Info file: RRA60.info, -*-Text-*- produced by latexinfo-format-buffer from file: RRA60.texinfo File: RRA60.info Node: Top, Prev: (dir), Up: (dir), Next: Introduction Revised Report on the Algorithmic Language Algol 60 ===================================================== By J.W. Backus, F.L. Bauer, J.Green, C. Katz, J. McCarthy P. Naur, A.J. Perlis, H. Rutishauser, K. Samelson, B. Vauquois J.H. Wegstein, A. van Wijngaarden, M. Woodger Edited by Peter Naur Dedicated to the memory of William Turanski * Menu: * Introduction:: Introduction * Description:: Description of the reference language * Editors note:: Note by the Editor Summary --------- The report gives a complete defining description of the international algorithmic language Algol 60. This is a languge suitable for expressing a large class of numerical processes in a form sufficiently concise for direct automatic translation into the language of programmed automatic computers. The introduction contains an account of the preparatory work leading up to the final conference, where the language was defined. In addition the notions reference language, publication language, and hardware representations are explained. In the first chapter a survey of the basic constituents and features of the language is given, and the formal notation, by which the syntactic structure is defined, is explained. The second chapter lists all the basic symbols, and the syntactic units known as identifiers, numbers, and strings are defined. Further some important notions such as quantity and value are defined. The third chapter explains the rules for forming expressions and the meaning of these expressions. Three different types of expressions exist: arithmetic, Boolean (logical), and designational. The fourth chapter describes the operational units of the language, known as statements. The basic statements are: assignment statements (evaluation of a formula), go to statements (explicit break of the sequence of execution of statements), dummy statements, and procedure statements (call for execution of a closed process, defined by a procedure declaration). The formation of more complex structures, having statement character, is explained. These include: conditional statements, for statements, compound statements, and blocks. In the fifth chapter the units known as declarations, serving for defining permanent properties of the units entering into a process described in the language, are defined. The report ends with two detailed examples of the use of the language and an alphabetic index of definitions.  File: RRA60.info Node: Introduction, Prev: Top, Up: Top, Next: Description Introduction ============== Background ------------ After the publication (1)(2) of a preliminary report on the algorithmic language Algol, as prepares at the conference in Zuerich in 1958, much interest in the Algol language developed. (1) Preliminary report - International Algebraic Language, Comm. Assoc. Comp. Mach. 1, No. 12 (1958), 8. (2) Report on the Algorithmic Language Algol by the ACM Committee on Programming Languages and the GAMM Committee on Programming, edited by A. J. Perlis and K. Samuelson, Numerische Mathematik Bd. 1, S. 41-60 (1959). As a result of an informal meeting held at Mainz in November 1958, about forty interested persons from several European countries held an Algol implementation conference in Copenhagen in February 1959. A "hardware group" was formed for working cooperatively right down to the level of the paper tape code. This conference also led to the publication by Regnecentralen, Copenhagen, of an `Algol Bulletin', edited by Peter Naur, which seved as a forum for further discussion. During the June 1959 ICIP Conference in Paris several meetings, both formal and informal ones, were held. These meetings revealed some misunderstandings as to the intent of the group which was primarily responsible for the formulation of the language, but at the same time made it clear that there exists a wide appreciation of the effort involved. As a result of the discussions it was decided to hold an international meeting in January 1959 for improving the Algol language and preparing a final report. At a European Algol Conference in Paris in November 1959 which was attended by about fifty people, seven European representatives were selected at attend the January 1960 Conference, and they represent the following organisations: Association Francaise de Calcul, British Computer Society, Gesellschaft fuer