Cantor Topology Research Articles

Topologies for Finite Words: Compatibility with the Cantor Topology

Infinite words are often considered as limits of finite words. As topological methods have been proved to be useful in the theory of ω-languages it seems to be providing to include finite and infinite words into one (topological) space. In most cases this results in a poor topological structure induced on the subspace of finite words. In the present paper we investigate the possibility to link topologies in the space of finite words with a topology in the space of infinite words via a natural mapping. A requirement in this linking of topologies consists in the compatibility of the topological properties (openness, closedness etc.) of images with preimages and vice versa. Here, we show that choosing for infinite words the natural topology of the CANTOR space and the δ-limit as linking mapping there are several natural topologies on the space of finite words compatible with the topology of the CANTOR space. It is interesting to observe that besides the well-known prefix topology there are at least two more whose origin is from language theory – centers and supercenters of languages. We show that several of these topologies on the space of finite words fit into a class of ℒ-topologies and exhibit their special properties w.r.t. to the compatibility with the CANTOR topology.

Safety and Liveness, Weakness and Strength, and the Underlying Topological Relations

We present a characterization that shows what it means for a formula to be a weak or strong version of another formula. We show that the weak version of a formula is not the same as Alpern and Schneider's safety component, but can be achieved by taking the closure in the Cantor topology over an augmented alphabet in which every formula is satisfiable. The resulting characterization allows us to show that the set of semantically weak formulas is exactly the set of nonpathological safety formulas. Furthermore, we use the characterization to show that the original versions of the ieee standard temporal logics psl and sva are broken, and we show that the source of the problem lies in the semantics of the sere intersection and fusion operators. Finally, we use the topological characterization to show the internal consistency of the alternative semantics adopted by the latest version of the psl standard.

Topology, monitorable properties and runtime verification

We review concepts like safety, liveness, and monitorability from a rigorous topological viewpoint. Thus, monitorability of an ω-language means that the boundary in the Cantor topology has an empty interior. We show that all ω-regular languages which are deterministic and co-deterministic are monitorable, but certain deterministic liveness properties like “infinitely many a's” cannot be written as a countable union of monitorable languages. We briefly discuss model checking with LTL, its three-valued variant LTL3 and monitor constructions based upon LTL3.

A Curtis–Hedlund–Lyndon theorem for Besicovitch and Weyl spaces

Global functions of cellular automata on state spaces equipped with the Cantor topology are well characterized by the Curtis–Hedlund–Lyndon theorem. In this paper, we develop a characterization of global functions of cellular automata on Z , if the state space is equipped by Weyl and Besicovitch topology. The necessary and sufficient condition for a function to be the global map of a cellular automaton is (1) a strong localization property, a condition that strengthen Lipschitz continuity, (2) the set of (Cantor) periodic states are positively invariant and (3) the function commutes (in the Weyl/Besicovitch sense) with the shift operator.

Some results about the chaotic behavior of cellular automata

We study the behavior of cellular automata (CA for short) in the Cantor, Besicovitch and Weyl topologies. We solve an open problem about the existence of transitive CA in the Besicovitch topology. The proof of this result has some interest of its own since it is obtained by using Kolmogorov complexity. To our knowledge it is the first result about discrete dynamical systems obtained using Kolmogorov complexity. We also prove that in the Besicovitch topology every CA has either a unique periodic point (thus a fixed point) or an uncountable set of periodic points. This result underlines the fact that CA have a great degree of stability; it may be considered a further step towards the understanding of CA periodic behavior. Moreover, we prove that in the Besicovitch topology there is a special set of configurations, the set of Toeplitz configurations, that plays a role similar to that of spatially periodic configurations in the Cantor topology, that is, it is dense and has a central role in the study of surjectivity and injectivity. Finally, it is shown that the set of spatially quasi-periodic configurations is not dense in the Weyl topology.

Remark on cellular automata and shift preserving maps

The main goal of this work is to show an extension of well known Hedlund’s theorem which states that in the Cantor topology the cellular automata are the continuous shift preserving maps. This extension maintains the topological structure and uses the concept of barriers of Set Theory to generalize the notion of local rule in the definition of cellular automata.

Optimization of optical cross-connects with wave-mixing conversion

This paper presents new constructions of multistage wave-mixing networks with arbitrary b/spl times/b space-switching elements, where b /spl ges/ 2. In these networks, for a size of F fiber links and W wavelengths per link, converter requirements are O(Flog/sub b/W) or O(FW/b) for rearrangeable nodes, and O(Flog/sub b/Wlog/sub b/(FW)) or O(FWlog/sub b/(FW)/b) for different types of strictly nonblocking nodes inspired from the Cantor topology. In all cases, the worst case number of cascaded conversion is O(log/sub b/W). When b=W /spl les/ F, the required number of converters, and the worst case number of cascaded conversions are respectively O(F) and O(1), and are both optimal up to a constant. The new networks generalize and improve upon previous wave-mixing networks based on 2/spl times/2 space switches.

A topological characterization of random sequences

The set of random sequences is large in the sense of measure, but small in the sense of category. This is the case when we regard the set of infinite sequences over a finite alphabet as a subset of the usual Cantor space. In this note we will show that the above result depends on the topology chosen. To this end we will use a relativization of the Cantor topology, the U δ -topology introduced by Staiger [RAIRO Inform. Théor. 21 (1987) 147–173]. This topology is also metric, but the distance between two sequences does not depend on their longest common prefix (Cantor metric), but on the number of their common prefixes in a given language U. The resulting space is complete, but not always compact. We will show how to derive a computable set U from a universal Martin-Löf test such that the set of non-random sequences is nowhere dense in the U δ -topology. As a byproduct we obtain a topological characterization of the set of random sequences. We also show that the Law of Large Numbers, which fails with respect to the usual topology, is true for the U δ -topology.

If not empty, NP — P is topologically large

In the classical Cantor topology or in the superset topology, NP and, consequently, classes included in NP are meagre. However, in a natural combination of the two topologies, we prove that NP — P, if not empty, is a second category class, while NP-complete sets form a first category class. These results are extended to different levels in the polynomial hierarchy and to the low and high hierarchies. P-immune sets in NP, NP-simple sets, P-bi-immune sets and NP-effectively simple sets are all second category (if not empty). It is shown that if C is any of the above second category classes, then for all B∈NP there exists an A∈ C such that A is arbitrarily close to B infinitely often.