Composition mechanisms for retrenchment

Retrenchment is a flexible model evolution formalism that arose as a reaction to the limitations imposed by refinement, and for which the proof obligations feature additional predicates for accommodating design data. Composition mechanisms for retrenchment are studied. Vertical, horizontal, dataflow, parallel and fusion compositions are described. Of particular note are the means by which the additional predicates compose. It is argued that all of the compositions introduced are associative, and that they are mutually coherent. Composition of retrenchment with refinement, so important for the smooth interworking of the two techniques, is discussed. Decomposition, allowing finer grained retrenchments to be extracted from a single large grained retrenchment, is also investigated

Charge stripes due to electron correlations in the two-dimensional spinless Falicov-Kimball model

We calculate the restricted phase diagram for the Falicov-Kimball model on a two-dimensional square lattice. We consider the limit where the conduction electron density is equal to the localized electron density, which is the limit related to the S_z=0 states of the Hubbard model. After considering over 20,000 different candidate phases (with a unit cell of 16 sites or less) and their thermodynamic mixtures, we find only about 100 stable phases in the ground-state phase diagram. We analyze these phases to describe where stripe phases occur and relate these discoveries to the physics behind stripe formation in the Hubbard model.Comment: (34 pages, 9 figures, submitted to Journal of Statistical Physics to celebrate Elliott Lieb's 70th birthday

Term graph rewriting and garbage collection using opfibrations

AbstractThe categorical semantics of (an abstract version of) the general term graph rewriting language DACTL is investigated. The operational semantics is reformulated in order to reveal its universal properties. The technical dissonance between the matchings of left-hand sides of rules to redexes, and the properties of rewrite rules themselves, is taken as the impetus for expressing the core of the model as a Grothendieck opfibration of a category of general rewrites over a base of general rewrite rules. Garbage collection is examined in this framework in order to reconcile the treatment with earlier approaches. It is shown that term rewriting has particularly good garbage-theoretic properties that do not generalise to all cases of graph rewriting and that this has been a stumbling block for aspects of some earlier models for graph rewriting

DPO Rewriting and Abstract Semantics via Opfibrations

AbstractThe classical DPO graph rewriting construction is re-expressed using the opfibration approach introduced originally for term graph rewriting. Using a skeleton category of graphs, a base of canonical graphs-in-context, with DPO rules as arrows, and with categories of redexes over each object in the base, yields a category of rewrites via the discrete Grothendieck construction. The various possible ways of combining rules and rewrites leads to a variety of functors amongst the various categories formed. Categories whose arrows are rewriting sequences have counterparts where the arrows are elementary event structures, and an event structure semantics for arbitrary graph grammars emerges naturally

Quantum Stability of (2+1)-Spacetimes with Non-Trivial Topology

Quantum fields are investigated in the (2+1)-open-universes with non-trivial topologies by the method of images. The universes are locally de Sitter spacetime and anti-de Sitter spacetime. In the present article we study spacetimes whose spatial topologies are a torus with a cusp and a sphere with three cusps as a step toward the more general case. A quantum energy momentum tensor is obtained by the point stripping method. Though the cusps are no singularities, the latter cusps cause the divergence of the quantum field. This suggests that only the latter cusps are quantum mechanically unstable. Of course at the singularity of the background spacetime the quantum field diverges. Also the possibility of the divergence of topological effect by a negative spatial curvature is discussed. Since the volume of the negatively curved space is larger than that of the flat space, one see so many images of a single source by the non-trivial topology. It is confirmed that this divergence does not appear in our models of topologies. The results will be applicable to the case of three dimensional multi black hole\cite{BR}.Comment: 17 pages, revtex, 3 uuencoded figures containe

Formalising the Continuous/Discrete Modeling Step

Formally capturing the transition from a continuous model to a discrete model is investigated using model based refinement techniques. A very simple model for stopping (eg. of a train) is developed in both the continuous and discrete domains. The difference between the two is quantified using generic results from ODE theory, and these estimates can be compared with the exact solutions. Such results do not fit well into a conventional model based refinement framework; however they can be accommodated into a model based retrenchment. The retrenchment is described, and the way it can interface to refinement development on both the continuous and discrete sides is outlined. The approach is compared to what can be achieved using hybrid systems techniques.Comment: In Proceedings Refine 2011, arXiv:1106.348

Requirements Validation by Lifting Retrenchments in B

Simple retrenchment is briefly reviewed in the B specification language of J.-R.Abrial (Abrial,1996) as a liberalization of classical refinement, for the formal description of application developments too demanding for refinement. The looser relationships allowed by retrenchment between adjacent models in the development process may capture some of the requirements information of the development. This can make requirements validation more difficult to understand since the locus of requirements should be the models, and not their interrelationships, as far as possible. Hence the universal construction of (Banach,2000), originally proposed for simple transition systems, is reformulated in B, in order to "lift" a given retrenchment conceptually, thus retracting such requirements information back to the level of abstraction of the abstract, ideal model. Examples demonstrate the cognitive value of retracting requirements to the abstract level, articulated in a well-understood formal language. This is also seen to yield a more understandable way of comparing alternative retrenchment designs. Some new B syntax in the pre- and postcondition style is presented to facilitate expression of the lifted requirements