The expressive power of modal logic with inclusion atoms

Modal inclusion logic is the extension of basic modal logic with inclusion atoms, and its semantics is defined on Kripke models with teams. A team of a Kripke model is just a subset of its domain. In this paper we give a complete characterisation for the expressive power of modal inclusion logic: a class of Kripke models with teams is definable in modal inclusion logic if and only if it is closed under k-bisimulation for some integer k, it is closed under unions, and it has the empty team property. We also prove that the same expressive power can be obtained by adding a single unary nonemptiness operator to modal logic. Furthermore, we establish an exponential lower bound for the size of the translation from modal inclusion logic to modal logic with the nonemptiness operator.Comment: In Proceedings GandALF 2015, arXiv:1509.0685

One-dimensional fragment of first-order logic

We introduce a novel decidable fragment of first-order logic. The fragment is one-dimensional in the sense that quantification is limited to applications of blocks of existential (universal) quantifiers such that at most one variable remains free in the quantified formula. The fragment is closed under Boolean operations, but additional restrictions (called uniformity conditions) apply to combinations of atomic formulae with two or more variables. We argue that the notions of one-dimensionality and uniformity together offer a novel perspective on the robust decidability of modal logics. We also establish that minor modifications to the restrictions of the syntax of the one-dimensional fragment lead to undecidable formalisms. Namely, the two-dimensional and non-uniform one-dimensional fragments are shown undecidable. Finally, we prove that with regard to expressivity, the one-dimensional fragment is incomparable with both the guarded negation fragment and two-variable logic with counting. Our proof of the decidability of the one-dimensional fragment is based on a technique involving a direct reduction to the monadic class of first-order logic. The novel technique is itself of an independent mathematical interest

The European Natural Gas Market: Imports to Rise Considerably

Since the late 1990s, natural gas has been the second most important source of energy in the European Union after oil. Current forecasts predict that the demand for natural gas will continue to rise over the long term. As the production of natural gas in the EU has already passed its peak and will drop in coming years, natural gas imports will have to rise considerably. To ensure according supplies is possible in the long run, as approximately 80% of the world's natural gas reserves are located within the economic reach of the EU-primarily in the former Soviet Union and Middle East. A supply of imported natural gas adequate to cover declines in EU production is contractually guaranteed through 2020. Additional projects will permit an increase in natural gas consumption in the EU of 20 to 30%. By 2020, 80% of the EU's supply will come from imports. Russia will remain by far the largest supplier, followed by Norway and Algeria. Security concerns are raised by the importation of natural gas from a small handful of suppliers. While the regional diversification of imports is only possible to a limited extent, the construction of a pipeline to connect Europe with natural gas rich nations on the Caspian Sea and in the Middle East would be a step in the right direction. The expansion of capacities for production, transport and storage of liquified natural gas (LNG) could also help to augment diversification.Natural gas, LNG, Import dependence, Reserves, Pipelines