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# Syntax the Algorithmic Language Algol 60

This file was generated from a copy of the Revised Report on the Algorithmic Language Algol 60'' produced by Erik Schoenfelder (schoenfr@ibr.cs.tu-bs.de) who wrote "It is typed-in by me, but I am not the author. This note is taken from the end of the report:"

## Copywrite Note

Note. This report is published in Numerische Mathematik, in the Communications of the ACM, and in the Journal of the British Computer Soc. Reproduction of this report for any purpose is explicitly permitted; reference should be made to this issue of Numerische Mathematik and to the respective issues of the Communications and the Journal of the British Computer Soc. as the source.

Technical University Delft Delft, Holland W. L. van der Poel, (Chairman of Working Group 2.1 on Algol of the International Federation for Information Processing)

## Notes from Translators

### Sources

The original report is well worth study and can be found at the Museum of RetroComputing //locke.ccil.org/pub/retro/retromuseum.html or via [ 366193.366201 ] in the ACM Digital Library.

The original BNF is defined in [ BNF in comp.text.Meta ]

### Omissions

This document is for quick reference. Some sections have been omitted as no longer needed: Some history, the definition of BNF, the extended examples at the end, and the Alpbabetical Index have been removed. I(RJB) have also used some modern abbreviations for some of the recursive definitions in the original... O(_) for optional and #(_) for any number of(including none), and N(_) for some number(but at lest one).

### Special Algol 60 Symbols

Translating the Algol 60 Report into MATHS is made difficult because Algol 60 preceeded by the American Symbolic Code for Information Interchange
1. ASCII::= See http://cse.csusb.edu/dick/samples/comp.text.ASCII.html Indeed The Algol60 group was consulted by the group working on ASCII.

I am currently rewriting this Wed Jul 11 13:47:37 PDT 2007

Algol 60 programs are made up of basic symbols -- what we now call lexemes. Some are represented by words in a special typeface and some by special symbols in a specially designed font. Sometimes I've used the ΤΕΧ symbol. I have also included Erik Schoenfelder LaTeX descriptions of these below.

The basic symbols that were printed like words where actually single symbols, not identifiers! In Algol60 a variable could be called "do" and not be confused with the basic symbol printed with the letters "d" and "o" in bold case. This caused implementers a hard time.

Here is complete list of these alphabetical basic symbols:

1. array::lexeme, indicates a type of data -- an n-dimensional array.
2. begin::lexeme, indicates the start of a block of code
3. Boolean::lexeme, name of the Boolean data type.
4. comment::lexeme, indicates that start of a comment.
5. do::lexeme, indicates the end of a for clause.
6. else::lexeme, separates the else from the then in a conditional statement
7. end::lexeme, indicates the end of a block of code.
8. false::lexeme, one of the Boolean data types.
9. for::lexeme, indicates the start of a loop.
10. goto::lexeme, indicates an unconditional transfer of control to a label.
11. if::lexeme, starts a selection statement of conditional expression.
12. integer::lexeme, name of a fixed point data type.
13. label::lexeme, indicate a label parameter.
14. own::lexeme, indicates a variable with static life time but local scope.
15. procedure::lexeme, indicates a sub-program or function.
16. real::lexeme, name of a floating point type.
17. step::lexeme, used in a for clause to indicate an increment
18. string::lexeme, a type of character string data type.
19. switch::lexeme, declares a variables that is something like an array of labels.
20. then::lexeme, Separates a condition from the true part of a selection.
21. true::lexeme, a Boolean value
22. until::lexeme, indicates the final value in for clause.
23. value::lexeme, indicate pass by value in a parameter.
24. while::lexeme, indicates a conditional loop part of a for clause.

I have also replaced the symbols in the special type face by some ASCII Symbols or TeX strings with a related meaning... I have just used the word in examples and replaced the ΤΕΧ symbols by ASCII strings. In eeffect the syntax describes the Algol60 reference language but the examples are in hypothetical hardware representation for a machine that uses ASCII.

### Special Lexemes

Most lexemes have matching ASCII strings: "+", "-", ":=", ... but others have to be finessed.

A small subscripted ten is used for the exponent in a real-type number. Erik Schoenfelder wrote "(i'd like to use $$_{10}$$, but it does not work in TeX)"

2. ten::lexeme=TeX(~{{\tt\tiny\kern-.1667em\lower.5ex\hbox{10}\kern-.125em}}).

The Algol 60 sign for a blank space in a string. Printed like a half box.

3. |_|::lexeme=TeX({{\hbox{\rule[-.2ex]{.1ex}{.3ex}\rule[-.2ex]{.8ex}{.1ex}\rule[-.2ex]{.1ex}{.3ex}\,}}).

4. power::lexeme=up arrow, shown as "^" in examples and ↑ in the reference language.

5. times::lexeme=The times sign: a cross like an x. Shown as an asterisk "*" in examples and \times in the reference language.

6. div::=The integer division operator: a - with a dot above and below. Shown as a "%" in example code in ΤΕΧ \div -- not in HTML.

7. le::lexeme= less or equal. Shown as <= in code and ≤ in the reference language.

8. ge::lexeme=greater or equal. Shown as ">=" in examples and ≥ in the reference language.

9. ne::lexeme=not equal to, "<>" in examples and ≠ in the reference language.

10. iff::lexeme=true when two expressions have the same truth-value, shown as "==" in examples and ≡ in the reference language.

11. hook::lexeme=logical implication, shown as "=>" in examples and ⊃ in the reference language.

12. or::lexeme=logical or, shown as "\/" in examples and ∨ in the reference language.

13. and::lexeme=logical and, shown as "/\" in examples and ∧ in the reference language.

14. not::lexeme=logical negation, shown as "~" in examples and \lnot in the reference language.

### The Empty String

15. empty::lexeme= "". In the Algol60 report empty was left undefined but understood to refer to a string with zero symbols in it.

### Special Glossary

16. procedure::= a subprogram or function that does not return a value that should have have a unique procedure_identifier and may be called in a procedure_statement and must be declared in a procedure_declaration.

17. selection::=conditional_statement. Algol 60 had no case statement - they hadn't been invented! Instead you used a kind of array of labes called a switch. See switch_declaration

18. loop::=for_statement.

