Untangled: A Complete Dynamic Topological Logic

Dynamic topological logic ($\mathbf{DTL}$) is a trimodal logic designed for reasoning about dynamic topological systems. It was shown by Fern\'andez-Duque that the natural set of axioms for $\mathbf{DTL}$ is incomplete, but he provided a complete axiomatisation in an extended language. In this paper, we consider dynamic topological logic over scattered spaces, which are topological spaces where every nonempty subspace has an isolated point. Scattered spaces appear in the context of computational logic as they provide semantics for provability and enjoy definable fixed points. We exhibit the first sound and complete dynamic topological logic in the original trimodal language. In particular, we show that the version of $\mathbf{DTL}$ based on the class of scattered spaces is finitely axiomatisable over the original language, and that the natural axiomatisation is sound and complete

Dynamic Cantor Derivative Logic

Topological semantics for modal logic based on the Cantor derivative operator gives rise to derivative logics, also referred to as d-logics. Unlike logics based on the topological closure operator, d-logics have not previously been studied in the framework of dynamical systems, which are pairs (X,f) consisting of a topological space X equipped with a continuous function f: X ? X. We introduce the logics wK4C, K4C and GLC and show that they all have the finite Kripke model property and are sound and complete with respect to the d-semantics in this dynamical setting. In particular, we prove that wK4C is the d-logic of all dynamic topological systems, K4C is the d-logic of all T_D dynamic topological systems, and GLC is the d-logic of all dynamic topological systems based on a scattered space. We also prove a general result for the case where f is a homeomorphism, which in particular yields soundness and completeness for the corresponding systems wK4H, K4H and GLH. The main contribution of this work is the foundation of a general proof method for finite model property and completeness of dynamic topological d-logics. Furthermore, our result for GLC constitutes the first step towards a proof of completeness for the trimodal topo-temporal language with respect to a finite axiomatisation - something known to be impossible over the class of all spaces

Non-finite axiomatizability of Dynamic Topological Logic

Dynamic topological logic (DTL) is a polymodal logic designed for reasoning about {\em dynamic topological systems. These are pairs (X,f), where X is a topological space and f:X->X is continuous. DTL uses a language L which combines the topological S4 modality [] with temporal operators from linear temporal logic. Recently, I gave a sound and complete axiomatization DTL* for an extension of the logic to the language L*, where is allowed to act on finite sets of formulas and is interpreted as a tangled closure operator. No complete axiomatization is known over L, although one proof system, which we shall call $\mathsf{KM}$, was conjectured to be complete by Kremer and Mints. In this paper we show that, given any language L' between L and L*, the set of valid formulas of L' is not finitely axiomatizable. It follows, in particular, that KM is incomplete.Comment: arXiv admin note: text overlap with arXiv:1201.5162 by other author

Dynamic Topological Logic of Metric Spaces

Dynamic Topological Logic (DT L) is a modal framework for reasoning about dynamical systems, that is, pairs hX; fi where X is a topological space and f : X ! X a continuous function. In this paper we consider the case where X is a metric space. We rst show that any formula which can be satis ed on an arbitrary dynamic topological system can be satis ed on one based on a metric space; in fact, this space can be taken to be countable and have no isolated points. Since any metric space with these properties is homeomorphic to the set of rational numbers, it follows that any formula can be satis ed on a system based on Q. We then show that the situation changes when considering complete metric spaces, by exhibiting a formula which is not valid in general but is valid on the class of systems based on a complete metric space. While we do not attempt to give a full characterization of the set of valid formulas on this class we do give a relative completeness result; any formula which is satis able on a dynamical system based on a complete metric space is also satis ed on one based on the Cantor spac

Topological Approximate Dynamic Programming under Temporal Logic Constraints

In this paper, we develop a Topological Approximate Dynamic Programming (TADP) method for planningin stochastic systems modeled as Markov Decision Processesto maximize the probability of satisfying high-level systemspecifications expressed in Linear Temporal Logic (LTL). Ourmethod includes two steps: First, we propose to decompose theplanning problem into a sequence of sub-problems based on thetopological property of the task automaton which is translatedfrom the LTL constraints. Second, we extend a model-freeapproximate dynamic programming method for value iterationto solve, in an order reverse to a causal dependency of valuefunctions, one for each state in the task automaton. Particularly,we show that the complexity of the TADP does not growpolynomially with the size of the product Markov DecisionProcess (MDP). The correctness and efficiency of the algorithmare demonstrated using a robotic motion planning example.Comment: 8 pages, 6 figures. Accepted by 58th Conference on Decision and Contro

Quantum computation with quasiparticles of the Fractional Quantum Hall Effect

We propose an approach that enables implementation of anyonic quantum computation in systems of antidots in the two-dimensional electron liquid in the FQHE regime. The approach is based on the adiabatic transfer of FQHE quasiparticles in the antidot systems, and uses their fractional statistics to perform quantum logic. Advantages of our scheme over other semiconductor-based proposals of quantum computation include the energy gap in the FQHE liquid that suppresses decoherence, and the topological nature of quasiparticle statistics that makes it possible to entangle two quasiparticles without their direct dynamic interaction.Comment: 4 pages, 2 figure