Angewandte Mathematik und Mechanik, and the Nederlands Rekenmachine Genootschap. The seven representatives held a final preparatory meeting at Mainz in December 1959. Meanwhile, in the United States, anyone who wished to suggest changes or corrections to Algol was requested to send his comments to the `Communications of the ACM', where they were published. These comments then became the basis of consideration for changes in the Algol language. Both the SHARE and USE organisations established Algol working groups, and both organisations were represented on the ACM Committee on Programming Languages. The ACM Committee met in Washington in November 1959 and considered all comments on Algol that had been sent to the ACM `Communications'. Also, seven representatives were selected to attend the January 1960 international conference. The seven representatives held a final preparatory meeting in Boston in December 1959. January 1960 Conference ------------------------- The thirteen representatives (1), from Denmark, England, France, Germany, Holland, Swiutzerland, and the United States, conferred in Paris from January 11 to 16, 1960. (1) William Turanski of the American group was killed by an automobile just prior to the January 1960 Conference. Prior to this meeting a completely new draft report was worked out from the preliminary report and the recommendations of the preparatory meetings by Peter Naur and the Conference adopted this new form as the basis for its report. The Conference then proceeded to work for agreement on each item of the report. The present report represents the union of the Committee's concepts and the intersection of its agreements. April 1962 Conference [Edited by M. Woodger] ============================================== A meeting of some of the authors of Algol 60 was held on 2nd - 3rd April in Rome, Italy, through the facilities and courtesy of the International Computation Centre. The following were present: Authors Advisers Observer F. L. Bauer M. Paul W. L. van der Poel J. Green R. Franciotti (Chairman, IFIP TC 2.1 C. Katz P. Z. Ingerman Working Group Algol) R. Kogon (representing J.W. Backus) P. Naur K. Samuelson G. Seegemueller J. H. Wegstein R.E. Utman A. van Wijngaarden M. Woodger P. Landin The purpose of the meeting was to correct known errors in, attempt to eliminate apparent ambiguities in, and otherwise clarify the Algol 60 Report. Extensions to the language were not considered at the meeting. Various proposals for correction and clarification that were submitted by interested parties in response to the Questionnaire in Algol Bulletin No. 14 were used as a guide. This report constitutes a supplement to the Algol 60 Report which should resolve a number of difficulties therein. Not all of the questions raised concerning the original report could be resolved. Rather than risk of hastily drawn conclusions on a number of subtle points, which might create new ambiguities, the committee decided to report only those points which thay unanimously felt could be stated in clear and unambiguous fashion. Questions concerned with the following areas left for further consideration by Working Group 2.1 of IFIP, in the expectation that current work on advanced programming languages will lead to better resolution: 1. Side effects of functions. 2. The call by name concept. 3. 'own': static or dynamic. 4. For statement: static or dynamic. 5. Conflict between specification and declaration. The authors of the Algol 60 Report present at the Rome Conference, being aware of the formation of a Working Group on Algol by IFIP, accepted that any collective responsibility which they might have with respect to the development, specification, and refinement of the Algol language will from now on be transferred to that body. This report has been reviewed by IFIP TC 2 on Programming Languages in August 1962 and has been approved by the Council of the International Federation for Information Processing. As with the preliminary Algol report, three different levels of language are recognized, namely a Reference Language, a Publication Language, and several Hardware Representations. Reference Language 1. It is the working language of the committee. 2. It is the defining language. 3. The characters are determined by ease of mutual understanding and not by any computer limitations, coders notation, or pure mathematical notation. 4. It is the basic reference and guide for compiler builders. 5. It is the guide for all hardware representaions. 