. . . . . . . . . ( end of section Notes from Translators) <<Contents | End>>

## Introduction

### Authors

        J.W. Backus, F.L. Bauer, J.Green, C. Katz, J. McCarthy
    P. Naur, A.J. Perlis, H. Rutishauser, K. Samuelson, B. Vauquois
             J.H. Wegstein, A. van Wijngaarden, M. Woodger

### Editor

                               Peter Naur

### Dedication

              Dedicated to the memory of William Turanski

### Summary

The report gives a complete defining description of the international algorithmic language Algol 60. This is a language 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.

1. ...

. . . . . . . . . ( end of section Introduction,) <<Contents | End>>

## Description

### Introduction to Description

1. [Omitted history]

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

It is the working language of the committee.

It is the defining language.

The characters are determined by ease of mutual understanding and not by any computer limitations, coders notation, or pure mathematical notation.

It is the basic reference and guide for compiler builders.

It is the guide for all hardware representations.

It is the guide for transliterating from publication language to any locally appropriate hardware representations.

The main publications of the Algol language itself will use the reference representation.

#### Publication Language

The publication language admits variations of the reference language according to usage of printing and handwriting (e.g. subscripts, spaces, exponents, Greek letters).

It is used for stating and communicating process.

The characters used may be different in different countries, but univocal correspondence with reference representation must be secured.

#### Hardware Representations

Each of these is a condensation of the reference language enforced by the limited number of characters on the standard input equipment.

Each one of these uses the character set of a particular computer and is the language accepted by a translator for that computer.

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.

 Parentheses ()                  Any form of parentheses, brackets,
                                braces.

 Basis of ten \ten                  Raising of the ten and of the following
                                integral number, inserting of the
                                intended multiplication sign.

#### Description of the reference language

##### Quotation
                     Was sich ueberhaupt sagen laesst, laesst sich
                     klar sagen; und wovon man nicht reden
                     kann, darueber muss man schweigen.
                                         Ludwig Wittgenstein

##### 1. Structure of the language

As stated in the introduction, the algorithmic language has three different kinds of representations -- reference, hardware, and publication -- and the development described in the sequel 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

###### Foot note 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.

. . . . . . . . . ( end of section 1. Structure of the language) <<Contents | End>>

##### 2. Basic symbols, identifiers, numbers, and strings.

###### Basic concepts
The reference language is built up from the following basic symbols:
1. basic_symbol::= letter | digit | logical_value | delimiter

###### 2.1. Letters

2. letter::= ASCII.letter.

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 [ Foot note 2 ] (cf. sections [ 2.4. Identifiers ] , [ 2.6. Strings ] ).

###### Foot note 2
It should be particularly noted that throughout the reference language underlining (underlined) 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.
3. digit::= ASCII.digit

Digits are used for forming numbers, identifiers, and strings.

###### 2.2.2 Logical values.
4. logical_value::= true | false.

The logical values have a fixed obvious meaning.

###### 2.3. Delimiters
5. delimiter::= operator | separator | bracket | declarator | specificator.
6. operator::= arithmetic_operator | relational_operator | logical_operator | sequential_operator.
7. arithmetic_operator::= "+" | "-" | times | "/" | div | power .
8. relational_operator::= "<" | le | "=" | ge | ">" | ne .
9. logical_operator::= iff | hook | or | and | not .
10. sequential_operator::= goto | if | then | else | for | do.
11. separator::= comma | dot | ten | colon | semicolon | ":=" | "_" | step | until | while | comment.
12. bracket::= "(" | " | "[" | "]" | "" | "'" | begin | end
13. declarator::= own | Boolean | integer | real | array | switch | procedure
14. specificator::= string | label | value

Delimiters have a fixed meaning which for the most part is obvious or else will be given at the appropriate place in the sequel.

###### Layout
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:
15. statement_separator::= ";" comment #non(";");"
(comment1): statement_separator = ";".
16. block_begining::=begin comment #non(";");"
(comment2): block_begining = begin.
17. block_ending::=end #(non(end | ";" | else))
(comment3): block_ending = end.

By equality is here meant that any of the structures shown on the left hand side may be replaced, in any occurrence outside of strings, by the symbol shown on the right hand side 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.

###### 2.4.1. Syntax.
1. identifier::= letter #( letter | digit).
###### 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 chosen 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).

. . . . . . . . . ( end of section 2.4. Identifiers) <<Contents | End>>

##### 2.5. Numbers
###### 2.5.1 Syntax.
1. unsigned_integer::= digit | unsigned_integer digit.
2. integer::= unsigned_integer | "+" unsigned_integer | "-" unsigned_integer.
3. decimal_fraction::= "." unsigned_integer.
4. exponential_part::= \ten integer.
5. decimal_number::= unsigned_integer | decimal_fraction | unsigned_integer decimal_fraction.

6. unsigned_number::= decimal_number | exponential_part | decimal_number exponential_part.

7. number::= unsigned_number | "+" unsigned_number | "-" unsigned_number.
###### 2.5.2. Examples.

  0               -200.084                -.083\ten -02
  177               + 07.43\ten 8                -\ten 7
   .5384             9.34\ten +10               \ten -4
  +0.7300             2\ten -4                  +\ten +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).

. . . . . . . . . ( end of section 2.5. Numbers) <<Contents | End>>

##### 2.6. Strings
###### 2.6.1. Syntax.
1. proper_string::=#non("" | "'").
2. open_string::= proper_string "" open_string"'" | open_string open_string.
3. string_literal::= "" open_string"'."
###### 2.6.2. Examples.
 5k,,-[[[ /\  =/:'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).

. . . . . . . . . ( end of section 2.6. Strings) <<Contents | End>>

##### 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 their 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.

#### 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, function designators, and elementary arithmetic, relational, logical, and sequential operators. Since the syntactic definition of both variables and function designators contains expressions, the definition of expressions, and their constituents, is necessarily recursive.

1. expression::= arithmetic_expression | Boolean_expression | designational_expression.

#### 3.1. Variables

##### 3.1.1. Syntax
1. variable_identifier::= identifier.
2. simple_variable::= variable_identifier.
3. subscript_expression::= arithmetic_expression.
4. subscript_list::= subscript_expression | subscript_list "," subscript_expression.
5. array_identifier::= identifier.
6. subscripted_value::= array_identifier "["subscripted_list "]".
7. variable::= simple_variable | subscripted_variable.
##### 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).

