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 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

The intuitionistic temporal logic of dynamical systems

A dynamical system is a pair $(X,f)$, where $X$ is a topological space and $f\colon X\to X$ is continuous. Kremer observed that the language of propositional linear temporal logic can be interpreted over the class of dynamical systems, giving rise to a natural intuitionistic temporal logic. We introduce a variant of Kremer's logic, which we denote ${\sf ITL^c}$, and show that it is decidable. We also show that minimality and Poincar\'e recurrence are both expressible in the language of ${\sf ITL^c}$, thus providing a decidable logic expressive enough to reason about non-trivial asymptotic behavior in dynamical systems

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

Towards Semantic Modeling of Contradictions and Disagreements: A Case Study of Medical Guidelines

We introduce a formal distinction between contradictions and disagreements in natural language texts, motivated by the need to formally reason about contradictory medical guidelines. This is a novel and potentially very useful distinction, and has not been discussed so far in NLP and logic. We also describe a NLP system capable of automated finding contradictory medical guidelines; the system uses a combination of text analysis and information retrieval modules. We also report positive evaluation results on a small corpus of contradictory medical recommendations.Comment: 5 pages, 1 figure, accepted at 12th International Conference on Computational Semantics (IWCS-2017