Let $\{u_{t}(x)\colon t\geq 0{,}\ x\in \mathbb{R}\}$ be a
random string taking values in $\mathbb{R}^{d}$. It is specified
by the following stochastic partial differential equation,
\begin{equation*}
\frac{\partial u_{t}(x)}{\partial t}
=\frac{\partial^{2}u_{t}(x)}{\partial x^{2}}+\dot{W},
\end{equation*}
where $\dot{W}(x,t)$ is two-parameter
white noise. The objective of the present paper is to study
the fractal properties of the algebraic sum of the image sets
for the random string process $\{u_{t}(x)\colon t\geq 0{,}\
x\in \mathbb{R}\}$. We obtain the Hausdorff and packing dimensions
of the algebraic sum of the image sets of the string. We also
consider the existence of the local times of the process
$\{u_{s}(y)-u_{t}(x)\colon s,t\geq 0\colon x, y\in \mathbb{R}\}$,
and find the Hausdorff and packing dimensions of the level
sets for the process $\{u_{s}(y)-u_{t}(x)\colon s,t\geq 0;
x, y\in \mathbb{R}\}$.