. . . . . . . . . ( end of section 3.1.4. Subscripts.) <<Contents | End>>

. . . . . . . . . ( end of section 3.1. Variables) <<Contents | End>>

#### 3.2. Function designators

##### 3.2.1. Syntax
1. procedure_identifier::= identifier.
2. actual_parameter::= string_literal | expression | array_identifier | switch_identifier | procedure_identifier.
3. letter_string::= letter | letter_string letter.
4. parameter_delimiter::= "," | ")" letter_string ":" "(".
5. actual_parameter_list::= actual_parameter | actual_parameter_list parameter_delimiter actual_parameter.
6. actual_parameter_part::= empty | "(" actual_parameter_list} ")".
7. function_designator::= procedure_identifier actual_parameter_part.
##### 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. Semantics.
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:

8. Reccommended_mathematical_functions::=following
Net
For E: Arithmetic_expression.
1. abs(E)::=the modulus (absolute value) of the value of the expression E .
2. sign(E)::=the sign of the value of E (+1 for E < 0, 0 for E=0, -1 for E < 0).
3. sqrt(E)::=the square root of the value of E.
4. sin(E)::=the sine of the value of E.
5. cos(E)::=the cosine of the value of E.
6. arctan(E)::=the principal value of the arctangent of the value of E.
7. ln(E)::=the natural logarithm of the value of E.
8. exp(E)::=the exponential function of the value of E (e ^ E).

(End of Net)

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.

. . . . . . . . . ( end of section 3.2. Function designators) <<Contents | End>>

#### 3.3. Arithmetic expressions

##### 3.3.1. Syntax
2. multiplying_operator::= timess | "/" | div .
3. primary::= unsigned_number | variable | function_designator | "(" arithmetic_expression ")".
4. factor::= primary | factor power primary.
5. term::= factor | term multiplying_operator factor.
7. if_clause::= if Boolean_expression then.
8. arithmetic_expression::= simple_arithmetic_expression | if_clause simple_arithmetic_expression else arithmetic_expression.
##### 3.3.2. Examples.
Primaries:
 	7.394\ten -8
 	sum
 	w[i+2,8]
 	cos(y+z *  3)
 	(a-3/y+vu ^  8)

Factors:

 	omega
 	sum ^  cos(y+z *  3)
 	7.394\ten -8 ^  w[i+2,8] ^  (a-3/y+vu ^  8)

Terms:

 	U
 	omega *  sum ^  cos(y+z *  3)/7.394\ten -8 ^  w[i+2,8] ^  (a-3/y+vu ^  8)

 	Simple arithmetic expressions:

 	U-Yu+omega *  sum ^  cos(y+z *  3)/7.394\ten -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\ten 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:

is equivalent to the construction:

##### 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.

In 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 term / factor and term % factor 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 % is defined only for two operands of type integer and will yield a result of type integer, mathematically defined as follows: .

1. a % b::= sign(a/b) * entier(abs(a/b))

Compare with sections 3.2.4 and 3.2.5.

