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Rationalized APL\PC Rel. T1.4 7 February 1989 Network Distribution (c) Copyright 1988, 1989 by NGARG, Zurich All rights reserved. Author: A. Werder Rationalized APL\PC is an experimental APL interpreter model that implements the concept of Rationalized APL and its successor described in "A Dictionary of APL" by K.E. Iverson (APL Quote Quad, Volume 18, #1, ACM). Although it does not entirely implement all of the Dictionary APL the interpreter provides the basic functionalities of a future 3rd generation APL. DISCLAIMER: This experimental software and documentation is provided on a "as is" basis with no warranty whatsoever. This product is provided as freeware; its use for commercial advantage is strictly prohibited. 1. Installing the Interpreter The requirements for running the Rationalized APL\PC interpreter are: an IBM/PC-XT or AT (or compatible) with 640kB main memory, a floppy disk (for the installation, if source is obtained from disk) and a hard disk. Any PC/DOS release above 2.1 is acceptable. PC Files for Interpreter There are 3 files required for running the interpreter: RATAPL.EXE RATAPL.SYS PROLOG.ERR The actual code which includes a minimal TurboProlog run time system is contained in the file of type .EXE. The definitions for the interpreter such as function and operator parts are contained in the file RATAPL.SYS. This is in fact the prolog data base in the form of an ASCII text file. The third file simply contains error messages required for the TurboProlog run time system in case of a failure. It is suggested that these files are installed in a suitable directory; for example, C:\RATAPL. This documentation assumes that Rationalized APL\PC is installed in that directory. To start the Rationalized APL\PC, simply type <CD \RATAPL> and then <RATAPL>. To see a short demonstration, start Rationalized APL\PC and type <)load demo>. 2. Implementation Details APL Keyboard and Display The keyboard mapping for APL and ASCII characters is based on the so called "union"-keyboard. The first alphabet is ASCII lowercase, the second alphabet (normally APL underbarred characters) is upper case. The APL characters can be generated by pressing ALT plus the corresponding key. A number of APL characters and overstrikes can be generated by pressing a key together with the CTRL-key. The four cursor keys as well as the <HOME>, <END>, <INS> and <DEL> key provide some simple editing facilities. In particular, cursor <UP> recalls the previously entered line for re-editing. TurboProlog resets the EGA character set to the default. The supplied workspace "ega" may be used to load the APL character set. Type <)load ega> to load the EGA APL character set. APL Variables: Limitations on Type, Rank and Size Only the 4 basic data types "character", "integer", "real" and "enclosed" have been implemented. For the purpose of internal rationalization the data types "string" and "token list" have been added. The rules for typing have been relaxed so that heterogeneous arrays are now allowed. Arrays may be heterogeneous in class and type. It is possible for instance to mix numbers and functions. Class and type of an array are given by the class and type of the first element in the array. Note that still most of the primitives require homogeneous arguments. Boolean variables are represented as integers. Integers are 2 bytes long. Type coercion between integer and real normally occurs if a scalar function is used. Some functions such as modulo or binomial which use built-in TurboProlog predicates however do not detect an integer overflow so that the result of such an operation can lead to incorrect results. The absence of integer arithmetics for real numbers in the underlying TurboProlog compiler unfortunately also limits the domain of a number of functions (for instance encode, decode etc.) to 2 byte-integers. Complex numbers are not implemented and consequently they neither are within the argument domain of scalar functions. Infinity and negative infinity are represented by the symbol ² and ²² respectively. Internally they are handled as big reals. Not all scalar functions can handle them correctly and care should be taken that no overflow occurs. In general the formatting facilities of the interpreter are very limited. In particular the display of enclosed arrays is very rudimentary. Also, in order to take advantage of the built-in TurboProlog predicates, input and display of real numbers can deviate from standard APL. The maxmimum rank of arrays has been fixed to 63, exceeding it will lead to a <limit error>. Since the data part of APL variables is always represented in a list of data elements and those lists are processed within the interpreter in recursive predicates the amount of stack memory required is considerable. Variables with more than 500 elements almost always lead to problems with the stack and should be avoided. Note: If a behaviour of the interpreter is