Weak theories of concatenation and minimal essentially undecidable theories

We consider weak theories of concatenation, that is, theories for strings or texts. We prove that the theory of concatenation , which is a weak subtheory of Grzegorczyk's theory , is a minimal essentially undecidable theory, that is, the theory is essentially undecidable and if one omits an axiom scheme from , then the resulting theory is no longer essentially undecidable. Moreover, we give a positive answer to Grzegorczyk and Zdanowski's conjecture that 'The theory is a minimal essentially undecidable theory'. For the alternative theories and which have the empty string, we also prove that the each theory without the neutrality of is to be such a theory too.