Abstract
Automata over infinite words, also known as omega-automata, play a key role in the verification and synthesis of reactive systems. The spectrum of omega-automata is defined by two characteristics: the acceptance condition (e.g. B\"uchi or parity) and the determinism (e.g., deterministic or nondeterministic) of an automaton. These characteristics play a crucial role in applications of automata theory. For example, certain acceptance conditions can be handled more efficiently than others by dedicated tools and algorithms. Furthermore, some applications, such as synthesis and probabilistic model checking, require that properties are represented as some type of deterministic omega-automata. However, properties cannot always be represented by automata with the desired acceptance condition and determinism. In this paper we study the problem of approximating linear-time properties by automata in a given class. Our approximation is based on preserving the language up to a user-defined precision given in terms of the size of the finite lasso representation of infinite executions that are preserved. We study the state complexity of different types of approximating automata, and provide constructions for the approximation within different automata classes, for example, for approximating a given automaton by one with a simpler acceptance condition.
Highlights
The specification of linear-time properties is a key ingredient of all typical frameworks for the verification and synthesis of reactive systems
Efficient synthesis algorithms exist for the class GR(1) of linear-time temporal logic specifications [2], which defines properties that are expressible as deterministic parity automata with three colors
We study the properties of n-lasso-precise approximations across the three dimensions of the complexity of the automata for such languages: size, acceptance condition, and determinism
Summary
The specification of linear-time properties is a key ingredient of all typical frameworks for the verification and synthesis of reactive systems. The choice of language approximation is inspired by applications in bounded model checking [5] and bounded synthesis [9] These methods are based on the observation that for finite-state systems, it suffices to consider lasso-shaped executions of bounded size. We study the approximation of nondeterministic by deterministic automata, and show that the worst-case exponential blow-up in the size is unavoidable for n-lasso-precise approximations As another example, consider the property described by the LTL formula ( p)∧( q), where p and q are some atomic propositions. In addition to the fact that their method applies to languages over finite words, the key difference to our work is that while their goal is to optimize precision within a state budget, we approximate automata with ones with simpler acceptance conditions that guarantees a desired precision. Our approximation allows us to approximate temporal properties with other temporal properties that are not necessarily safety
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