  In this paper, we investigate the synthesis problem of terminating reactive
systems from quantitative specifications. Such systems are modeled as finite
transducers whose executions are represented as finite words in $(I\times
O)^*$, where $I,O$ are finite sets of input and output symbols, respectively. A
weighted specification $S$ assigns a rational value (or $-\infty$) to words in
$(I\times O)^*$, and we consider three kinds of objectives for synthesis,
namely threshold objectives where the system's executions are required to be
above some given threshold, best-value and approximate objectives where the
system is required to perform as best as it can by providing output symbols
that yield the best value and $\varepsilon$-best value respectively w.r.t. $S$.
We establish a landscape of decidability results for these three objectives and
weighted specifications with partial domain over finite words given by
deterministic weighted automata equipped with sum, discounted-sum and average
measures. The resulting objectives are not regular in general and we develop an
infinite game framework to solve the corresponding synthesis problems, namely
the class of (weighted) critical prefix games.
