Demonic Lattices and Semilattices in Relational Semigroups with Ordinary Composition

Relation algebra and its reducts provide us with a strong tool for reasoning about nondeterministic programs and their partial correctness. Demonic calculus, introduced to model the behaviour of a machine where the demon is in control of nondeterminism, has also provided us with an extension of that reasoning to total correctness.We formalise the framework for relational reasoning about total correctness in nondeterministic programs using semigroups with ordinary composition and demonic lattice operations. We show that the class of representable demonic join semigroups is not finitely axiomatisable and that the representation class of demonic meet semigroups does not have the finite representation property for its finite members.For lattice semigroups (with composition, demonic join and demonic meet) we show that the representation problem for finite algebras is undecidable, moreover the finite representation problem is also undecidable. It follows that the representation class is not finitely axiomatisable, furthermore the finite representation property fails

Synchronous Kleene algebra

AbstractThe work presented here investigates the combination of Kleene algebra with the synchrony model of concurrency from Milner’s SCCS calculus. The resulting algebraic structure is called synchronous Kleene algebra. Models are given in terms of sets of synchronous strings and finite automata accepting synchronous strings. The extension of synchronous Kleene algebra with Boolean tests is presented together with models on sets of guarded synchronous strings and the associated automata on guarded synchronous strings. Completeness w.r.t. the standard interpretations is given for each of the two new formalisms. Decidability follows from completeness. Kleene algebra with synchrony should be included in the class of true concurrency models. In this direction, a comparison with Mazurkiewicz traces is made which yields their incomparability with synchronous Kleene algebras (one cannot simulate the other). On the other hand, we isolate a class of pomsets which captures exactly synchronous Kleene algebras. We present an application to Hoare-like reasoning about parallel programs in the style of synchrony

On the Fine-Structure of Regular Algebra

Regular algebra is the algebra of regular expressions as induced by regular language identity. We use Isabelle/HOL for a detailed systematic study of the regular algebra axioms given by Boffa, Conway, Kozen and Salomaa. We investigate the relationships between these systems, formalise a soundness proof for the smallest class (Salomaa’s) and obtain completeness for the largest one (Boffa’s) relative to a deep result by Krob. As a case study in formalised mathematics, our investigations also shed some light on the power of theorem proving technology for reasoning with algebras and their models, including proof automation and counterexample generation

A Preliminary Report on Students’ Reflections about Their Learning in an Active Learning Classroom

In the past decades, College Algebra has become a big hurdle for students to graduate or further pursue STEM or related careers. For most of the student population, College Algebra is a terminal course and only a small portion of students take it for further mathematics courses. The traditional content of College Algebra does not serve either group of students well (Mathematical Association of America, 2004; Mathematical Association of America & National Council of Teachers of Mathematics, 2012). In recent years, at a large Hispanic-serving university, the course design for liberal arts students has been changed. An active-learning curriculum has been implemented, namely Quantitative Reasoning. This curriculum engages students with opportunities to learn math concepts from relevant everyday sources and even their own personally collected data. Students build their own understanding of mathematics by relevant data-based situations experienced through preview activities designed to prepare students for class, collaboration in class discussion and discovery through relevant problem situations, and practice activities extending learning after each lesson. The purpose of this study was to investigate student perceptions of their learning outcomes from an active-learning structured course. That is, what impact does a course design with pre-assignment tasks, authentic problem solving through collaboration in class, and practice assignments after lessons have on diverse student populations in a quantitative reasoning course

Proving abstract algebra skills with problem-based learning integrated with videos and worksheets

Previous research showed students faced difficulties in solving the given problems in the abstract algebra course. The research aimed to describe the effect of the method of problem-based learning integrated with videos and worksheets to improve the proving skills of mathematics education students in one of the universities in Central Kalimantan, Indonesia. The researcher developed and uploaded the videos on YouTube. The research design was an experimental study. The researcher implemented the method in an experimental class. The control class students learned by using the usual method of the past three years that emphasized acquiring the abstract algebra concepts. The researcher selected the experimental class randomly. The numbers of students in the experimental and control classes were 32 and 28, respectively. The students of both classes solved the same problems in the post-test at end of the implementation. The post-test contained five problems to prove. The research results showed that the transactive reasoning activities in the experimental class enabled the students to prove at an appropriate abstraction level. The students’ scores in the abstract algebra for the experimental class were greater than those in the control class. Therefore, the method affected students’ ability to solve abstract algebra problems

Probabilistic Guarded KAT Modulo Bisimilarity: Completeness and Complexity

We introduce Probabilistic Guarded Kleene Algebra with Tests (ProbGKAT), an extension of GKAT that allows reasoning about uninterpreted imperative programs with probabilistic branching. We give its operational semantics in terms of special class of probabilistic automata. We give a sound and complete Salomaa-style axiomatisation of bisimilarity of ProbGKAT expressions. Finally, we show that bisimilarity of ProbGKAT expressions can be decided in O(n3 log n) time via a generic partition refinement algorithm

A Compositional Proof System for the Modal mu-Calculus

We present a proof system for determining satisfaction betweenprocesses in a fairly general process algebra and assertions of the modal mu-calculus. The proof system is compositional in the structure of processes. It extends earlier work on compositional reasoning within the modal mu-calculus and combines it with techniques from work on local model checking. The proof system is sound for all processes and complete for a class of finite-state processes