###### 3.3.4.3.
The operation denotes exponentiation, where the factor is the base and the primary is the exponent. Thus for example 2 ^ n ^ k means
2. (2^n)^k while
3. 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. . . . . . . . . . ( end of section 3.3.4. Operators and types.) <<Contents | End>> ##### 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: * / % 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 operations 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. Similarly, the possibility of the occurrence 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. . . . . . . . . . . ( end of section 3.3. Arithmetic expressions) <<Contents | End>> #### 3.4. Boolean expressions ##### 3.4.1. Syntax. 1. relational_operator::= "<" | le | "=" | ge | ">" | ne. 2. relation::= simple_arithmetic_expression relational_operator simple_arithmetic_expression. 3. Boolean_primary::= logical_value | variable | function_designator | relation | "(" Boolean_expression ")". 4. Boolean_secondary::= Boolean_primary | not Boolean_primary. 5. Boolean_factor::= Boolean_secondary | Boolean_factor and Boolean_secondary. 6. Boolean_term::= Boolean_factor | Boolean_term or Boolean_factor. 7. implication::= Boolean_term | implication hook Boolean_term. 8. simple_Boolean::= implication | simple_Boolean iff implication. 9. Boolean_expression::= simple_Boolean | if_clause simple_Boolean else Boolean_expression. ##### 3.4.2. Examples.  x=-2  Y>V \/ z<q  a+b>-5 /\ z-d>q^2  p /\ q \/ x<>y  g==~a /\ b /\ ~c \/ d \/ e=> ~f  if k<1 then s<w else h <= c  if if if a then b else c then d else f then g else h < k ##### 3.4.3. Semantics. A Boolean expression is a rule for computing a logical value. The principles of evaluation are entirely analogous to those given for arithmetic expressions in section 3.3.3. ##### 3.4.4. Types. Variables and function designators entered as Boolean primaries must be declared Boolean (cf. section 5.1. Type declarations and section 5.4.4. Value of function designators). ##### 3.4.5. The operators. Relations take on the value true whenever the corresponding relation is satisfied for the expressions involved, otherwise false. The meaning of the logical operators ~ (not), /\ (and), \/ (or), => (hook), and == (iff), is given by the following function table.  b1 | false | false | true | true  b2 | false | true | false | true  ----------------------------------------------------------------  ~ b1 | true | true | false | false  b1 /\ b2 | false | false | false | true  b1 \/ b2 | false | true | true | true  b1 => b2 | true | true | false | true  b1 == b2 | true | false | false | true ##### 3.4.6. Precedence of operators. The sequence of operations within one expression is generally from left to right, with the following additional rules: ###### 3.4.6.1. According to the syntax given in section 3.4.1 the following rules of precedence hold: first: arithmetic expressions according to section 3.3.5. second: < <= = >= > <> third: /\ fourth: /\ fifth: \/ sixth: => seventh: == ###### 3.4.6.2. The use of parentheses will be interpreted in the sense given in section 3.3.5.2. . . . . . . . . . ( end of section 3.4.6. Precedence of operators.) <<Contents | End>> . . . . . . . . . ( end of section 3.4. Boolean expressions) <<Contents | End>> #### 3.5. Designational expressions ##### 3.5.1. Syntax. 1. label::= identifier | unsigned_integer. 2. switch_identifier::= identifier. 3. switch_designator::= switch_identifier l_bracket subscript_expression r_bracket. 4. simple_designational_expression::= label | switch_designator | l_paren designational_expression r_paren. 5. designational_expression::= simple_designational_expression | if_clause simple_designational_expression else designational_expression. ##### 3.5.2. Examples.  17  p9  Choose[n-1]  Town [if y<0 then N else N+1]  if Ab<c then 17 else q[if w <= 0 then 2 else n] ##### 3.5.3. Semantics. A designational expression is a rule for obtaining a label of a statement (cf. section 4. Statements). Again the principle of the evaluation is entirely analogous to that of arithmetic expressions (section 3.3.3). In the general case the Boolean expression of the if clauses will select a simple designational expression. If this is a label the desired result is already found. A switch designator refers refers to the corresponding switch declaration (cf. section 5.3. Switch declarations) and by the actual numerical value of its subscript expression selects one of the designational expressions listed in the switch declaration by counting these from left to right. Since the designational expression thus selected may again by a switch designator this evaluation is obviously a recursive process. ##### 3.5.4. The subscript expression. The evaluation of the subscript expression is analogous to that of subscripted variables (cf. section 3.1.4.2). The value of a switch designator is defined only if the subscript expression assumes one of the positive values 1, 2, 3, ..., n, where n is the number of entries in the switch list. ##### 3.5.5. Unsigned integers as labels. Unsigned integers used as labels have the property that leading zeroes do not affect their meaning, e.g. 00127 denotes the same label as 217. . . . . . . . . . ( end of section 3.5. Designational expressions) <<Contents | End>> . . . . . . . . . ( end of section 3. Expressions) <<Contents | End>> ### 4. Statements The units of operation within the language are called statements. The will normally be executed consecutively as written. However, this sequence of operations may be broken by go to statements, which define their successor explicitly, and shortened by conditional statements, which may cause certain statements to be skipped. In order to make it possible to define a specific dynamic succession, statements may be provided with labels. Since sequences of statements may be grouped together into compound statements and blocks the definition of statement must necessarily be recursive. Also since declarations, described in section 5, enter fundamentally into the syntactic structure, the syntactic definition of statements must suppose declarations to be already defined. #### 4.1. Compound statements and blocks ##### 4.1.1 Syntax 1. unlabelled_basic_statement::= assignment_statement | go_to_statement | dummy_statement | procedure_statement. 2. basic_statement::= unlabelled_basic_statement | label colon basic_statement. 3. unconditional_statement::= basic_statement | compound_statement | block. 4. statement::= unconditional_statement | conditional_statement | for_statement. 5. compound_tail::= statement end | statement ";" compound_tail. 6. block_head::= begin declaration | block_head ";" declaration 7. unlabelled_compound::= begin compound_tail. 8. compound_statement::= unlabelled_compound | label colon compound_statement. 9. block::= unlabelled_block | label colon block. 10. program::= block | compound_statement. This syntax may be illustrated as follows: Denoting arbitrary statements, declarations, and labels, by the letters S, D, L, respectively, the basic syntactic units take the forms: Compound statement:  L:L: ... begin S; S; ... S; S end Block:  L:L: ... begin D; D; .. D; S; S; ... S; S end It should by kept in mind that each of the statements S may again be a complete compound statement or a block. ##### 4.1.2. Examples. ###### Basic statements:  a:=p+q  goto Naples  Start: Continue: W:=7.993 ###### Compound statements:  begin x:=0; for y:=1 step 1 until n do x:=x+A[y];  if x>q then goto STOP else if x>w-2 then goto S;  Aw: St: W:=x+bob end ###### Block:  Q: begin integer i, k; real w;  for i:=1 step 1 until m do  for k:=i+1 step 1 until m do  begin w:=A[i,k];  A[i,k]:=A[k,i];  A[k,i]:=w end for i and k  end block Q . . . . . . . . . ( end of section 4.1.2. Examples.) <<Contents | End>> ##### 4.1.3. Semantics. Every block automatically introduces a new level of nomenclature. This is realized as follows: Any identifier occurring within the block my through a suitable declaration (cf. section [ 5. Declarations ] ) be specified to be local to the block in question. This means (a) that the entity represented by this identifier inside the blocks has no existence outside it and (b) that any entity represented by this identifier outside the block is completely inaccessible inside the block. Identifiers (except those representing labels) occurring within a block and not being declared to this block will be non-local to it, i.e. will represent the same entity inside the block and in the level immediately outside it. A label separated by a colon from a statement, i.e. labelling that statement, behaves as though declared in the head of the smallest embracing block, i.e. the smallest block whose brackets begin and end enclose that statement. In this context a procedure body must be considered as if it were enclosed by begin and end and treated as a block. Since a statement of a block may again itself be a block the concepts local and non-local to a block must be understood recursively. Thus an identifier, which is non-local to a block A, may or may not be non-local to the block B in which A is one statement. . . . . . . . . . ( end of section 4.1. Compound statements and blocks) <<Contents | End>> #### 4.2. Assignment statements ##### 4.2.1. Syntax. 1. left_part::= variable ":=" | procedure_identifier ":=". 2. left_part_list::= left_part | left_part_list left_part. 3. assignment_statement::= left_part_list arithmetic_expression | left_part_list Boolean_expression. ##### 4.2.2. Examples.  s:=p[0]:=n:=n+1+s  n:=n+1  A:=B/C-v-q*S  S[v,k+2]:=3-arctan(s*zeta)  V:=Q>Y/\Z ##### 4.2.3. Semantics. Assignment statements serve for assigning the value of an expression to one or several variables or procedure identifiers. Assignment to a procedure_identifier may only occur within the body of a procedure defining the value of a function designator (cf. section 5.4.4). The process will in the general case be understood to take place in three steps as follows: ###### 4.2.3.1. Any subscript expression occurring in the left part variables are evaluated in sequence from left to right. ###### 4.2.3.2. The expression of the statement is evaluated. ###### 4.2.3.3. The value of the expression is assigned to all the left part variables, with any subscript expressions having values as evaluated in step 4.2.3.1. . . . . . . . . . ( end of section 4.2.3. Semantics.) <<Contents | End>> ##### 4.2.4. Types. The type associated with all variables and procedure identifiers of a left part list must be the same. If the type is Boolean, the expression must likewise be Boolean. If the type is real or integer, the expression must be arithmetic. If the type of the arithmetic expression differs from that associated with the variables and procedure identifiers, appropriate transfer functions are understood to be automatically invoked. For transfer from real to integer type the transfer function is understood to yield a result equivalent to entier(E+0.5) where E is the value of the expression. The type associated with a procedure identifier is given by the declarator which appears as the first symbol of the corresponding procedure declaration (cf. section 5.4.4). . . . . . . . . . ( end of section 4.2. Assignment statements) <<Contents | End>> #### 4.3. Go to statements ##### 4.3.1. Syntax 1. go_to_statement::= goto designational_expression. ##### 4.3.2. Examples.  goto 8  goto exit [n+1]  goto Town [ if y<0 then N else N+1]  goto if Ab<c then 17 else q [ if w\mlt0 then 2 else n] ##### 4.3.3. Semantics. A go to statement interrupts the normal sequence of operations, defined by the write-up of statements, by defining its successor explicitly by the value of a designational expression. Thus the next statement to be executed will be the one having this value as its label. ##### 4.3.4. Restriction. Since labels are inherently local, no go to statement can lead from outside into a block. A go to statement may, however, lead from outside into a compound statement. ##### 4.3.5. Go to an undefined switch designator. A go to statement is equivalent to a dummy statement if the designational expression is a switch designator whose value is undefined. . . . . . . . . . ( end of section 4.3. Go to statements) <<Contents | End>> #### 4.4. Dummy statements ##### 4.4.1. Syntax 1. dummy_statement::= empty. ##### 4.4.2. Examples.  L:  begin ....; John: end ##### 4.4.3. Semantics. A dummy statement executes no operation. It may serve to place a label. . . . . . . . . . ( end of section 4.4. Dummy statements) <<Contents | End>> #### 4.5. Conditional statements ##### 4.5.1. Syntax 1. if_clause::= if Boolean_expression then. 2. unconditional_statement::= basic_statement | compound_statement | block. 3. if_statement::= if_clause unconditional_statement. 4. conditional_statement::= if_statement | if_statement else statement | if_clause for_statement | label colon conditional_statement. ##### 4.5.2. Examples.  if x>0 then n:=n+1  if s>u then V: q:=n+m else goto R  if s<0 \/ P<=Q then AA: begin if q\mltv then a:=v/s  else y:=2*a end else if v>s then a:=v-q  else if v>s-1 then goto S ##### 4.5.3. Semantics of if. Conditional statements cause certain statements to be executed or skipped depending on the running values of specified Boolean expressions. ###### 4.5.3.1. If statement. The unconditional statement of an if statement will be executed if the Boolean expression of the if clause is true. Otherwise it will be skipped and the operation will be continued with the next statement. ###### 4.5.3.2. Conditional statement. According to the syntax two different forms of conditional statements are possible. These may be illustrated as follows:  if B1 then S1 else if B2 then S2 else S3; S4  if B1 then S1 else if B2 then S2 else if B3 then S3; S4 Here B1 to B3 are Boolean expressions, while S1 to S3 are unconditional statements. S4 is the statement following the complete conditional statement. The execution of a conditional statement may be described as follows: The Boolean expression of the if clause are evaluated one after the other in sequence from left to right until one yielding the value true is found. Then the unconditional statement following this Boolean is executed. Unless this statement defines its successor explicitly the next statement to be executed will be S4, i.e. the statement following the complete complete conditional statement. Thus the effect of the delimiter else may be described by saying that it defines the successor of the statement it follows to be the statement following the complete conditional statement. The construction  else$unconditional_statement
is equivalent to
  else   if   true   then  \$unconditional_statement

