Lazy regular sensing

A complexity measure for regular languages based on the sensing required to recognize them was recently introduced by Almagor, Kuperberg, and Kupferman. 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, when the input letters composing the word are truth assignments to a finite set of signals. We introduce the notion of lazy sensing, where the signals are not sensed simultaneously. Rather, the signals are ordered, and a signal is sensed only if the values of the signals sensed so far have not determined the successor state. We study four classes of lazy sensing, induced by distinguishing between the cases where the order of the signals is static or dynamic (that is, fixed in advance or depends on the values of the signals sensed so far), and the cases where the order is global or local (that is, the same for all states of the automaton, or not). We examine the different classes of lazy sensing and the saving they enable, with respect to each other and with respect to (non-lazy) sensing. We also examine the trade offs between sensing cost and size. Our results show that the good properties of sensing are preserved in the lazy setting. In particular, saving sensing does not conflict with saving size: in all four classes, the lazy-sensing cost of a regular language can be attained in the minimal automaton recognizing the language.