said to be problematic this means that in case of a failure the Rationalized APL\PC interpreter will terminate with an error message (in the lucky case) but it may also cause the PC to hang (in the unlucky case). In such a situation the active workspace is definitely lost. Frequent )save of the workspace can save the user from loss of valuable data. Some work has been done in order to make the interpreter safer. In particular, stack overflow should now be reported as workspace full error (due to some problems with the compiler's error trapping feature errors other than ws full may also occur though). The use of TurboProlog version 2.0 compiler has helped to sort out some problems. Yet the interpreter is still not 100% reliable... Active and Saved Workspaces The user's workspace is kept in the interpreter's (Prolog-) data base. The workspace which mainly consists of the symbol table is read and written every time the user is prompted in immediate execution mode. If a user defined function is called, a new local symbol table is set up that also contains the global environment. Upon completion of a function the local environment is dissolved and the previous more global environment is re-established. Global objects are not kept within the symbol table functor. They are stored as separate objects in the data base. This mechanism is very slow and limits considerably the use of user defined (in particular recursive) functions. Defined functions consisting only of one line are special cased and run a bit faster. Saved workspaces are unloaded versions of the relevant parts of the data base. It is not guaranteed that saved workspaces are compatible with future releases of the Rationalized APL\PC. Workspaces saved under release T1.0 are not compatible with this release. Function and Operator Parts The definition of functions and operators (the basic function and operator parts) are also taken from the interpreter's data base. This data base is initially set up at program start (this is done with a <consult> of the file <RATAPL.SYS>). Provided the experimentor is familiar with the internals of the Rationalized APL\PC and has some knowledge of TurboProlog, it is possible to change such parts in order to vary the behaviour of the interpreter or some part of it. Such changes however should be made with extreme caution. A set of files for maintaining the part definitions is available separately from the author. Parsing rules are not defined dynamically through the data base. For reasons of speed they have been coded into the interpreter. Modifying those syntax rules is therefore not possible. 3. Functionalities of the Rationalized APL\PC The Rationalized APL Syntax The Rationalized APL\PC interpreter is based on the Rationalized APL syntax. The syntax rules have been been slightly modified however. Functions derived from a monadic operator which evaluate to class data (for instance from a functional enclose operator) do not require an additional pair of parentheses (thus allowing the expression f+$, -$ if $ is the functional enclose operator). The current version implements most of the APL primitives listed in "A Dictionary of APL". Note that the assignment of symbols to primitives deviates in a few cases from the APL Dictionary. Available Primitive Functions + identity plus 0 0 0 - negative minus 0 0 0 Æ sign times 0 0 0 ÷ reciprocal divide 0 0 0 * exponential power 0 0 0 natural log log 0 0 0 ì ceiling maximum 0 0 0 Å floor minimum 0 0 0 ■ magnitude absolute 0 0 0 ! factorial binomial 0 0 0 ? roll deal 0 0 0 not implemented ~ not difference 0 0 0 difference not implemented ^ and 0 0 lcm not implemented · or 0 0 gcd not implemented Ω nand 0 0 σ nor 0 0 = equal 0 0 å not equal 0 0 < enclose less ² 0 0 > disclose greater 0 0 0 ≥ not less 0 0 ≤ not greater 0 0 ∙ Pi times circle 0 0 0 circle functions: αε1 2 3 ²3 Γ index gen. index of ² 1 ² , ravel catenate ² ² ² ₧ ravel catenate ² ² ² functions along first axis weak enclose link ² ² ² pass dex ² ² ² stop lev ² ² ² √ shape reshape ² 1 ² nub take ² 1 ² raze drop ² 1 ² { cart.prod. from ² 0 ² no complementary indexing φ transpose transpose ² 1 ² dyadic transpose: α≡α Φ reverse rotate 1 0 1 Θ reverse rotate ² ² ² reverse along first axis ε member 0 ² τ encode ² ² en- and decode: 2-Byte integers µ tokenise decode 1 ² ² only; tokenise not implemented grade grade ² 1 ² no secondary ordering gradedown gradedown ² 1 ² no secondary ordering ÿ mat.inverse mat.divide 2 ² 2 not implemented ≡ match ² ² ⌠ format format ² ² ² very limited implementation ⌡ execute execute 1 1 1 extended result range I-beam ² ² system functions Operators f/ reduction derived function: monadic only fδ reduction along 1st axis derived function: monadic only f\ scan fº scan along 1st axis α/ compress fδ compress along 1st axis f\ expand not implemented fº expand along 1st axis not implemented f swap f} select, merge monadic form not implemented f[ yoke always f] yoke conditionally experimental α.f tie, outer product forces fπ0; α: ° treated as 0 f.g inner product derived function: dyadic only f.