# Concatenation theory

**Concatenation theory**, also called **string theory**, **character-string theory**, or **theoretical syntax**, studies character strings over finite alphabets of characters, signs, symbols, or marks. String theory is foundational for formal linguistics, computer science, logic, and metamathematics especially proof theory.[1] A generative grammar can be seen as a recursive definition in string theory.

The most basic operation on strings is concatenation; 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 1956 Alonzo Church wrote: "Like any branch of mathematics, theoretical syntax may, and ultimately must, be studied by the axiomatic method".[2] Church was evidently unaware that string theory already had two axiomatizations from the 1930s: one by Hans Hermes and one by Alfred Tarski.[3] 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.[4] 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, not to be confused with similar distinctions underlying other type-token distinctions.

## References

- John Corcoran and Matt Lavine, "Discovering string theory".
*Bulletin of Symbolic Logic*. 19 (2013) 253–4. - Alonzo Church,
*Introduction to Mathematical Logic*, Princeton UP, Princeton, 1956 - John Corcoran, William Frank and Michael Maloney, "String theory",
*Journal of Symbolic Logic*, vol. 39 (1974) pp. 625– 637 - Pages 173–4 of Alfred Tarski,
*The concept of truth in formalized languages*, reprinted in*Logic, Semantics, Metamathematics*, Hackett, Indianapolis, 1983, pp. 152–278