If none of the Boolean expressions of the if clauses is true, the effect of the whole conditional statement will be equivalent to that of a dummy statement.

1. [Omitted pictures]

. . . . . . . . . ( end of section 4.5.3. Semantics of if.) <<Contents | End>>

##### 4.5.4. Go to into a conditional statement.
The effect of a go to statement leading into a conditional statement follows directly from the above explanation of the effect of else.

. . . . . . . . . ( end of section 4.5. Conditional statements) <<Contents | End>>

#### 4.6. For statements

##### 4.6.1. Syntax
1. for_list_element::= arithmetic_expression | arithmetic_expression step arithmetic_expression until arithmetic_expression | arithmetic_expression while Boolean_expression
2. for_list::= for_list_element #( comma for_list_element).
3. for_clause::= for variable ":=" for_list do.
4. for_statement::= #( label colon) for_clause statement.

##### 4.6.2. Examples.
  for  q:=1  step  s  until  n  do  A[q]:=B[q]
  for  k:=1,V1*2  while  V1<N  do
  for  j:=I+G,L,1  step  1  until  N, C+D  do  A[k,j]:=B[k,j]

##### 4.6.3. Semantics.
A for clause causes the statement S which it precedes to be repeatedly executed zero or more times. In addition it performs a sequence of assignments to its controlled variable. The process may be visualized by means of the following picture:

[Omitted picture]

In this picture the word initialize means: perform the first assignment of the for clause. Advance means: perform the next assignment of the for clause. Test determines if the last assignment has been done. If so, the execution continues with the successor of the for statement. If not, the statement following the for clause is executed.

##### 4.6.4. The for list elements.
The for list gives a rule for obtaining the values which are consecutively assigned to the controlled variable. This sequence of values is obtained from the for list elements by taking these one by one in order in which they are written. The sequence of values generated by each of the three species of for list elements and the corresponding execution of the statement S are given by the following rules:

###### 4.6.4.1. Arithmetic expression.
This element gives rise to one value, namely the value of the given arithmetic expression as calculated immediately before the corresponding execution of the statement S.

###### 4.6.4.2. Step-until-element.
An element of the form A step B until C, where A, B, and C are arithmetic expressions, gives rise to an execution which may be described most concisely in terms of additional Algol statement as follows:
      V := A
 L1:   if  (V-C)*sign(B) > 0  then   goto  Element exhausted'';
      Statement S;
      V := V+B;
       goto  L1;
where V is the controlled variable of the for clause and Element exhausted' points to the evaluation according to the next element in the for list, or if the step-until-element is the last of the list, to the next statement in the program.

