The F5 Criterion revised

The purpose of this work is to generalize part of the theory behind Faugere's "F5" algorithm. This is one of the fastest known algorithms to compute a Groebner basis of a polynomial ideal I generated by polynomials f_{1},...,f_{m}. A major reason for this is what Faugere called the algorithm's "new" criterion, and we call "the F5 criterion"; it provides a sufficient condition for a set of polynomials G to be a Groebner basis. However, the F5 algorithm is difficult to grasp, and there are unresolved questions regarding its termination. This paper introduces some new concepts that place the criterion in a more general setting: S-Groebner bases and primitive S-irreducible polynomials. We use these to propose a new, simple algorithm based on a revised F5 criterion. The new concepts also enable us to remove various restrictions, such as proving termination without the requirement that f_{1},...,f_{m} be a regular sequence.Comment: Originally submitted by Arri in 2009, with material added by Perry since 2010. The 2016 editions correct typographical issues not caught in previous editions bring the theory of the body into conformity with the published version of the pape

ACKERMANN’S FUNCTION IN ITERATIVE FORM: A PROOF ASSISTANT EXPERIMENT

Ackermann's function can be expressed using an iterative algorithm, which essentially takes the form of a term rewriting system. Although the termination of this algorithm is far from obvious, its equivalence to the traditional recursive formulation--and therefore its totality--has a simple proof in Isabelle/HOL. This is a small example of formalising mathematics using a proof assistant, with a focus on the treatment of difficult recursions.ERC Advanced Grant ALEXANDRIA (Project GA 742178

Graceful Termination -- Graceful Resetting

Correct — let alone graceful — termination of parallel systems is sometimes thought to be a difficult problem. This is particularly imagined to be so under the pure message-passing MIMD discipline of occam and transputer networks, where global operations (like setting a shared flag or abortions) are not allowed and where time-outs cannot be set for every communication. This paper describes some common, but erroneous, occam approaches to this problem and contrasts them with what can be done in Ada [0, 1, 2]. These methods are all rejected on the grounds of insecurity and performance overheads. A simple, legal, secure and efficient occam method is then presented. This method also solves a much more important problem — the general (or partial) resetting of a parallel system (or sub-system). The resetting mechanism is quite independent of the parallel application algorithm, which can therefore be developed without worrying about such matters. This separation of concerns is good software engineering and is fully supported by the occam philosophy. Finally, an application of this resetting mechanism is described that permits the dynamic reconstruction of occam network topologies

Analysis of the electromagnetic scattering from an inlet geometry with lossy walls

One of the primary goals is to develop an approximate but sufficiently accurate analysis for the problem of electromagnetic (EM) plane wave scattering by an open ended, perfectly-conducting, semi-infinite hollow circular waveguide (or duct) with a thin, uniform layer of lossy or absorbing material on its inner wall, and with a simple termination inside. The less difficult but useful problem of the EM scattering by a two-dimensional (2-D), semi-infinite parallel plate waveguide with an impedance boundary condition on the inner walls was chosen initially for analysis. The impedance boundary condition in this problem serves to model a thin layer of lossy dielectric/ferrite coating on the otherwise perfectly-conducting interior waveguide walls. An approximate but efficient and accurate ray solution was obtained recently. That solution is presently being extended to the case of a moderately thick dielectric/ferrite coating on the walls so as to be valid for situations where the impedance boundary condition may not remain sufficiently accurate

The F5 Criterion Revised

The purpose of this work is to generalize part of the theory behind Faugere\u27s F5 algorithm. This is one of the fastest known algorithms to compute a Gröbner basis of a polynomial ideal I generated by polynomials f1,…,fm. A major reason for this is what Faugere called the algorithm\u27s new criterion, and we call the F5 criterion : it provides a sufficient condition for a set of polynomialsGto be a Gröbner basis. However. the F5 algorithm is difficult to grasp, and there are unresolved questions regarding its termination. This paper introduces some new concepts that place the criterion in a more general setting:S-Gröbner bases and primitive S-irreducible polynomials. We use these to propose a new, simple algorithm based on a revised F5 criterion. The new concepts also enable us to remove various restrictions, such as proving termination without the requirement that f1,…,fm be a regular sequence. (C) 2011 Elsevier Ltd. All rights reserved

The F5 Criterion Revised

The purpose of this work is to generalize part of the theory behind Faugere\u27s F5 algorithm. This is one of the fastest known algorithms to compute a Gröbner basis of a polynomial ideal I generated by polynomials f1,…,fm. A major reason for this is what Faugere called the algorithm\u27s new criterion, and we call the F5 criterion : it provides a sufficient condition for a set of polynomialsGto be a Gröbner basis. However. the F5 algorithm is difficult to grasp, and there are unresolved questions regarding its termination. This paper introduces some new concepts that place the criterion in a more general setting:S-Gröbner bases and primitive S-irreducible polynomials. We use these to propose a new, simple algorithm based on a revised F5 criterion. The new concepts also enable us to remove various restrictions, such as proving termination without the requirement that f1,…,fm be a regular sequence. (C) 2011 Elsevier Ltd. All rights reserved

Inference Systems with Corules for Fair Subtyping and Liveness Properties of Binary Session Types

Many properties of communication protocols stem from the combination of safety and liveness properties. Characterizing such combined properties by means of a single inference system is difficult because of the fundamentally different techniques (coinduction and induction, respectively) usually involved in defining and proving them. In this paper we show that Generalized Inference Systems allow for simple and insightful characterizations of (at least some of) these combined inductive/coinductive properties for dependent session types. In particular, we illustrate the role of corules in characterizing weak termination (the property of protocols that can always eventually terminate), fair compliance (the property of interactions that can always be extended to reach client satisfaction) and also fair subtyping, a liveness-preserving refinement relation for session types