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From:
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
Project MAC
Memorandum MACM347 April 28, 1967
Project MAC participants
J. J. Donovan, k. F. Ledgard Subject: Canonic Systems1and Their Application to Programming Languages
Abstract: I i
This paper presents two basic results: the use of established method,s of recursive defini1tio.n to
1. specify the syntax of computer languages (including contextsensitive require~ents),
2. specify the translation of programs in one computer language into programs in another language.
The method can be used to write one specification for both the syntax of a language (e.g. a source language) and its translation into a target language (e.g. an assembler ! language). A syntactically legal program and its translation into the target language can then be generated using the specification. If the target language is understood, the semantics of the first language is specified.
The paper develops the method of recursi.ve definition in conjunction with an example specifying the syntax of a limited subset of PL/I and its translation into IBM System/360 assembler language.
I The application of the method to a generalized translator for computer
languages is discussed.
1. Introduction
Our objective is to present a single method for expressing the syntax
and translation of computer languages.
The objective to develop either methods for specifying the syntax or
methP,ds for specifying the translation of computer lanugages is not new.
In response to the demand for numerous problemoriented computer languages to
meet the needs of diverse fields, there has been considerable activity to ease
the effort required to implement a language.
Much of this activity has led to the development of methods for
specifying, at least in part, the syntax of computer languages.< 7 12)
The methods for specifying syntax have facilitated the description of
computer languages to members of the comp~ting field and have led to t he
development of syntaxdirected translators.
. (25 26) Other activity has been directed to developing tabledriven compilers '
(18,19) h and programming languages for expressing string transformations. Te
table  driven compilers have generally been limited to a particular type of ,,
target language and have required excessive detail in writing the specifications
to fill the tables for a particular source language. The string transformation
languages have been limited to special types of string transformations and ,,, .t have not been found generally useful for translating computer languages .:~
d (15,16,17) Approaches to the ormalization of semantics have also been ma e.
Here we present a single, formal method for specifying completely the
syntax and translation of a computer language. The method is independent
of both the source and target languages. The method uses an uncluttered,
readable notation. The method recursively classifies sets of strings.
The syntax of a computer language is characterized by specifying a set
where each element is a syntactically legal program. the translation of a
( computer language is characterized by specifying a set of ordered pairs, where the first element of each pair is a legal program in the source
language and the second element is a corresponding program in the target language
that preserves the meaning of the source language program. If the target
language is understood, the semantics of the source language has bee.n specified.
The paper develops the method of recursive definition with an example
specifying the syntax of a limited subset of PL/I and its translation into
IBM System/360 assembler language. A discussion of the power of the method
and of its application to a generalized translator is presented. An
ordered set of appendices is also presented. The appendices present:
1) a brief summary of the notation for the method of recursive definition,
2) two programs in the subset of PL/I and their translation into System/360
assembler language, 3) a BackusNaur Form specification of the syntax of
the subset of PL/I, 4) a complete specification of the syntax of the subset

0
using the method of recursive definition, 5) a complete specification of the
syntax of the subset artd its translation into System/360 assembler language using
the method of recursive definition, and 6) a derivation of a syntactically
legal program in the subset and its translation into assembler language .
2. Basis of Formalization
The formalization for the method presented here evolved from Post's
canonical systems, (l)and hence will be called canpnic systems. Smullyan( 2)
used an applied variant of the canonical systems of Post in his definition
of elementary formal systems . In class notes on the application of
elementary formal systems to the definition of self contained mathematical
systems, Trenchard More(J) modified the definition of elementary formal
systems . Elementary formal systems (now called canonic systems in recognition
of the earlier work by Post) were further modified to meet the definitional
needs of computer languages and applied to the definition of syntax by
Donovan.
(
(
which assex:ts tni:1t the logical notion of "recursive function" or "recursively I
enumerable set" '(which is encompassed by canonic systems) is capable of fully l
'r describing ~e intuitive concept of an algorithm. >) ,t . H H
3. Canonict Systems
A canonic system is a finite sequence of rules for recursively defining
sets. The elements of the sets are strings of symbols selected from some
finite alphabet. Each rule is called a canon. A canon generally has the
form
set A b set B
which may be interpreted informally:
If 11a1
11 is a member of the set named "set A111 ,and 11 a2
11 is a member of
the set named "set A2
11 ; and "a II is a member of the set , n
named "set A", then we can assert that "b" is a member of the !l
set named "set B" .
The "a. 11 and "b" represent symbols from the finite alphsbet; the "set A." 1 ~~i
and "set B" are the names of the sets defined.
In the remainder of this section we will elaborate on this notation.
A synopsis of the notation is given in Appendix 1. and may be used as a
reference throughout the text. The notation will be developed by a series
of examples taken from the canonic system specifying the syntax of a subset
of PL/I, called Little PL/I. This subset includes limited f orms of PL/I
GO TO statements, IF statements, label assignment statements, label declaration
statements, and arithmetic assignment statements. The Backus Naur Form
description of the subset is given_ in Appendix 3. Two example syntactically
legal programs in the subset are given in Appendix 2. One of these examples
is repeated here:
0
PROCEDURE ; DECLARE LX LABEL; L: I= I+l:A*IBIC;
LX=L; GO TO CHECK;
M: I = I+lj LX=M;
CHECK: IF I
(
(
These canons may be read :
(From no premises) we can assert that the symbol "A" is a member of the set named " letter" .
(From no pr emises) we can assert that the symbol "B" is a member of the set named "letter" .
And so on.
The canons specify a set named "letter" comprising the capital letters of
the English alphabet. The sign 11 ~ " is the assertion sign. The strings "A letter" , "B letter", , and "Z letter" are conclusions . The capital
English letters A through Z are members of the object language. The underlined,
lower case characters are predicates. A predicate, here "letter" , is the
name of a s et .
The set named "identifier" may be specified in terms of the set
named "letter":
11 letter r .l1 letter 1
,/ 1
identifier
12 letter f /l /2 identifier
/ 1 letter + i letter 1 2
These canons may be read:
If "4" is a member of the set named "letter", then we can assert that 11/
111 is a member of the set named " identifier".
If " i " is a member of the set named 1 the set named " letter" , then we
of the set named "identifier".
And so on.
*
"letter" and 111 " is a member of 2 can assert that 11 )' f' " is a member 1 2
The canons specify a set consisting of identifiers of one to eight capital
* Actually, these canons shoul d properly be called canon schema, which denote in~r~nc.P.~ of c.~non~.
(27)
(28)
(34)
r
English letters. The lower case (possibly subscripted or superscripted)
English letters 1 1 through L8 are variables that represent members of the
set named "letter" . The strin= 11 I " " I J " are terms. A term is a string er 1 , 1 2 of variables or symbols from the object alphabet (e.g. 111
111 , 11 i
1J
211 , or
"A.11 1211
) . The sign "t" is the conjunction sign. The strings 11 .f1
letter",
"P2 letter", "13 letter" are premises. The " 't " before and 11 ~ " separates premises, all of which must be satisfied to assert the conclusion. The
strings ")'~ letter", 11i 1 identifier", 11J
2 letterit, 11,.t
1 ,/
2 identifier" are
remarks.
The vocabulary used to describe a canon is sumarized as follows:
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