Good-Enough Synthesis
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In the classical synthesis problem, we are given an LTL formula $\psi$ over sets of input and output signals, and we synthesize a system $\T$ that realizes $\psi$: with every input sequences $x$, the system associates an output sequence $\T(x)$ such that the generated computation $x \otimes \T(x)$ satisfies $\psi$. In practice, the requirement to satisfy the specification in all environments is often too strong, and it is common to add assumptions on the environment.
We introduce and study a new type of relaxation on this requirement. In good-enough synthesis (GE-synthesis), the system is required to generate a satisfying computation only if one exists. Formally, an input sequence $x$ is hopeful if there exists some output sequence $y$ such that the computation $x \otimes y$ satisfies $\psi$, and a system GE-realizes $\psi$ if it generates a computation that satisfies $\psi$ on all hopeful input sequences. GE-synthesis is particularly relevant when the notion of correctness is multi-valued (rather than Boolean), and thus we seek systems of the highest possible quality, and when synthesizing autonomous systems, which interact with unexpected environments and are often only expected to do their best.
We study GE-synthesis in Boolean and multi-valued settings. In both, we suggest and solve various definitions of GE-synthesis, corresponding to different ways a designer may want to take hopefulness into account. We show that in all variants, GE-synthesis is not computationally harder than traditional synthesis, and can be implemented on top of existing tools. Our algorithms are based on careful combinations of nondeterministic and universal automata. We augment systems that GE-realize their specifications by monitors that provide satisfaction information. In the multi-valued setting, we provide both a worst-case analysis and an expectation-based one, the latter corresponding to an interaction with a stochastic environment.