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 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.

中文翻译：

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足够好的综合

在经典综合问题中，我们在输入和输出信号的集合上给出了一个 LTL 公式 \psi，我们合成了一个实现 \psi 的系统 T：对于每个输入序列 x，系统关联一个输出序列 T(x)生成的计算 x \otimes T(x) 满足 \psi。在实践中，在所有环境中都满足规范的要求往往过于强烈，在环境中添加假设是很常见的。我们针对这一要求引入了一种新的放松方式。在足够好的综合（GE 综合）中，系统需要生成一个令人满意的计算，只有在存在的情况下。形式上，如果存在某个输出序列 y 使得计算 x \otimes y 满足 \psi，则输入序列 x 是有希望的，并且系统 GE 实现了 \psi，如果它生成的计算满足所有有希望的输入序列的 \psi。当正确性的概念是多值的（而不是布尔值）时，GE 综合特别相关，因此我们寻求最高质量的系统，以及在合成与意外环境相互作用并且通常只期望做的自治系统时他们最好的。我们研究布尔和多值设置中的 GE 合成。在这两个方面，我们建议并解决了 GE 综合的各种定义，对应于设计人员可能希望考虑希望的不同方式。我们表明，在所有变体中，GE 合成在计算上并不比传统合成难，并且可以在现有工具之上实现。我们的算法基于非确定性和通用自动机的仔细组合。我们通过提供满意度信息的监视器来增强 GE 实现其规格的系统。在多值设置中，我们提供最坏情况分析和基于期望的分析，后者对应于与随机环境的交互。