6. It is the guide for transliterating from publication language to any locally appropriate hardware representations. 7. The main publications of the Algol language itself will use the reference representation. Publication Language 1. The publication language admits variations of the reference language according to usage of printing and handwriting (e.g. subscripts, spaces, exponents, Greek letters). 2. It is used for stating and communicating process. 3. The caracters used may be different in different countries, but univocal correspondence with reference representaion must be secured. Hardware Representations 1. Each of these is a condensation of the reference language enforced by the limited number of characters on the standard input equipment. 2. Each one of these uses the character set of a particular computer and is the language accepted by a translator for that computer. 3. Each of these must by accompanied by a special set of rules for transliterating from publication or reference language. For transliteration between the reference language and a language suitable for publications, among others, the following rules are recommended. Reference Language Publication Language Subscript brackets [ ] Lowering of the line between the brackets and removal of the brackets. Exponentiation ^ Raising the exponent. Parantheses () Any form of parantheses, brackets, braces. Basis of ten '10' Raising of the ten and of the following integral number, inserting of the intended multiplication sign.  File: RRA60.info Node: Description, Prev: Introduction, Up: Top, Next: Structure Description of the reference language ======================================= Was sich ueberhaupt sagen laesst, laesst sich klar sagen; und wovon man nicht reden kann, darueber muss man schweigen. Ludwig Wittgenstein * Menu: * Structure:: 1. Structure of the language * Basic:: 2. Basic symbols, identifiers, numbers, and strings * Expressions:: 3. Expressions * Statements:: 4. Statements * Declarations:: 5. Declarations * Examples:: Examples of procedure declarations * Index:: Alphabetic index of concepts and syntactic units * Note:: Note  File: RRA60.info Node: Structure, Prev: Description, Up: Description, Next: Basic 1. Structure of the language ------------------------------ As stated in the introdution, the algorithmic language has three different kinds of representations -- reference, hardware, and publication -- and the development described in the squel is in terms of the language are represented by a given set of symbols -- and it is only in the choice of symbols that the other two representations may differ. Structure and content must be the same for all representations. The purpose of the algorithmic language is to describe computational processes. The basic concept used for the description of calculating rules is the well known arithmetic expression containing as constituents numbers, variables, and functions. From such expressions are compounded, by applying rules of arithmetic composition, self-contained units of the language -- explicit formulae -- called assignment statements. To show the flow of computational processes, certain non-arithmetic statements and statement clauses are added which may describe e.g., alternatives, or iterative repetitions of computing statements. Since it is necessary for the function of the statements that one statement refers to another, statements may be provided with labels. A sequence of statements may be enclosed between the statement brackets 'begin' and 'end' to form a compound statement. Statements are supported by declarations which are not themselves computing instructions, but inform the translator of the existence and certain properties of objects appearing in statements, such as the class of numbers taken on as values by a variable, the dimension of an array of numbers, or even the set of rules defining a function. A sequence of declarations followed by a sequence of statements and enclosed between 'begin' and 'end' constitutes a block. Every declaration appears in a block in this way and is valid only for that block. A program is a block or compound statement which is not contained within another statement and which makes no use of other statements not contained within it. In the sequel the syntax and semantics of the language will be given (1). (1) Whenever the precision of arithmetic is stated as being in general not specified, or the outcome of a certain process is left undefined or said to be undefined, this is to be interpreted in the sense that a program only fully defines a computational process if the accompanying information specifies the precision assumed, the kind of arithmetic assumed, and the course of action to be taken in all such cases as may occur during the execution of the computation. 