Morphism: a Symbolic List Processing Language

With the formal study of natural languages, we have obtained some residual results which greatly increase our power of logical manipulation. Coupled with a computer, our language extends enormously the external logic of our computing device. This paper is concerned with a well-defined symbolic list processing language and a set of operators for this language. These operators, which we call morphisms, are themselves operators thereby giving us an indefinite nesting ability. We have defined a schematic representation of'our operators (called a comb-scheme) and have then proceeded to describe a numbering technique which allows us to use these operators with a computer. Since our operators and lists are defined at the time of execution, our ability to change them is unlimited. The pattern of this paper is to build up from an alphabet to words, then non-connected words, then lists and the operations on lists and finally operations on the operations. We have taken, what we hope to be, descriptive examples in linguistics and non-linguistical applications in the hope of being representative of our language and easily understandable. V A N D E N B U R G H 1. MOP,.PHISM: A S Y M B O L I C LIST P R O C E S S I N G LANGUAGE I n t r o d u c t i o n We at the Facult~ des Sciences of the Umverslte de Rennes are interested in the mathematical study of languages. Since we have a computer (1620), it is only natural that we attempt to combine the two. With languages being our major interest, we find that an extensive study of lists and symbolic list processing techniques will yield a most powerful and broad spectrum of results when applied to a computer. It appears that problems involving natural languages and certain types of mathematical problems are computer solvable when list processing techniques are applied to them. There are also linguistical problems which yield beneficial results when they are approached in this manner. We are therefore going to define: lists, operations on lists (which we call morphisms), and a comb-scheme which allows us to manipulate the lists and morphisms; this scheme gives us the ability to manipulate almost any pattern that can be handled in a logical manner (in linguistic usage, the term logical will refer to letters, words, sentences, phonemes; in a predetermined or random occurrence). We have defined our operators in such a manner that we allow for an indefinite nesting; adding to a computing machine an external logical system. Also, our operators, in the computer, are defined at proces~ ing time so our ability to change them at execution time is unlimited. This paper can be considered to be divided into four sections. Part one defines our terms and gives definitions for such things as an alphabet, lists, words, non-connected words, etc. Part two defines a morphism and our operators on lists. Part three shows our combscheme. Part four consists of examples: in linguistics; an example in the logical generation of a flow diagram into a computer program; and finally, an example of the solution of a mathematical problem (the arithmetic operations on polynominials of n variables). V A N D E N B U R G H 2. I. Definitions I. 1 Words and Symbols Since we are using a computer, composed of: let us define our alphabet to be the Latin capital letters the blank (#) for notational purposes but on printed page a blank. the punctuation marks . and , the digits excluding: = $ / + @ ( ) and 0 which is also acceptable to the computer. I. Z Non-connected words Let us put in parallel the Latin phrase: nec adulatoribus latus praebes and its French equivalent: et tune pretes pas le flanc aux flatteurs; The Latin word nec corresponds, in French, to the three nonconsecutive words e_t ... n.e ... pa_.~s and the Latin word praebes corresponds to tu ... prates Thus it appears that for the words of a language, on the same level as words, there is a series of words not necessarily consecutive: the non-connected words. For a non-connected word, the components will be separated by the symbol: . For example: A T A BP,_IQUE RI DO are non-connected words. The first has two components and the second has three components. 1. 3 C o m b s c h e m e In t h e f o r m u l a s , a s i n a l g e b r a , we d e s i g n a t e a v a r i a b l e b y x ( a n d , e v e n t u a l l y i n t e r v e n i n g v a r i a b l e s b y Xl , x 2 ' ' " ); a n o n c o n n e c t e d w o r d of N c o m p o n e n t s w i l l be s c h e m a t i z e d b y a c o m b o r c o m b s c h e m e