Abstract Quantitative semantics of Signal Temporal Logic (STL) play an important role in both the falsification and control synthesis for dynamical systems by assigning numerical quantities to truth values. Recently, several different quantitative semantics have been proposed, offering better performance in many cases. Yet a general, systematic understanding of the structure and properties of quantitative semantics is missing. In this paper, we develop a general framework to model quantitative semantics. We focus mainly on soundness, which requires that the quantitative semantics of a statement is positive when the statement is true, and negative when the statement is false. This ensures that counterexamples will not be missed during verification. We derive simple, necessary conditions in our framework for soundness. We show how several recently proposed quantitative semantics fit in our framework, and how others do not, typically because they do not strictly satisfy soundness. We implement various quantitative semantics, including existing semantics from literature, in our framework and compare their effectiveness as objective functions for optimization-based falsification on both novel and existing benchmarks.

Cyber-Physical Systems (CPS) consist of software interacting with the physical world, such as robots, vehicles, and industrial processes. CPS are frequently responsible for the safety of lives, property, or the environment, and so software correctness must be determined with a high degree of certainty. To that end, simply testing a CPS is insufficient, as its interactions with the physical world may be difficult to predict, and unsafe conditions may not be immediately obvious. Formal verification can provide stronger safety guarantees but relies on the accuracy of the verified system in representing the real system. Bringing together verification and implementation can be challenging, as languages that are typically used to implement CPS are not easy to formally verify, and languages that lend themselves well to verification often abstract away low-level implementation details. Translation between verification and implementation languages is possible, but requires additional assurances in the translation process and increases software complexity; having both in a single language is desirable. This paper presents a formalization of MARVeLus, a CPS language which combines verification and implementation. We develop a metatheory for its synchronous refinement type system and demonstrate verified synchronous programs executing on real systems.

Several quantitative semantics of temporal logics have been investigated recently. We propose a general form to model those quantitative semantics, establish requirements for soundness, and evaluate the framework on a few examples.