α power not implemented fπα set function rank fπg on (function composition) απf cut not implemented αf glue fα glue fg with (function composition) αΣf prefer ~α≡α fΣα defer ~α≡α fΣg upon (function composition) fh[g yoke always fh]g yoke conditionally (recursion operator) α∞ direct definition f∞g function application α∞f class definition αε4 5 Derived Functions Functions can be recursively derived and there is virtually no limitation on the depth of recursion (except the interpreter's stack area). Some examples might illustrate the power of such recursions: (,π2)Σ1 Ravel rank-2 cells after deferring axis +.Æδ Reduction on inner product +/Σ2 Reduction along axis 2 ₧>/ The raze function ⌡Σ(0{) Execute after selection Internally a number of derived functions are used to implement functions of high complexitiy with combinations of functions of less complexity. For instance, the cartesian product has been implemented as the derived function >Σ(0.(,>)>/Σ(<π0>)) Again, the complexity of the definition restricts the domain of the arguments to relatively small arrays in such cases. User defined functions can also be derived: ('+' ∞ 'α+')π0 With the exception functions derived from the del-operator the execution of derived functions is deferred until they are actually evaluated. The vector representation of derived functions for displays corresponds to the constituting APL expression after names have been replaced by their values. Function Rank The RATAPL.SYS file delivered with the interpreter defines some default function ranks different from the APL Dictionary. For instance an expression like Γ2 3 results in a rank error. Using the function rank operator explicitly will produce a result rather than an error message: Γπ(0) 2 3 0 1 0 0 1 2 A modification of the RATAPL.SYS file can change the behaviour to conform with the Dictionary. Function Definition Functions may be defined using the function definition operator (del): plus '+' ∞ 'α+' fac ('(≤1){1 2' 'Æ± -1' '1') ∞ ° The operator always requires both definitions, the monadic and the dyadic. The arguments may be a character array of rank>2 or an enclosed vector of character arrays of rank<2. Instead of an empty character vector the jot (°) can be used to designate a non existing case of a definition. Execution of an undefined case is identical to the execution of the identity function. The following rules apply within the function definition: - The symbol α denotes the optional left, the right argument. - The result of the last line executed is propagated as the function's overall result. - Branching is done with (the interpreter replaces the goto by an assignment to òlc). - If a branch is done with an vector of line numbers then those lines are executed in the corresponding order. - The function terminates upon exhaustion of òlc or if an invalid line number is detected. - Self reference of a function is achieved with the symbol ±. - One line functions should not begin with a branch. Resuming a suspended function is not possible. A naked branch usually pops up one level and so allows to escape from it. Note that a naked branch in immediate execution mode on the top level causes the interpreter to terminate normally: the next level is PC/DOS. Defined functions functionally behave like primitive and derived functions. In particular they may appear as arguments to operators (for instance in inner products; likewise the argument rank of a defined function may be set with the rank operator). Defined functions build up a new level in the state indicator stack and provide a new environment for local variables. This process is unfortunately is extremely slow so that the execution defined functions is quite limited. Defined Operators The del-hyper operator may be used for changing the object class. With a left argument of 4 the operator turns a function (primitive, derived or defined) into a monadic operator. A left argument of 5 turns it into a dyadic operator. 4 ∞ f Change class of f from function to monadic operator 5 ∞ f Change class of f from function to dyadic operator Examples: fenc 4 ∞ < ª functional enclose each 4 ∞ ('>'∞°) ª each operator ª Hodgkinson's if-else construct (Hodgkinson, 1987) if 5 ∞ (°∞'>(å0){(<∞),(<∞α)') else 5 ∞ (°∞'>((<∞α)≡(<∞)){(<∞α),(<∞)') + if (n>0) else - ª conditionally select a function The Yoke-Operators The current release includes the definitions for two new monadic operators: Yoke always and yoke conditionally. Yoke always conforms to the definitions given in the Dictionary of APL. Both accept a (dyadic) function argument. The derived form however is not of class function but of class dyadic operator. The derived dyadic operators accept a left and right operator argument of class function. Both yoke operators perform function composition except that they combine three functions instead of two like their known counterparts (for instance Σ). The definition is the following: kf h[ g yoke always: k(x,y): h(f(x,y), g(x,y)) k(x): h(f(x), g(x)) kf h] g yoke conditionally: k(x,y): f(x,y)å0: h(x, k(x, g(x,y))) f(x,y)=0: h(x, g(x,y)) k(x): f(x)å0: h(x, k(g(x))) f(x)=0: h(x, g(x)) As can be seen from the definiton, the conditional yoke applies the derived function recursively until a condition expression (identified by the left derived operator argument f) evaluates to 0. A final function composition on h and g terminates the recursion. The faculty function fac: Æfac -1: ≤1 : 1 could be written as fac 1< Æ] (1ìΣ(-1)) whereas the more complex scan4 ∞ ('1<Σ√(₧[(δ))](²1)' ∞ °) provides an alternative definition of the monadic scan-first operator. The use of the simpler variant yoke always is more obvious. This can be shown by a few examples: sort {[ ª apply from on the results of grade and identity ª an alternative form of the average function sum+/ shape√ dividedBy÷ avg sum dividedBy[ shape System Constants, Variables and Functions The only system constant ° is an undisclosable scalar. System variables which are a part of the workspace environment are available but not fully implemented. With the exception of the variables òapl and òdbg, the system variables (òio, òrl, òct, òpp, òps, òpw) can be referenced but assigning a value to them has no influence (note that for instance òio is always 0 and cannot be changed). System variables can be localised on every new SI stack level. òapl APL dialect (initially 0); 0 forces the interpreter to use Rationalized APL rules, 1 forces AIDA rules (see Gfeller 86). òav Atomic vector; this system variable is available (Note: display of 0{òav causes subsequent characters to be discarded). òio Index origin (always 0). The index origin cannot be varied. òct Comparison tolerance (always 0). The concept of comparison tolerance and fuzzy functions is not implemented, however the variable is available. òdbg Debugging mode (initially 0). This variable can be used to inspect the behaviour of the interpreter during syntax parsing and operator and function execution. It acceptes the following values: 1 - Syntax parsing, display of tokens in left and right stack (see APL Dictionary). 2 - Same as 1 but display of full internal representation of tokens in right stack. 3 - Same as 2 plus additional display of operator execution stack during evaluation phase. òlc Line counter; the line counter can be set with the branch (). Only the top level of the state indicator is reported. The result is always an integer vector of line numbers (or empty). òpp, òps, System variables that normally govern the òpw, òfc formatting during output. These system variables are not used for this purpose, they are only available for completeness. òrl Random link; not implemented. ò, ù Quad, quote-quad; quad is only defined for output, quote-quad only for input. System Functions òai Result is a 3-element vector and which reports the user identification and the elapsed time since the program start (the 3rd element is unused). òcr Canonical representation (monadic argument rank 1); returns the canonical representation of user defined functions. The result is a two element enclosed array with the first element containing the monadic and the second containing the dyadic definition of . òcmd DOS command interface. Executes the DOS command passed in and restores APL environment upon completion. òedit Edit vector of enclosed character vectors using TurboPrologs built in editor. òex Expunge object (monadic argument rank: 1); removes the most local value bound to name from the symbol table. ònc Name class (monadic argument rank: 1); reports the class of a named object. The result is an integer array of rank and shape depending on the rank and shape of the argument. The classes are assigned as follows: 0 - Invalid name 2 - Variable 3 - Function 4 - Monadic operator 5 - Dyadic operator ònl Name list (monadic argument rank: 0); returns a list of all objects of class in the form of a character matrix. Note that ²1 denotes all classes. òsa Storage area; reports the available free stack, heap and tail area in bytes. òts System time stamp in the form of a 7-element integer vector (year, month, day, hour, minute, second, second/100). òwa Work area; reports the available free heap area in bytes (same as 1{òsa). For the purpose of internal design rationalization the I-beam primitive has been implemented to provide a further number of system functions. The following functions are available: 0 Convert string or token list to character vector 6 Convert character vector to string scalar 8 Tokenise string; result is a token list 20 Edit string vector in current window 21 Switch or define new window on screen √√ 0: switch to window (must be defined) √ 7: define window [0] with attributes [1 2] and an offset and size of 3 220 Resize the currently active window 23 Resize the screen (EGA/VGA cards only) 0{ rows, 1{ columns 24 Display a menu in current