Inherent Vacuity in Lattice Automata

Vacuity checking is traditionally performed after model checking has terminated successfully. It ensures that all the elements of the specification have played a role in its satisfaction by the system. The need to check the quality of specifications is even more acute in property-based design, where the specification is the only input, serving as a basis to the development of the system. Inherent vacuity adapts the theory of vacuity in model checking to the setting of property-based design. Essentially, a specification is inherently vacuous if it can be mutated into a simpler equivalent specification, which is known, in the case of specifications in linear temporal logic, to coincide with the fact the specification is satisfied vacuously in all systems. A recent development in formal methods is an extension of the Boolean setting to a multi-valued one. In particular, instead of Boolean automata, which either accept or reject their input, there is a growing interest in weighted automata, which map an input word to a value from a semiring over a large domain. A distributive finite lattice is a special case of a semiring, and lattice automata are used in several methods for reasoning about multi-valued objects. We study inherent vacuity in the setting of lattice automata, namely the ability to mutate the value of a transition in the automaton without changing its language. We define the concept of inherent vacuity in lattice automata, study the complexity of deciding different types of vacuity, and relate the setting to the one known for linear temporal logics.