Applications of Finite Model Theory: Optimisation Problems, Hybrid Modal Logics and Games.

There exists an interesting relationships between two seemingly distinct fields: logic from the field of Model Theory, which deals with the truth of statements about discrete structures; and Computational Complexity, which deals with the classification of problems by how much of a particular computer resource is required in order to compute a solution. This relationship is known as Descriptive Complexity and it is the primary application of the tools from Model Theory when they are restricted to the finite; this restriction is commonly called Finite Model Theory. In this thesis, we investigate the extension of the results of Descriptive Complexity from classes of decision problems to classes of optimisation problems. When dealing with decision problems the natural mapping from true and false in logic to yes and no instances of a problem is used but when dealing with optimisation problems, other features of a logic need to be used. We investigate what these features are and provide results in the form of logical frameworks that can be used for describing optimisation problems in particular classes, building on the existing research into this area. Another application of Finite Model Theory that this thesis investigates is the relative expressiveness of various fragments of an extension of modal logic called hybrid modal logic. This is achieved through taking the Ehrenfeucht-Fraïssé game from Model Theory and modifying it so that it can be applied to hybrid modal logic. Then, by developing winning strategies for the players in the game, results are obtained that show strict hierarchies of expressiveness for fragments of hybrid modal logic that are generated by varying the quantifier depth and the number of proposition and nominal symbols available

Allan Brotsky, 1920-2015

Allan Brotsky\u27s obituary, published by Golden Gate University

With the People, Against Polluters

As one of the first law school clinics in the United States to prioritize environmental justice in its work. the ELJC has been widely recognized as a provider of high quality pro bono legal services to neighborhoods suffering the most from pollution. giving underrepresented communities a voice in the legal system. The Clinic offers free legal services to · community groups, public interest organizations. and residents who are working to promote environmental equity

With the People, Against Polluters

As one of the first law school clinics in the United States to prioritize environmental justice in its work. the ELJC has been widely recognized as a provider of high quality pro bono legal services to neighborhoods suffering the most from pollution. giving underrepresented communities a voice in the legal system. The Clinic offers free legal services to · community groups, public interest organizations. and residents who are working to promote environmental equity

The homotopy of Гq and classifying spaces

Let γ≥1 and Гq ^ be the topological groupoid of germs or orientation preserving local C(^v) diffeomorphisms of/R(^z). Then a E(^2) spectral sequence is constructed with the E(^2) terms computed from the homological properties of Г(_2), and E(^oo) is the bigraded module associated to the filtration of the homology of the classifying space, BГ given by Haefliger in [HA3]. Let S≥1 and Δ - rr [1,2,.....S+1] be the objects of the category C(_Δ)s with morphisms ≤ and Ѓ(_2)(Δ(^s) the space of functors from C(_Δ)-s to Г(_2) with the usual topology on Ѓ(_q). We prove that a) H(_t)(Ѓ(_q)(Δ(^s))(A')) = 0 for t > q. b) If Gl(_2) is the group of linear transformations of R(^2) with positive determinant then if V: Ѓ(_2)(Δ(^s))→(Gl(_2))(^s) is the map obtained by taking derivatives of ɤ (1 ≤ i) for 2 ≤ i ≤ s +1, ɤϵЃ(_2) is an isomorphism for t <q. These calculations go a long way towards calculating the E(^2) terms of the above spectral sequence. The spectral sequence is constructed for a large class of topological groupoids referred to as well formed topological groupoids, and the corresponding theorem on the high dimensional homologies of topological groupoids is proved for a special class of well formed topological groupoids which include the known topological groupoids associated with foliations

Home Video Evangelism in the Local Church

The video boom is but one aspect of the modern communication revolution that has found a place in the privacy of millions of homes.The church has been conscious of the potential of the mass media for evangelism. Effective communication is an integral component of effective evangelism. Radio, television, motion pictures, audio cassettes and multi-media have been utilized with varying degrees of success and usually at great cost. Contributing to video\u27s popularity is its comparative inexpensiveness