Russell J. Bradford

Applying parallel discrete event simulation to network emulation

The simulation of wide area computer networks is one area where the benefits of parallel simulation have been clearly demonstrated. We present a description of a system that uses a parallel discrete event simulator to act as a high speed network emulator. With this, real Internet Protocol (IP) traffic generated by application programs running on user workstations can interact with modelled traffic in the emulator thus providing a controlled test environment for distributed applications. The network emulator uses the TasKit conservative parallel discrete event simulation (PDES) kernel. TasKit has been shown to be able to achieve improved parallel performance over existing conservative and optimistic PDES kernels, as well as improved sequential performance over an existing central-event-list based kernel. This paper explains the modifications that have been made to TasKit to enable real-time operation along with the emulator interface that allows the IP network simulation running in the TasKit kernel to interact with real IP clients. Initial emulator performance data is included.

Cylindrical algebraic decompositions for boolean combinations

This article makes the key observation that when using cylindrical algebraic decomposition (CAD) to solve a problem with respect to a set of polynomials, it is not always the signs of those polynomials that are of paramount importance but rather the truth values of certain quantifier free formulae involving them. This motivates our definition of a Truth Table Invariant CAD (TTICAD). We generalise the theory of equational constraints to design an algorithm which will efficiently construct a TTICAD for a wide class of problems, producing stronger results than when using equational constraints alone. The algorithm is implemented fully in Maple and we present promising results from experimentation.

Optimising problem formulation for cylindrical algebraic decomposition

Cylindrical algebraic decomposition (CAD) is an important tool for the study of real algebraic geometry with many applications both within mathematics and elsewhere. It is known to have doubly exponential complexity in the number of variables in the worst case, but the actual computation time can vary greatly. It is possible to offer different formulations for a given problem leading to great differences in tractability. In this paper we suggest a new measure for CAD complexity which takes into account the real geometry of the problem. This leads to new heuristics for choosing: the variable ordering for a CAD problem, a designated equational constraint, and formulations for truth-table invariant CADs (TTICADs). We then consider the possibility of using Grobner bases to precondition TTICAD and when such formulations constitute the creation of a new problem.

Truth table invariant cylindrical algebraic decomposition

When using cylindrical algebraic decomposition (CAD) to solve a problem with respect to a set of polynomials, it is likely not the signs of those polynomials that are of paramount importance but rather the truth values of certain quantifier free formulae involving them. This observation motivates our article and definition of a Truth Table Invariant CAD (TTICAD).In ISSAC 2013 the current authors presented an algorithm that can efficiently and directly construct a TTICAD for a list of formulae in which each has an equational constraint. This was achieved by generalising McCallums theory of reduced projection operators. In this paper we present an extended version of our theory which can be applied to an arbitrary list of formulae, achieving savings if at least one has an equational constraint. We also explain how the theory of reduced projection operators can allow for further improvements to the lifting phase of CAD algorithms, even in the context of a single equational constraint.The algorithm is implemented fully in Maple and we present both promising results from experimentation and a complexity analysis showing the benefits of our contributions.

Reasoning about the Elementary Functions of Complex Analysis

There are many problems with the simplification of elementary functions, particularly over the complex plane, though not exclusively – see (20). Systems tend to make “howlers” or not to simplify enough. In this paper we outline the “unwinding number” approach to such problems, and show how it can be used to prevent errors and to systematise such simplification, even though we have not yet reduced the simplification process to a complete algorithm. The unsolved problems are probably more amenable to the techniques of artificial intelligence and theorem proving than the original problem of complex-variable analysis.

Truth Table Invariant Cylindrical Algebraic Decomposition by Regular Chains

A new algorithm to compute cylindrical algebraic decompositions (CADs) is presented, building on two recent advances. Firstly, the output is truth table invariant (a TTICAD) meaning given formulae have constant truth value on each cell of the decomposition. Secondly, the computation uses regular chains theory to first build a cylindrical decomposition of complex space (CCD) incrementally by polynomial. Significant modification of the regular chains technology was used to achieve the more sophisticated invariance criteria. Experimental results on an implementation in the RegularChains Library for Maple verify that combining these advances gives an algorithm superior to its individual components and competitive with the state of the art.

A parallel discrete event IP network emulator

Testing distributed applications over the Internet is fraught with problems: due to the inability to control a wide area network consistent, reproducible performance experiments are not possible. A system is described that uses a parallel discrete event simulator that can act as a real-time network emulator. Real Internet Protocol (IP) traffic generated by application programs running on user workstations can interact with modelled traffic in the emulator; thus providing a controlled test environment for distributed applications. Parallel execution enables the emulator to simulate large virtual networks and to model traffic interactions that could not be done in real-time sequentially. This paper gives an overview of the emulator and explores the various external data routing methods that the emulator supports. These routing methods allow the emulator to be operated in shared environments with certain constraints, as well as in dedicated test environments. Preliminary performance results are included.

A pi-calculus Model of a Spanish Fish Market - Preliminary Report

This paper reports an educational exercise in using the π-calculus to model components of an electronic marketplace. Specifically, we are looking at the Spanish fish market, since we have participated in the construction of several simulations of this scenario over the past 18 months and now feel it is time to prepare a more precise description. Our objectives in doing this were (i) to gain familiarity with the π-calculus (ii) to find out whether the π-calculus might provide a suitable basis for defining the behaviour of components in an electronic marketplace. It is not our intention at this stage to establish the correctness of the components or the completeness of the model: these will be addressed later using existing tools and by developing new ones. In summary, this is an experience report.

Problem Formulation for Truth-Table Invariant Cylindrical Algebraic Decomposition by Incremental Triangular Decomposition

Cylindrical algebraic decompositions (CADs) are a key tool for solving problems in real algebraic geometry and beyond. We recently presented a new CAD algorithm combining two advances: truth-table invariance, making the CAD invariant with respect to the truth of logical formulae rather than the signs of polynomials; and CAD construction by regular chains technology, where first a complex decomposition is constructed by refining a tree incrementally by constraint. We here consider how best to formulate problems for input to this algorithm. We focus on a choice (not relevant for other CAD algorithms) about the order in which constraints are presented. We develop new heuristics to help make this choice and thus allow the best use of the algorithm in practice. We also consider other choices of problem formulation for CAD, as discussed in CICM 2013, revisiting these in the context of the new algorithm.

The Bath algebraic number package

This paper describes a package implemented in REDUCE 3.2 for the manipulation of algebraic numbers. The package regards algebraic numbers as elements of abstract extensions of the rational numbers, not as particular real or complex numbers. We describe in this paper the various design choices that were made, and the current state of the package, as well as future possibilities for enhancement.