  Reactive synthesis automatically derives a strategy that satisfies a given
specification. However, requiring a strategy to meet the specification in every
situation is, in many cases, too hard of a requirement. Particularly in
compositional synthesis of distributed systems, individual winning strategies
for the processes often do not exist. Remorsefree dominance, a weaker notion
than winning, accounts for such situations: dominant strategies are only
required to be as good as any alternative strategy, i.e., they are allowed to
violate the specification if no other strategy would have satisfied it in the
same situation. The composition of dominant strategies is only guaranteed to be
dominant for safety properties, though; preventing the use of dominance in
compositional synthesis for liveness specifications. Yet, safety properties are
often not expressive enough. In this paper, we thus introduce a new winning
condition for strategies, called delay-dominance, that overcomes this weakness
of remorsefree~dominance: we show that it is compositional for many safety and
liveness specifications, enabling a compositional synthesis algorithm based on
delay-dominance for general specifications. Furthermore, we introduce an
automaton construction for recognizing delay-dominant strategies and prove its
soundness and completeness. The resulting automaton is of single-exponential
size in the squared length of the specification and can immediately be used for
safraless synthesis procedures. Thus, synthesis of delay-dominant strategies
is, as synthesis of winning strategies, in 2EXPTIME.