1.1 Formalism for syntactic description. .......................................... The syntax will be described with the aid of metalinguistic formulae (1). (1) Cf. J. W. Backus, The syntax and semantics of the proposed international algebraic language of the Zuerich ACM-GRAMM conference. ICIP Paris, June 1959. Their interpretation is best explained by an example: ::= ( | [ | ( | Sequences of characters enclosed in the bracket <> represent metalinguistic variables whose values are sequences of symbols. The marks ::= and | (the latter with the meaning of or) are metalinguistic connectives. Any mark in a formula, which is not a variable or a connective, denotes itself (or the class of marks which are similar to it). Juxta position of marks and/or variables in a formula signifies juxtaposition of the sequences denoted. Thus the formula above gives a recursive rule for the formation of values of the variable . It indicates that may have the value ( or [ or that given some legitimate value of , another may be formed by following it with the character ( or by following it with some value of the variable . If the values of are the decimal digits, some values of are: [(((1(37( (12345( ((( [86 In order to facilitate the study, the symbols used for distinguishing the metalinguistic variables (i.e. the sequence of characters appearing within the brackets <> as AB in the above example) have been choosen to be words describing approximately the nature of the corresponding variable. Where words which have appeared in this manner are used elsewhere in the text they will refer to the corresponding syntactic definition. In addition some formulae have been given in more than one place. Definition: ::= (i.e. the null string of symbols).  File: RRA60.info Node: Basic, Prev: Structure, Up: Description, Next: Expressions 2. Basic symbols, identifiers, numbers, and strings. ------------------------------------------------------ Basic concepts ----------------- The reference language is built up from the following basic symbols: ::= | | | 2.1. Letters .............. ::= a | b | c | d | e | f | g | h | i | j | k | l | m | n | o | p | q | r | s | t | u | v | w | x | y | z | A | B | C | D | E | F | G | H | I | J | K | L | M | N | O | P | Q | R | S | T | U | V | W | X | Y | Z This alphabet may be arbitrarily restricted, or extended with any other distinctive character (i.e. character not coinciding with any digit, logical value or delimiter). Letters do not have individual meaning. They are used for forming identifiers and strings (1) (cf. sections 2.4. Identifiers, 2.6. Strings). (1) It should be particularly noted that throughout the reference language underlining [here this looks like 'underlined' - (schoenfr)] is used for defining independent basic symbols (see sections 2.2.2 and 2.3). These are understood to have no relation to the individual letters of which they are composed. Within the present report underlining will be used for no other purposes. 2.2.1 Digits. ::= 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 Digits are used for forming numbers, identifiers, and strings. 2.2.2 Logical values. ::= 'true' | 'false' The logical values have a fixed obvious meaning. 2.3. Delimiters ................. ::= | | | | ::= | | | ::= + | - | * | / | 'div' | ^ ::= < | <= | = | >= | > | != ::= 'equiv' | 'impl' | 'or' | 'and' | 'not' ::= 'goto' | 'if' | 'then' | 'else' | 'for' | 'do' (2) ::= , | . | '10' | : | ; | := | _ | 'step' | 'until' | 'while' | 'comment' ::= ( | ) | [ | ] | ` | ' | 'begin' | 'end' ::= 'own' | 'Boolean' | 'integer' | 'real' | 'array' | 'switch' | 'procedure' ::= 'string' | 'label' | 'value' (2) 'do' is used in for statements. It has no relation to the do of the preliminary report, which is not included in Algol60. Delimiters have a fixed meaning which for the most part is obvious or else will be given at the appropriate place in the sequel. Typographical features such as blank space or change to a new line have no significance in the reference language. They, however, be used freely for facilitating reading. For the purpose of including text among the symbols of a program the following "comment" conventions hold: The sequence of basic symbols: is equivalent to ; 'comment' ; ; 'begin' 'comment' ; 'begin' 'end' 'end' By equivalence is here meant that any of the three structures shown in the left hand column may be replaced, in any occurence outside of strings, by the symbol shown in the same line in the right hand column without any effect on the action of the program. It is further understood that the comment structure encountered first in the text when reading from left to right has precedence in being replaced over later structures contained in the sequence. 