  A system with sporadic errors (SSE) is a controller which produces high
quality output but it may occasionally violate a critical requirement REQ(I,O).
A run-time enforcement shield is a controller which takes (I,O) (coming from
SSE) as its input, and it produces a corrected output O' which guarantees the
invariance of requirement REQ(I,O'). Moreover, the output sequence O' must
deviate from O "as little as possible" to maintain the quality. In this paper,
we give a method for logical specification of shields using formulas of logic
Quantified Discrete Duration Calculus (QDDC). The specification consists of a
correctness requirement REQ as well as a hard deviation constraint HDC which
must both be mandatorily and invariantly satisfied by the shield. Moreover, we
also use quantitative optimization to give a shield which minimizes the
expected value of cumulative deviation in an H-optimal fashion. We show how
tool DCSynth implementing soft requirement guided synthesis can be used for
automatic synthesis of shields from a given specification. Next, we give
logical formulas specifying several notions of shields including the
k-Stabilizing shield of Bloem "et al." as well as the Burst-error shield of Wu
"et al.", and a new e,d-shield. Shields can be automatically synthesized for
all these specifications using the tool DCSynth. We give experimental results
showing the performance of our shield synthesis tool in relation to previous
work. We also compare the performance of the shields synthesized under diverse
hard deviation constraints in terms of their expected deviation and the worst
case burst-deviation latency.
