Approximate automata for omega-regular languages

Automata over infinite words, also known as ω -automata, play a key role in the verification and synthesis of reactive systems. The spectrum of ω -automata is defined by two characteristics: the acceptance condition (e.g. Büchi 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 ω -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

Lindelöf spaces C(X) over topological groups

Theorem 1 proves (among the others) that for a locally compact topological group X the following assertions are equivalent: (i) X is metrizable and sigma-compact. (ii) C-p(X) is analytic. (iii) C-p(X) is K-analytic. (iv) C-p(X) is Lindelof. (v) C-c(X) is a separable metrizable and complete locally convex space. (vi) C,(X) is compactly dominated by irrationals. This result supplements earlier results of Corson, Christensen and Calbrix and provides several applications, for example, it easily applies to show that: (1) For a compact topological group X the Eberlein, Talagrand, Gul'ko and Corson compactness are equivalent and any compact group of this type is metrizable. (2) For a locally compact topological group X the space C-p(X) is Lindelof iff C-c(X) is weakly Lindelof. The proofs heavily depend on the following result of independent interest: A locally compact topological group X is metrizable iff every compact subgroup of X has countable tightness (Theorem 2). More applications of Theorem 1 and Theorem 2 are provided

Weak developments and metrization

AbstractThe notions of a weak k-development and of a weak development, defined in terms of sequences of open covers, were recently introduced by the first and the third authors. The first notion was applied to extend in an interesting way Michael's Theorem on double set-valued selections. The second notion is situated between that of a development and of a base of countable order. To see that a space with a weak development has a base of countable order, we use the classical works of H.H. Wicke and J.M. Worrell.We also introduce and study the new notion of a sharp base, which is strictly weaker than that of a uniform base and strictly stronger than that of a base of countable order and of a weakly uniform base, and which is strongly connected to the notion of a weak development. Several examples are exhibited to prove that the new notions do not coincide with the old ones. In short, our results show that the notions of a weak development and of a sharp base fit very well into already existing system of generalized metrizability properties defined in terms of sequences of open covers or bases. Several open questions are formulated