2.4. Identifiers .................. 2.4.1. Syntax. ::= LETTER> | | 2.4.2. Examples. q Soup V17a a34kTMNs MARILYN 2.4.3. Semantics. Identifiers have no inherent meaning, but serve for the identification of simple variables, arrays, labels, switches, and procedures. They may be choosen freely (cf. however section 3.2.4. Standard functions). The same identifiers cannot be used to denote two different quantities except when these quantities have disjoint scopes as defined by the declarations of the program (cf section 2.7. Quantities, kinds and scopes and section 5. Declarations). 2.5. Numbers .............. 2.5.1 Syntax. ::= | ::= | + | - ::= . ::= '10' ::= | | ::= | | ::= | + | - 2.5.2. Examples. 0 -200.084 -.083'10'-02 177 + 07.43'10'8 -'10'7 .5384 9.34'10'+10 '10'-4 +0.7300 2'10'-4 +'10'+5 2.5.3. Semantics. Decimal numbers have their conventional meaning. The exponent part is scale factor expressed as an integral power of 10. 2.5.4. Types. Integers are of the type 'integer'. All other numbers are of type 'real' (cf. section 5.1 Type declarations). 2.6. Strings .............. 2.6.1. Syntax. ::= | ::= `' | ::= `' 2.6.2. Examples. `5k,,-`[[[`'and'=/:'Tt" `This_is_a_`string" 2.6.3. Semantics. In order to enable the language to handle arbitrary sequences of basic symbols the string quotes ` and ' are introduced. The Symbol _ denotes a space. It has no significance outside strings. Strings are used as actual parameters of procedures (cf. sections 3.2. Function designators and 4.7. Procedure Statements). 2.7. Quantities, kinds and scopes ................................... The following kinds of quantities are distinguished: simple variables, arrays, labels, switches, and procedures. The scope of a quantity is the set of statements and expressions in which the declaration of the identifier associated with that quantity is valid. For labels see section 4.1.3. 2.8. Values and types ....................... A value is an ordered set of numbers (special case: a single number), an ordered set of logical values (special case: a single logical value), or a label. Certain of the syntactic units are said to possess values. These values will in general change during the execution of the program The values of expressions and thair constituents are defined in section 3. The value of an array identifier is the ordered set of values of the corresponding array of subscripted variables (cf. section 3.1.4.1). The various "types" ('integer', 'real', 'Boolean') basically denote properties of values. The types associated with syntactic units refer to the values of these units.  File: RRA60.info Node: Expressions, Prev: Basic, Up: Description, Next: Statements 3. Expressions ---------------- In the language the primary constituents of the programs describing algorithmic processes are arithmetic, Boolean, and designational expressions. Constituents of the expressions, except for certain delimiters, are logical values, numbers, variables, fuction designators, and elementary arithmetic, relational, logical, and sequenctial operators. Since the syntactic definition of both variables and function designators contains expressions, the definition of expressions, and their constituents, is necessarily recursive. ::= | | 3.1. Variables ................ 3.1.1. Syntax ::= ::= ::= ::= | , ::= ::= [ ] ::= | 3.1.2. Examples epsilon detA a17 Q[7,2] x[sin(n*pi/2),Q[3,n,4]] 3.1.3. Semantics. A variable is a designation given to a single value. This value may be used in expressions for forming other values and may be changed at will by means of assignment statements (section 4.2). The type of the value of a particular variable is defined in the declaration for the variable itself (cf. section 5.1. Type declarations) or for the corresponding array identifier (cf. section 5.2. Array declarations), 3.1.4. Subscripts. 