window and return the selected row index (where is a string variable; for matrices, use the expression: 246π><π1 ) To edit an array (an enclosed vector of character vectors for instance) the composed edit function can be used (in fact the system function òedit is defined the same way): edit0>Σ(20Σ(6π>)) aedit 'try this''example''for editing' Since the built-in TurboProlog editor is invoked APL characters can not be typed in. They are displayed however and can be composed with the proper ALT-number sequence. System Commands System commands are implemented in a very rudimentary form. Among the available commands, load and save are probably the most useful commands. )clear Clear workspace; only to be executed on the root level of the SI stack. )load Load Rationalized APL\PC workspace )off Leave Rationalized APL\PC (the same effect can be achieved with a naked branch on the top immediate execution level). )save Save Rationalized APL\PC workspace )si Display state indicator; note that an immediate execution level is always reported separately and that since defined functions and operators are always unnamed no names are displayed. )wsid Display or set name of the active workspace. Workspaces are saved in the form of a TurboProlog data base. They are ordinary ASCII text files containing only a few predicates (workspace information, state indicator and global values). Changing the content of such a file outside of Rationalized APL\PC can cause a message "workspace incompatible" at the next load. Assignment Direct and indirect assignment is supported. Direct assignment can be local or global. If the assignment arrow is preceded by a space then the variable is assigned a global scope. Otherwise the variable will have local scope. aΓ10 ª a local variable fenc 4∞< ª a global operator Note that within defined functions variables are always assumed to be global until they are assigned a value locally. In the example inc('+AA+1' ∞ °) the first occurrence of A from right binds the value of the global variable A and assigns it (after having it incremented) to a local variable named A. The global variable A remains unchanged if the function inc is executed. Indirect assignment is achieved with an expression within parentheses to the left of the assignment arrow. If the expression evaluates to an enclosed array then multiple assignment is provided. In that case the outer frames of the expressions to the left and the right of the assignment arrow must correspond. ('jan''feb''mar') φqtr881 ª assign each column ª individually (<> 'plus''a') ⌡>'+''100' ª mixed class assigment If the expression on the left of the assigment arrow contains recursively enclosed elements then the corresponding cells are disclosed prior to the assigment. 4. Miscellanous Functionalities Non-data Arrays The interpreter allows the construction of non-data arrays. Currently only function arrays are implemented. The simplest method the create a function arrays is to use the execute function with an argument of rank grater than 1: ⌡π(0) 2 2 √'+-Æ÷' ª 2 by 2 function matrix Refer to Werder 1988 for more details. Function arrays may be applied to data arrays in a similar way as proposed by Bernecky (Bernecky 84): 2 far 4 ª note the scalar extension 6 ²2 8 0.5 (⌡π0 'Φ') 2 3√Γ6 2 1 0 3 4 5 In the current implementation the shape of the outer frame resulting from applying the function rank must conform with the shape of the function array (unless the function array is a scalar in which case scalar extension applies). The function array's function rank is derived from the first function. Uniformity in function ranks is not required for the array. APL Language Dialects In a clear workspace, the system variable òapl is set to 0. This causes the interpreter to use normal Rationalized APL rules for syntax parsing. If the variable òapl is set to 1 the interpreter is forced to use the parsing rules of the APL incompatible dialect AIDA (see Gfeller 1986 for details). Note that only the syntax is changed, and no additional functionalities are provided. òapl1 (/+) (Γ10) 45 5√+,- + - + - + Since de-tokenisation is always based on òapl0 displaying for instance the derived function (/+) results in the normal order +/. Note that no other value except 0 or 1 should ever be assigned to òapl. A value different from 0 or 1 forces the interpreter to use inexistant parsing rules - and therefore results always in an undefined syntax. 5. References Gfeller 86 Gfeller, Martin: Operators considered harmful, APL Quote Quad, Vol. 17, No. 1, September 1986 Hodgkinson 87 Hodgkinson, Robert: Practical Uses of Operators in SHARP APL/HP, APL 87 Conference Proceedings, Dallas, May 1987 Iverson 83 Iverson, K.E.: Rationalized APL, I.P. Sharp Research Report #1, Toronto, April 1983 Iverson 87 Iverson, K.E.: A Dictionary of APL, I.P. Sharp Associates, Toronto, March 1987 Werder 88 Werder, A.: Arrays of Objects in Rationalized APL, APL 88 Conference Proceedings, Sydney, February 1988