  In recent years there has been considerable interest in theories over string
equations, length function, and string-number conversion predicate within the
formal verification, software engineering, and security communities. SMT
solvers for these theories, such as Z3str2, CVC4, and S3, are of immense
practical value in exposing security vulnerabilities in string-intensive
programs. Additionally, there are many open decidability and
complexity-theoretic questions in the context of theories over strings that are
of great interest to mathematicians. Motivated by the above-mentioned
applications and open questions, we study a first-order, many-sorted,
quantifier-free theory $T_{s,n}$ of string equations, linear arithmetic over
string length, and string-number conversion predicate and prove three theorems.
First, we prove that the satisfiability problem for the theory $T_{s,n}$ is
undecidable via a reduction from a theory of linear arithmetic over natural
numbers with power predicate, we call power arithmetic. Second, we show that
the string-numeric conversion predicate is expressible in terms of the power
predicate, string equations, and length function. This second theorem, in
conjunction with the reduction we propose for the undecidability theorem,
suggests that the power predicate is expressible in terms of word equations and
length function if and only if the string-numeric conversion predicate is also
expressible in the same fragment. Such results are very useful tools in
comparing the expressive power of different theories, and for establishing
decidability and complexity results. Third, we provide a consistent
axiomatization ${\Gamma}$ for the functions and predicates of $T_{s,n}$.
Additionally, we prove that the theory $T_{\Gamma}$ , obtained via logical
closure of ${\Gamma}$, is not a complete theory.