###### 4.6.4.3. While-element.
The execution governed by a for list element of the form E while F, where E is an arithmetic and F a Boolean expression, is most concisely described in terms of additional Algol statements as follows:

 L3:  V := E
       if  ~ F  then   goto  Element exhausted'';
      Statement S;
       goto  L3;

where the notation is the same as in 4.6.4.2 above.

###### 4.6.5. The value of the controlled variable upon exit.
Upon exit out of the statement S (supposed to be compound) through a go to statement the value of the controlled variable will be the same as it was immediately preceding the execution of the go to statement.

If the exit is due to exhaustion of the for list, on the other hand, the value of the controlled variable is undefined after the exit.

###### 4.6.6. Go to leading into a for statement.
The effect of a go to statement, outside a for statement, which refers to a label within the for statement, is undefined.

. . . . . . . . . ( end of section 4.6.4. The for list elements.) <<Contents | End>>

. . . . . . . . . ( end of section 4.6. For statements) <<Contents | End>>

#### 4.7. Procedure statements

##### 4.7.1. Syntax
1. procedure_statement::= procedure_identifier actual_parameter_part.

##### 4.7.2. Examples.
 	Spur (A) Order: (7) Result to: (V)
 	Transpose (W, v+1)
 	Absmax (A, N, M, Yy, I, K)
 	Innerproduct (A [t,P,u], B [P], 10, P, Y)
These examples correspond to examples given in section 5.4.2. [ #5.4.2. Example Procedures ]

##### 4.7.3. Semantics.
A procedure_statement serves to invoke (call for) the execution of a procedure_body (cf. section 5.4. procedure declarations). Where the procedure_body is a statement written in Algol the effect of this execution will be equivalent to the effect of performing the following operations on the program at the time of execution of the procedure_statement.
###### 4.7.3.1. Value assignment (call by value).
All formal parameters quoted in the value part of the procedure declaration heading are assigned the values (cf. section 2.8. Values and types) of the corresponding actual parameters, these assignments being considers as being performed explicitly before entering the procedure_body. The effect is as though an additional block embracing the procedure_body were created in which these assignments were made to variables local to this fictitious block with types as given in the corresponding specifications (cf. section 5.4.5). As a consequence, variables called by value are to be considered as nonlocal to the body of the procedure, but local to the fictitious block (cf. section 5.4.3).
###### 4.7.3.2. Name replacement (call by name).
Any formal parameter not quoted in the value list is replaced, throughout the procedure_body, by the corresponding actual parameter, after enclosing this latter in parentheses wherever syntactically possible. Possible conflicts between identifiers inserted through this process and other identifiers already present within the procedure_body will be avoided by suitable systematic changes of the formal or local identifiers involved.
###### 4.7.3.3. Body replacement and execution.
Finally the procedure_body, modified as above, is inserted in place of the procedure_statement and executed. if the procedure is called from a place outside the scope of any non-local quantity of the procedure_body the conflicts between the identifiers inserted through this process of body replacement and the identifiers whose declarations are valid at the place of the procedure statement or function designator will be avoided through suitable systematic changes of the latter identifiers.

. . . . . . . . . ( end of section 4.7.3. Semantics.) <<Contents | End>>

##### 4.7.4. Actual-formal correspondence.
The correspondence between the actual parameters of the procedure_statement and the formal parameters of the procedure_heading is established as follows: The actual parameter list of the procedure_statement must have the same number of entries as the formal parameter list of the procedure declaration heading. The correspondence is obtained by taking the entries of these two lists in the same order.
##### 4.7.5. Restrictions.
For a procedure_statement to be defined it is evidently necessary that the operations on the procedure_body defined in sections 4.7.3.1 and 4.7.3.2 lead to a correct Algol statement.

This imposes the restriction on any procedure_statement that the kind and type of each actual parameter to be compatible with the kind and type of the corresponding formal parameter. Some important particular cases of this general rule are the following:

###### 4.7.5.1.
If a string is supplied as an actual parameter in a procedure statement or function designator, whose defining procedure_body is an Algol 60 statement (as opposed to non-Algol code, cf. section 4.7.8), then this string can only be used within the procedure_body as an actual parameter in further procedure calls. Ultimately it can only be used by a procedure_body expressed in non-Algol code.

###### 4.7.5.2.
A formal parameter which occurs as a left part variable in an assignment statement within the procedure_body and which is not called by value can only correspond to an actual parameter which is a variable (special case of expression).

###### 4.7.5.3.
A formal parameter which is used within the procedure_body as an array identifier can only correspond to an actual parameter which is an array identifier of an array of the same dimensions. In addition if the formal parameter is called by value the local array created during the call will have the same subscript bounds as the actual array.

###### 4.7.5.4.
A formal parameter which is called by value cannot in general correspond to a switch identifier or a procedure_identifier or a string, because these latter do not possess values (the exception is the procedure identifier of a procedure declaration which has an empty formal parameter part (cf. section 5.4.1) and which defines the value of a function designator (cf. section 5.4.4). This procedure identifier is in itself a complete expression).

###### 4.7.5.5.
Any formal parameter may have restrictions on the type of the corresponding actual parameter associated with it (these restrictions may, or may not, be given through specifications in the procedure heading). In the procedure_statement such restrictions must evidently be observed.

. . . . . . . . . ( end of section 4.7.5. Restrictions.) <<Contents | End>>

##### 4.7.7. Parameter delimiters.
All parameter delimiters are understood to be equivalent. No correspondence between the parameter delimiters used in a procedure_statement and those used in the procedure_heading is expected beyond their number is the same. Thus the information conveyed by using the elaborate ones is entirely optional.

##### 4.7.8. Procedure body expressed in code.
The restrictions imposed on a procedure statement calling a procedure having its body expressed in non-Algol code evidently can only be derived from the characteristics of the code used and the intent of the user and thus fall outside the scope of the reference language.

