The Influence of the Test Operator on the Expressive Power of PDL-like Logics

Berman and Paterson proved that test-free propositional dynamic logic (PDL) is weaker than PDL. One would raise questions: does a similar result also hold for extensions of PDL? For example, is test-free converse-PDL (CPDL) weaker than CPDL? In what circumstances the test operator can be eliminated without reducing the expressive power of a PDL-based logical formalism? These problems have not yet been studied. As the description logics $\mathcal{ALC}_{trans}$ and $\mathcal{ALC}_{reg}$ are, respectively, variants of test-free PDL and PDL, there is a concept of $\mathcal{ALC}_{reg}$ that is not equivalent to any concept of $\mathcal{ALC}_{trans}$. Generalizing this, we prove that there is a concept of $\mathcal{ALC}_{reg}$ that is not equivalent to any concept of the logic that extends $\mathcal{ALC}_{trans}$ with inverse roles, nominals, qualified number restrictions, the universal role and local reflexivity of roles. We also provide some results for the case with RBoxes and TBoxes. One of them states that tests can be eliminated from TBoxes of the deterministic Horn fragment of $\mathcal{ALC}_{reg}$.

Sub-national government and pathways to sustainable energy

In an effort to understand how to promote more sustainable forms of energy provision, researchers have begun addressing the scale of political and governance processes, yet the effects of sub-national government remain neglected. At the same time, analysts of political devolution, decentralisation and independence have rarely given attention to the energy sector. Papers in this special issue seek to better understand the relationship between sub-national government and pathways to sustainable energy: examining how city-regional and devolved governments have shaped agendas for building retrofit; elucidating the importance of decentralised governance in knitting together electricity, heat and transport energy markets; mapping the complex, fuzzy spatial organisation of legal powers to direct energy policy across multi-level polities; and analysing conflicts over the allocation of energy infrastructure consenting powers between national and devolved governments. The papers highlight the interdependencies of action in different governmental arenas, and reinforce arguments for greater central-to-local reflexivity in governance styles. Analysing the interface between sub-national government and energy transition also raises new questions about the meaning of ‘sovereignty’, the fragmentary nature of democratic control over energy systems, and the effects of boundaries.

On the Local Lifting Properties of Operator Spaces

In this paper, we mainly study operator spaces which have the locally lifting property (LLP). The dual of any ternary ring of operators is shown to satisfy the strongly local reflexivity, and this is used to prove that strongly local reflexivity holds also for operator spaces which have the LLP. Several homological characterizations of the LLP and weak expectation property are given. We also prove that for any operator space V, V** has the LLP if and only if V has the LLP and V* is exact.

ULTRAPOWERS OF BANACH ALGEBRAS AND MODULES

The Arens products are the standard way of extending the product from a Banach algebrato its bidual″. Ultrapowers provide another method which is more symmetric, but one that in general will only give a bilinear map, which may not be associative. We show that ifis Arens regular, then there is at least one way to use an ultrapower to recover the Arens product, a result previously known for C*-algebras. Our main tool is a principle of local reflexivity result for modules and algebras.