3.1.4.1. Subscripted variables designate values which are components of multidimensional arrays (cf. section 5.2. Array declarations). Each arithmetic expression of the subscript list occupies one subscript position of the subscripted variable and is called a subscript. The complete list of subscripts is enclosed in the subscript brackets [ ]. The array component referred to by a subscripted variable is specified by the actual numerical value of its subscripts (cf. section 3.3. Arithmetic expressions). 3.1.4.2. Each subscript position acts like a variable of type 'integer' and the evaluation of the subscript is understood to be equivalent to an assignment to this fictitious variable (cf. section 4.2.4). The value of the subscripted variable is defined only if the value of the subscript expression is within the subscript bounds of the array (cf. section 5.2. Array declarations). 3.2. Function designators ........................... 3.2.1. Syntax ::= ::= | | | | ::= | ::= , | ) : ( ::= | ::= | ( ) ::= 3.2.2. Examples sin(a-b) J(v+s,n) R S(s-5) Temperature: (T) Pressure: (P) Compile (`:=') Stack: (Q) 3.2.3. Sematics. Function designators define single numerical or logical values which result through the application of given sets of rules defined by a procedure declaration (cf. section 5.4. Procedure declarations) to fixed sets of actual parameters. The rules governing specification of actual parameters are given in section 4.7. Procedure statements. Not every procedure declaration defines the value of a function designator. 3.2.4. Standard functions. Certain identifiers should be reserved for the standard functions of analysis, which will be expressed as procedures. It is recommended that this reserved list should contain: abs (E) for the modulus (absolute value) of the value of the expression E sign (E) for the sign of the value of E (+1 for E>0, 0 for E=0, -1 for E<0) sqrt (E) for the square root of the value of E sin (E) for the sine of the value of E cos (E) for the cosine of the value of E arctan (E) for the principal value of the arctangent of the value of E ln (E) for the natural logarithm of the value of E exp (E) for the exponential function of the value of E (e ^ E) These functions are all understood to operate indifferently on arguments both of type 'real' and 'integer'. They will all yield values of type 'real', except for sign (E) which will have values of type 'integer'. In a particular representation these function may be available without explicit declarations (cf. section 5. Declarations). 3.2.5. Transfer functions. It is understood that transfer functions between any pair of quantities and expressions my be defined. Among the standard functions it is recommended that there be one, namely entier (E), which "transfers" an expression of real type to one of integer type, and assigns to it the value which is the largest integer not greater than the value of E. 3.3. Arithmetic expressions ............................. 3.3.1. Syntax ::= + | - ::= * | / | 'div' ::= | | | ( ) ::= | | ^ ::= | ::= | | ::= 'if' 'then' ::= | 'else' 3.3.2. Examples. Primaries: 7.394'10'-8 sum w[i+2,8] cos(y+z*3) (a-3/y+vu^8) Factors: omega sum^cos(y+z*3) 7.394'10'-8^w[i+2,8]^(a-3/y+vu^8) Terms: U omega*sum^cos(y+z*3)/7.394'10'-8^w[i+2,8]^(a-3/y+vu^8) Simple arithmetic expressions: U-Yu+omega*sum^cos(y+z*3)/7.394'10'-8^w[i+2,8]^ (a-3/y+vu^8) Arithmetic expressions: w*u-Q(S+Cu)^2 'if' q>0 'then' S+3*Q/A 'else' 2*S+3*q 'if' a<0 'then' U+V 'else' 'if' a*b>17 'then' U/V 'else' 'if' k!=y 'then' V/U 'else' 0 a*sin(omega*t) 0.57'10'12*a[N*(N-1)/2,0] (A*arctan(y)+Z)^(7+Q) 'if' q 'then' n-1 'else' n 'if' a<0 'then' A/B 'else' 'if' b=0 'then' B/A 'else' z 3.3.3. Semantics. An arithmetic expression is a rule for computing a numerical value. In case of simple arithmetic expressions this value is obtained by executing the indicated arithmetic operations on the actual numerical values of the primaries of the expression, as explained in detail in section 3.3.4 below. The actual numerical value for a primary is obvious in the case of numbers. For variables it is the current value (assigned last in the dynamic sense), and for function designators it is the value arising from the computing rules defining the procedure (cf. section 5.4.4. Values of function designators) when applied to the current values of the procedure parameters given in the expression. Finally, for arithmetic expressions enclosed in parentheses the value must through a recursive analysis be expressed in terms of the values of primaries of the other three kinds. In the