. . . . . . . . . ( end of section 4.7. Procedure statements) <<Contents | End>>

. . . . . . . . . ( end of section 4. Statements) <<Contents | End>>

### 5. Declarations

Declarations serve to define certain properties of the quantities used in the program, and to associate them with identifiers. A declaration of an identifier is valid for one block. Outside this block the particular identifier may be used for other purposes (cf. section 4.1.3).

Dynamically this implies the following: at the time of an entry into a block (through the begin since the labels inside are local and therefore inaccessible from outside) all identifiers declared for the block assume the significance implied by the nature of the declarations given. If these identifiers had already been defined by other declarations outside they are for the time being given a new significance. Identifiers which are not declared for the block, on the other hand, retain their old meaning.

At the time of an exit from an block (through end, or by a go to statement) all identifiers which are declared for the block lose their local significance.

A declaration my be marked with the additional declarator own. This has the following effect: upon a reentry into the block, the values of own quantities will be unchanged from their values at the last exit, while the values of declared variables which are not marked as own are undefined. Apart from labels and formal parameters of procedure declarations and with the possible exception of those for standard functions (cf. sections 3.2.4 and 3.2.5) all identifiers of a program must be declared. No identifier may be declared more than once in any one block head.

#### Syntax.

1. declaration::= type_declaration | array_declaration | switch_declaration | procedure_declaration.

#### 5.1. Type declarations

##### 5.1.1 Syntax.
1. type_list::= simple_variable | simple_variable , type_list.
2. type::= real | integer | Boolean.
3. local_or_own_type::= type | own type.
4. type_declaration::= local_or_own_type type_list.
##### 5.1.2. Examples.
  integer  p, q, s
  own  Boolean Acryl, n
##### 5.1.3. Semantics.
Type declarations serve to declare certain identifiers to represent simple variables of a given type. Real declared variables may only assume positive or negative values including zero. Integer declared variables may only assume positive and negative integral values including zero. Boolean declared variables may only assume the values true and false.

In arithmetic expressions any position which can be occupied by a real declared variable may be occupied by an integer declared variable.

For the semantics of own, see the fourth paragraph of section 5 above.

. . . . . . . . . ( end of section 5.1. Type declarations) <<Contents | End>>

#### 5.2. Array declarations

##### 5.2.1 Syntax.
1. lower_bound::= arithmetic_expression.
2. upper_bound::= arithmetic_expression.
3. bound_pair::= lower_bound colon upper_bound.
4. bound_pair_list::= bound_pair | bound_pair_list comma bound_pair.
5. array_segment::= array_identifier l_bracket bound_pair_list r_bracket | array_identifier comma array_segment
6. array_list::= array_segment | array_list comma array_segment
7. array_declaration::= array array_list | local_or_own_type array array_list
##### 5.2.2. Examples.
  array  a, b, c [7:n, 2:m], s [-2:10]
  own   integer   array  A [ if  c<0  then  2  else  1:20]
  real   array  q [-7:-1]
##### 5.2.3. Semantics of array declarations.
An array declaration declares one or several identifiers to represent multidimensional arrays of subscripted variables and gives the dimensions of the arrays, the bound of the subscripts, and the types of the variables.
###### 5.2.3.1. Subscript bounds.
The subscript bounds for any array are given in the first subscript bracket following the identifier of this array in the form of a bound pair list. Each item of this list gives the lower and upper bound of a subscript in the form of two arithmetic expressions separated by the delimiter :. The bound pair list gives the bounds of all subscripts taken in order from left to right.
###### 5.2.3.2. Dimensions.
The dimensions are given as the number of entries in the bound pair list.
###### 5.2.3.3. Types.
All arrays declared in one declaration are of the same quoted type. If no type declarator is given the type real is understood.

. . . . . . . . . ( end of section 5.2.3. Semantics of array declarations.) <<Contents | End>>

##### 5.2.4. Lower upper bound expressions.
###### 5.2.4.1.
The expressions will be evaluated in the same way as subscript expressions (cf. section 3.1.4.2).
###### 5.2.4.2.
The expressions can only depend on variables and procedures which are non-local to the block for which the array declaration is valid. Consequently in the outermost block of a program only array declarations with constant bounds may be declared.
###### 5.2.4.3.
An array identifier id defined only when the values of all upper subscript bounds are not smaller than those of the corresponding lower bounds.
###### 5.2.4.4.
The expressions will by evaluated once at each entrace into the block.

. . . . . . . . . ( end of section 5.2.4. Lower upper bound expressions.) <<Contents | End>>

##### 5.2.5. The identity of subscripted variables.
The identity of a subscripted variable is not related to the subscript bounds given in the array declaration. However, even if an array is declared own the values of the corresponding subscripted variables will, at any time, be defined only for those of these variables which have subscripts within the most recently calculated subscript bounds.

. . . . . . . . . ( end of section 5.2. Array declarations) <<Contents | End>>

#### 5.3. Switch declarations

##### 5.3.1 Syntax.

1. switch_list::= designational_expression | switch_list , designational_expression

2. switch_declaration::= switch switch_identifier ":=" switch_list

##### 5.3.2. Examples.

 switch  S:=S1,S2,Q[m],  if  v>-5  then  S3  else  S4
 switch  Q:=p1,w

##### 5.3.3. Semantics.
A switch declaration defines the set of values of the corresponding switch designators. These values are given one by one as the values of the designational expressions entered in the switch list. With each of these designational expressions there is associated a positive integer, 1, 2, ..., obtained by counting the items in the list from left to right. The value of the switch designator corresponding to a given value of the subscript expression (cf. section 3.5. Designational expressions) is the value of the designational expression in the switch list having this given value as its associated integer.
##### 5.3.4. Evaluation of expressions in the switch list.
An expression in the switch list will be evaluated every time the item of the list in which the expression occurs is referred to, using the current values of all variables involved.
##### 5.3.5. Influence of scopes.
If a switch designator occurs outside the scope of a quantity entering into a designational expression in the switch list, and an evaluation of this switch designator selects this designational expression, then the conflicts between the identifiers for the quantities in this expression and the identifiers whose declarations are valid at the place of the switch designator will be avoided through suitable systematic changes of the latter identifiers.

