Abstract
The size of deterministic automata required for recognizing regular and [Formula: see text]-regular languages is a well-studied measure for the complexity of languages. We introduce and study a new complexity measure, based on the sensing required for recognizing the language. Intuitively, the sensing cost quantifies the detail in which a random input word has to be read in order to decide its membership in the language. We study the sensing cost of regular and [Formula: see text]-regular languages, as well as applications of the study in practice, especially in the monitoring and synthesis of reactive systems.
Highlights
Studying the complexity of a formal language, there are several complexity measures to consider
For regular and ω-regular languages, given by means of finite-state automata, the classical complexity measure is the size of a minimal deterministic automaton that recognizes the language
Mathematical tools in signal processing are used to reconstruct information based on compressed sensing [6], and in the context of data streaming, one cannot store in memory the entire input, and has to approximate its properties according to partial “sketches” [13]
Summary
Studying the complexity of a formal language, there are several complexity measures to consider. We suggest definitions that capture the intuition of “required number of sensors” in these settings and solve the problems of generating monitors and transducers that minimize sensing For both settings, we focus on safety languages. Our goal is extended to designing a transducer that controls the traffic lights according to the speed of the traffic in each direction, and satisfies some specification (say, give priority to slow traffic), while minimizing the sensing of cars. Recall that the definition of sensing above assumes a uniform probability on the assignments to the signals, whereas in monitoring we want to consider instead more intricate probability spaces – ones that restrict attention to words in the language. The problems of computing the minimal sensing cost and finding a minimallysensing transducer are EXPTIME-complete even for specifications given by means of deterministic safety automata. A transducer that attains the minimal sensing cost always exists for safety specifications
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