more general arithmetic expression, which include if clauses, one out of several simple arithmetic expressions is selected on the basis of the actual values of the Boolean expression (cf. section 3.4. Boolean expressions). This selection is made as follows: The Boolean expressions of the if clauses are evaluated one by one in the sequence from left to right until one having the value 'true' is found. The value of the arithmetic expression is then the value of the first arithmetic expression following this Boolean (the largest arithmetic expression found in this position is understood). The construction: 'else' is equivalent to the construction: 'else' 'if' 'true' 'then' 3.3.4. Operators and types. Apart from the Boolean expressions of if clauses, the constituents of simple arithmetic expressions must be of types 'real' or 'integer' (cf. section 5.1. Type declarations). The meaning of the basic operators and the types of the expressions to which they lead are given by the following rules: 3.3.4.1. The operators +, -, and * have the conventional meaning (addition, subtraction, and multiplication). The type of the expression will by 'integer' if both of the operands are of 'integer' type, otherwise 'real'. 3.3.4.2. The operations / and 'div' both denote division, to be understood as a multiplication of the term by the reciprocal of the factor with due regard to the rules of precedence (cf. section 3.3.5). Thus for example a/b*7/(p-q)*v/s means ((((a*(b^-1))*7)*((p-q)^-1))*v)*(s^-1) The operator / is defined for all four combinations of types 'real' and 'integer' and will yield results of 'real' type in any case. The operator 'div' is defined only for two operands of type 'integer' and will yield a result of type 'integer', mathematically defined as follows: a 'div' b = sign(a/b) * entier(abs(a/b)) (cf. sections 3.2.4 and 3.2.5). 3.3.4.3. The operation ^ denotes exponentation, where the factor is the base and the primary is the exponent. Thus for example 2 ^ n ^ k means (2^n)^k while 2 ^ (n ^ m) means 2^(n^m) Writing i for a number of 'integer' type, r for a number of 'real' type, and a for a number of ether 'integer' or 'real' type, the result is given by the following rules: a ^ i if i>0: a*a*...*a (i times), of the same type as a. if i=0: if a!=0: 1, of the same type as a. if a=0: undefined. if i<0, if a!=0: 1/(a*a*a*...*a) (the denominator has -i factors), of type 'real'. if a=0: undefined. a ^ r if a>0: exp(r*ln(a)), of type 'real'. if a=0, if r>0: 0.0, of type 'real'. if r<=0: undefined. if a<0: always undefined. 3.3.5. Precedence of operators. The sequence of operations within one expression is generally from left to right, with the following additional rules: 3.3.5.1. According to the syntax given in section 3.3.1 the following rules of precedence hold: first: ^ second: * / 'div' third: + - 3.3.5.2. The expression between a left parenthesis and the matching right parenthesis is evaluated by itself and this value is used in subsequent calculations. Censequently the desired order of execution of opearations within an expression can always be arranged by appropriate positioning of parenthesis. 3.3.6. Arithmetics of 'real' quantities. Numbers and variables of type 'real' must be interpreted in the sense of numerical analysis, i.e. as entities defined inherently with only a finite accuracy. Similary, the possibility of the occurance of a finite deviation from the mathematically defined result in any arithmetic expression is explicitly understood. No exact arithmetic will be specified, however, and it is indeed understood that different hardware representations may evaluate arithmetic expressions differently. The control of the possible consequences of such differences must be carried out by the methods of numerical analysis. This control must be considered a part of the process to be described, and will therefore be expressed in terms of the language itself. 3.4. Boolean expressions .......................... 3.4.1. Syntax. ::= < | <= | = | >= | > | != ::= ::= | | | | ( ) ::= | 'not' ::= | 'and' ::= | 'or' ::= | 'impl' ::= | 'equiv' ::= | 'else' 3.4.2. Examples. x=-2 Y>V'or'z-5'and'z-d>q^2 p'and'q'or'x!=y g'equiv''not'a'and'b'and''not'c'or'd'or'e'impl''not' f 'if' k<1 'then' s= > != third: 'not' fourth: 'and' fifth: 'or' sixth: 'impl' seventh: 'equiv' 3.4.6.2. The use of parentheses will be interpreted in the sense given in section 3.3.5.2. 3.5. Designational expressions ................................ 3.5.1. Syntax.