. . . . . . . . . ( end of section 5.3. Switch declarations) <<Contents | End>>

#### 5.4. Procedure declarations

##### 5.4.1 Syntax.
1. formal_parameter::= identifier .
2. formal_parameter_list::= formal_parameter | formal_parameter_list parameter_delimiter formal_parameter.
3. formal_parameter_part::= empty | l_paren formal_parameter_list r_paren.
4. identifier_list::= identifier | identifier_list comma identifier.
5. value_part::= value identifier_list semicolon | empty.
6. specifier::= string | type | array | type array | label | switch | procedure | type procedure.
7. specification_part::= empty | specifier identifier_list semicolon | specification_part specifier identifier_list.
8. procedure_heading::= procedure_identifier formal_parameter_part semicolon value_part specification_part.
9. procedure_body::= statement | code.
##### 5.4.2. Example Procedures
 procedure  Spur (a) Order: (n); value  n;
 array  a; integer  n; real  s;
 begin  integer  k;
 s:=0;
 for  k:=1 step  1 until  n do  s:=s+a[k,k]
 end

 procedure  Transpose (a) Order: (n); value  n;
 array  a; integer  n;
 begin  real  w; integer  i, k;
 for  i := 1 step  1 until  n do
     for  k := 1+i step  1 until  n do
         begin  w:=a[i,k];
                 a[i,k]:=a[k,i];
                 a[k,i]:=w
         end
 end  Transpose

 integer  procedure  Step (u); real  u;
 Step:=if  0<=u/\u<=1 then  1 else  0

 procedure  Absmax (a) Size: (n, m) Result: (y) Subscripts: (i, k);
 comment  The absolute greatest element of the matrix a, of size n by m
 is transferred to y, and the subscripts of this element to i and k;
 array  a; integer  n, m, i, k; real  y;
 begin  integer  p, q;
 y := 0;
 for  p:=1 step  1 until  n do  for  q:=1 step  1 until  m do
 if  abs(a[p,q]])>y then  begin  y:=abs(a[p,q]);
     i:=p; k:=q end  end  Absmax

 procedure  Innerproduct (a, b) Order: (k, p) Result: (y); value  k;
 integer  k, p; real  y, a, b;
 s:=0;
 for  p:=1 step  1 until  k do  s:=s+a*b;
 y:=s
 end  Innerproduct

##### 5.4.3. Semantics.
A procedure declaration serves to define the procedure associated with a procedure_identifier. The principal constituent of a procedure declaration is a statement or a piece of code, the procedure_body, which through the use of procedure_statements and/or function designators may be activated from other parts of the block in the head of which the procedure declaration appears. Associated with the body is a heading, which specifies certain identifiers occurring within the body to represent formal parameters. Formal parameters in the procedure_body will, whenever the procedure is activated (cf. section 3.2. Function designators and section 4.7. Procedure statements) be assigned the values of or replaced by actual parameters. Identifiers in the procedure_body which are not formal will be either local or non-local to the body depending on whether they are declared within the body or not. Those of them which are non-local to the body may well be local to the block in the head of which the procedure declaration appears. The procedure_body always acts like a block, whether it has the form of one or not. Consequently the scope of any label labelling a statement within the body or the body itself can never extended beyond the procedure_body. In addition, if the identifier of a formal parameter is declared anew within the procedure_body (including the case of its use as a label in section 4.1.3), it is thereby given a local significance and actual parameters which correspond to it are inaccessible throughout the scope of its inner local quantity..

5.4.4. Values of function designators. For a procedure declaration to define the value of a function designator there must, within the procedure declaration body, occur one or more explicit assignment statements with the procedure_identifier in a left part; at least one of these must be executed, and the type associated with the procedure identifier must be declared through the appearance of a type declarator as the very first symbol of the procedure declaration. The last value so assigned is used to continue the evaluation of the expression in which the function designator occurs. Any occurrence of the procedure identifier within the body of the procedure other than in a left part in an assignment statement denotes activation of the procedure.

5.4.5. Specifications. In the heading a specification part, giving information about the kinds and types of the formal parameters by means of an obvious notation, may be included. In this part no formal parameter may occur more than once. Specification of formal parameters called by value (cf. section 4.7.3.1) must be supplied and specifications of formal parameters called by name (cf. section 4.7.3.2) may be omitted.

##### 5.4.6. Code as procedure body.
It is understood that the procedure body may be expressed in non-Algol language. Since it is intended that the use of this feature should be entirely a question of hardware representation, no further rules concerning this code language can be given within the reference language.

. . . . . . . . . ( end of section 5.4. Procedure declarations) <<Contents | End>>

. . . . . . . . . ( end of section 5. Declarations) <<Contents | End>>

### Examples

1. [Omitted]

. . . . . . . . . ( end of section Examples) <<Contents | End>>

. . . . . . . . . ( end of section Description) <<Contents | End>>

. . . . . . . . . ( end of section Syntax) <<Contents | End>>

# End Notes

## Translation of Quote

 From A.R.Diller@computer-science.birmingham.ac.uk Thu Mar  2 15:20 PST 1995
 Date: Thu, 2 Mar 95 17:05:14 GMT
 From: A.R.Diller@computer-science.birmingham.ac.uk
 Message-Id: <1711.9503021705@fat-controller.cs.bham.ac.uk>
 To: dick@blaze.csci.csusb.edu

Thanks for the information concerning the online Algol60 report. A colleague of mine is a Wittgenstein expert and concerning the quote and he informs me that:

The first bit

                     Was sich ueberhaupt sagen laesst, laesst sich
                     klar sagen;

means "what can be said at all can be said clearly" and is the 2nd sentence in Tractatus 4.116.

The second bit:

                          und wovon man nicht reden
                          kann, darueber muss man schweigen.`

means "whereof we cannot speak, thereover must we pass in silence", and is the famous Tractatus 7.

The two bits may be together in some other (perhaps) preparatory piece of writing.

I hope that helps,

Antoni Diller

## 2005-06-15 Wed Jun 15 08:06 Correction from Vladimir A Merzliakov

"Vladimir A Merzliakov" <wanderer@rsu.ru> found a broken link. I think it was caused by a bug in my mth2html translation script. I hope this version has lost it.

## 2005-06-15 Wed Jun 15 08:06 Working on some broken internal links

Some of the ned of section links are not correct so that you can not return rapidly to the head of a section from the end of it.

. . . . . . . . . ( end of section End Notes) <<Contents | End>>