Canonical Abstract Syntax Trees

This paper presents Gom, a language for describing abstract syntax trees and generating a Java implementation for those trees. Gom includes features allowing the user to specify and modify the interface of the data structure. These features provide in particular the capability to maintain the internal representation of data in canonical form with respect to a rewrite system. This explicitly guarantees that the client program only manipulates normal forms for this rewrite system, a feature which is only implicitly used in many implementations

Automated verification of refinement laws

Demonic refinement algebras are variants of Kleene algebras. Introduced by von Wright as a light-weight variant of the refinement calculus, their intended semantics are positively disjunctive predicate transformers, and their calculus is entirely within first-order equational logic. So, for the first time, off-the-shelf automated theorem proving (ATP) becomes available for refinement proofs. We used ATP to verify a toolkit of basic refinement laws. Based on this toolkit, we then verified two classical complex refinement laws for action systems by ATP: a data refinement law and Back's atomicity refinement law. We also present a refinement law for infinite loops that has been discovered through automated analysis. Our proof experiments not only demonstrate that refinement can effectively be automated, they also compare eleven different ATP systems and suggest that program verification with variants of Kleene algebras yields interesting theorem proving benchmarks. Finally, we apply hypothesis learning techniques that seem indispensable for automating more complex proofs

Retrenching the Purse: Finite Exception Logs, and Validating the Small

The Mondex Electronic Purse is an outstanding example of industrial scale formal refinement, and was the first verification to achieve ITSEC level E6 certification. A formal abstract model and a formal concrete model were developed, and a formal refinement was hand-proved between them. Nevertheless, certain requirements issues were set beyond the scope of the formal development, or handled in an unnatural manner. The retrenchment Tower Pattern is used to address one such issue in detail: the finiteness of the purse log (which records unsuccessful transactions). A retrenchment is constructed from the lowest level model of the purse system to a model in which logs are finite, and is then lifted to create two refinement developments of the purse, working at different levels of detail, and connected via retrenchments, forming the tower. The tower development is appropriately validated, vindicating the design used

Towards Stratification Learning through Homology Inference

A topological approach to stratification learning is developed for point cloud data drawn from a stratified space. Given such data, our objective is to infer which points belong to the same strata. First we define a multi-scale notion of a stratified space, giving a stratification for each radius level. We then use methods derived from kernel and cokernel persistent homology to cluster the data points into different strata, and we prove a result which guarantees the correctness of our clustering, given certain topological conditions; some geometric intuition for these topological conditions is also provided. Our correctness result is then given a probabilistic flavor: we give bounds on the minimum number of sample points required to infer, with probability, which points belong to the same strata. Finally, we give an explicit algorithm for the clustering, prove its correctness, and apply it to some simulated data.Comment: 48 page

Software Engineering and Complexity in Effective Algebraic Geometry

We introduce the notion of a robust parameterized arithmetic circuit for the evaluation of algebraic families of multivariate polynomials. Based on this notion, we present a computation model, adapted to Scientific Computing, which captures all known branching parsimonious symbolic algorithms in effective Algebraic Geometry. We justify this model by arguments from Software Engineering. Finally we exhibit a class of simple elimination problems of effective Algebraic Geometry which require exponential time to be solved by branching parsimonious algorithms of our computation model.Comment: 70 pages. arXiv admin note: substantial text overlap with arXiv:1201.434

Nonexistence of Chaotic Solutions of Nonlinear Differential Equations

We discuss some important issues arising from computational efforts in dynamical systems and fluid dynamics. Various individuals have misunderstood these issues since the onset of these problem areas; indeed, they have been routinely misinterpreted, and even viewed as "laws" by some. This paper hopes to stimulate appropriate corrections and to realign thinking, with the overall goal being sound future progress in dynamical systems and fluid dynamics.Comment: 12 page

Balancing classroom management with mathematical learning: Using practice-based task design in mathematics teacher education

In this paper we present the results from a study conducted in a UK institution in which 21mathematics pre-service teachers engage with two practice-based tasks featuring incidents where classroom management interferes with mathematical learning. We investigate their considerations when they make decisions in classroom situations and how these tasks can trigger their reflections on the teaching and learning of mathematics. In our analysis we used the constructs of social and sociomathematical norms (Cobb & Yackel, 1996) and Teaching Triad (Jaworski, 1994). Results indicate commendable norms pre-service teachers aspire to establish in their classroom, such as peer respect, value of discussion and investigative mathematical learning. However, they often miss the opportunity to engage students with metacognitive discussions and mathematical challenge as they focus on behavioural issues or endorse dichotomous and simplistic views of mathematical learning. We credit these tasks with allowing insight into pre-service teachers’ considerations and we propose their further implementation in teacher education programs