentiation Situation item selects a multiplication operator, the corresponding operands are (x+1) and (x**2), both have non-zero derivatives (assuming a differentiation with respect to x), the corresponding differentiation rule is the product rule D X (U*V) = DU*V+U*DV . With the aid of the leng fields, the phrases U,V,DU,DV can be located from the top of the respective stacks; the relevant pointers have been added to the figure. The 'algorithm* described in Figure 3 will produce the correct de- rivative (the figure shows the stepwise results in the derivative stack only). To complete processing the item '*', we now have to infix it into the formula stack. This, however, can be accomplished by the same routines which perform multiplication in the derivative stack (an overflow of the formula stack is impossible). 13 insert top formula phrase below top of derivative multiply below top of derivative 1,2 insert below top formula phrase on top of derivative multiply on top derivative 1,3 1 X ** 2 2 * X 1 ** * + tderiv X ** 2 2 * X ** * tderi V X ** 2 2 * X X + 1 ** * + tderiv X ** 2 2 * x * (x + 1) ** + tderiv X ** 2 + 2 * X * (x + 1) add on top_ derivative + t tderiv 1 any operation is done by infixing the operator between the relevant phrases 2 a multiplicative factor of '1' can be discarded. 3 since the precedence of the second phrase (principal type '+') is lower than the precedence of the operator to be infixed, '*', we add parentheses to the second phrase. Figure 3. Example of the Differentiation Process 14 The differentiation machine is built from eight processors for machine operations: perform the indicated binary operation to the indicated stack element and the stack element below as right and left operand respectively. perform the indicated function (unary operator) on the indicated stack element. copy the indicated formula stack element to the indicated position in the derivative stack. create the indicated node into the indicated position in the derivative stack. The first six processors can operate on either the function or the derivative stack--they don't know the difference. Each will first attempt an appropriate simplification (discard additive zero, multiplicative or power one, evaluate any operation on two constants, collapse inverse function composition) and if it fails, will infix the operator and correct the result by parenthesization if necessary. These processors are slightly more generalized than is needed for their application in the differentiation machine. This differentiation machine therefore is implemented with a restricted machine language to access the processors: (it should become clear below, why this machine only seems to operate on the derivative stack.) 15 add sub mul div pow top below top fun node copy top below top _ top below top . top below top of derivative stack kind top below top to derivative from formula stack stack The machine selects (by inspection of the item node and DU and DV, whichever is appropriate) one of 28 differentiation situations (compare the table below). The machine's driving table then contains a differentiation algorithm, written in 'machine language', for each differentiation situation. This algorithm is executed by driving the processors to produce the derivate phrase corresponding to item. 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Supplementary Notes (Abstracts calc2 is a lesson differentiation operating on the PLATO IV computer-aided instruction system. A symbolic differentiator guides the student through all steps of the differentiation of a student-chosen formula. The report describes the details of the table-driven symbolic differentiator. 1 Key *orls and Document Analysis. 17a. Descriptors calculus computer-aided instruction symbolic differentiation derivative Identifiers Open-Ended Terms II Field/l ■ iiemi-nt RELEASE UNLIMITED T»M NTlS-35 II0-7C 19. Security Class (Thi- Report i _, UNCLASSIFIED 20. Security Class (This Page UNCLASSIFIED I?}. >o. >[ I'M 84 ??. USCOMM-DC 4G329-F- "*. ' in iww H UNIVERSITY Of ILLINOIS URBANA 3 0112 064441527