We study first-order concatenation theory with bounded quantifiers. We give axiomatizations with interesting properties, and we prove some normal-form results. Finally, we prove a number of decidability and undecidability results.

We introduce two essentially undecidable first-order theories WT and T. The intended model for the theories is a term model. We prove that WT is mutually interpretable with Robinson's R. Moreover, we prove that Robinson's Q is interpretable in T.

In the language $$\lbrace 0, 1, \circ , \preceq \rbrace $$ { 0 , 1 , ∘ , ⪯ } , where 0 and 1 are constant symbols, $$\circ $$ ∘ is a binary function symbol and $$\preceq $$ ⪯ is a binary relation symbol, we formulate two theories, $$ \textsf {WD} $$ WD and $$ {\textsf {D}}$$ D , that are mutually interpretable with the theory of arithmetic $$ {\textsf {R}} $$ R and Robinson arithmetic $${\textsf {Q}} $$ Q , respectively. The intended model of $$ \textsf {WD} $$ WD and $$ {\textsf {D}}$$ D is the free semigroup generated by $$\lbrace {\varvec{0}}, {\varvec{1}} \rbrace $$ { 0 , 1 } under string concatenation extended with the prefix relation. The theories $$ \textsf {WD} $$ WD and $$ {\textsf {D}}$$ D are purely universally axiomatised, in contrast to $$ {\textsf {Q}} $$ Q which has the $$\varPi _2$$ Π 2 -axiom $$\forall x \; [ \ x = 0 \vee \exists y \; [ \ x = Sy \ ] \ ] $$ ∀ x [ x = 0 ∨ ∃ y [ x = S y ] ] .

We identify a number of decidable and undecidable fragments of first-order concatenation theory. We also give a purely universal axiomatization which is complete for the fragments we identify. Furthermore, we prove some normal-form results.

We show that there is no theory that is minimal with respect to interpretability among recursively enumerable essentially undecidable theories.