Concatenation theory, also called string theory, character-string theory, or theoretical

syntax In linguistics, syntax () is the study of how words and morphemes combine to form larger units such as phrases and sentences. Central concerns of syntax include word order, grammatical relations, hierarchical sentence structure ( constituenc ...

, studies character strings over finite alphabets of characters, signs, symbols, or marks. String theory is foundational for

formal linguistics Formal linguistics is the branch of linguistics which uses applied mathematical methods for the analysis of natural languages. Such methods include formal languages, formal grammars and first-order logical expressions. Formal linguistics also forms ...

, computer science, logic, and metamathematics especially proof theory. A

generative grammar Generative grammar, or generativism , is a linguistic theory that regards linguistics as the study of a hypothesised innate grammatical structure. It is a biological or biologistic modification of earlier structuralist theories of linguisti ...

can be seen as a recursive definition in string theory. The most basic operation on strings is

concatenation In formal language theory and computer programming, string concatenation is the operation of joining character strings end-to-end. For example, the concatenation of "snow" and "ball" is "snowball". In certain formalisations of concatenat ...

; connect two strings to form a longer string whose length is the sum of the lengths of those two strings. ABCDE is the concatenation of AB with CDE, in symbols ABCDE = AB ^ CDE. Strings, and concatenation of strings can be treated as an algebraic system with some properties resembling those of the addition of integers; in modern mathematics, this system is called a

free monoid In abstract algebra, the free monoid on a set is the monoid whose elements are all the finite sequences (or strings) of zero or more elements from that set, with string concatenation as the monoid operation and with the unique sequence of zero ele ...

. In 1956

Alonzo Church Alonzo Church (June 14, 1903 – August 11, 1995) was an American mathematician, computer scientist, logician, philosopher, professor and editor who made major contributions to mathematical logic and the foundations of theoretical computer scien ...

wrote: "Like any branch of mathematics, theoretical syntax may, and ultimately must, be studied by the axiomatic method". Church was evidently unaware that string theory already had two axiomatizations from the 1930s: one by Hans Hermes and one by

Alfred Tarski Alfred Tarski (, born Alfred Teitelbaum;School of Mathematics and Statistics, University of St Andrews ''School of Mathematics and Statistics, University of St Andrews''. January 14, 1901 – October 26, 1983) was a Polish-American logician a ...

. Coincidentally, the first English presentation of Tarski's 1933 axiomatic foundations of string theory appeared in 1956 – the same year that Church called for such axiomatizations.Pages 173–4 of Alfred Tarski, ''The concept of truth in formalized languages'', reprinted in ''Logic, Semantics, Metamathematics'', Hackett, Indianapolis, 1983, pp. 152–278 As Tarski himself noted using other terminology, serious difficulties arise if strings are construed as tokens rather than types in the sense of Pierce's type-token distinction.

References

{{Reflist Logic Syntax Philosophy of logic History of logic Philosophy of language