Semigroups with if-then-else and halting programs

The "if–then–else" construction is one of the most elementary programming commands, and its abstract laws have been widely studied, starting with McCarthy. Possibly, the most obvious extension of this is to include the operation of composition of programs, which gives a semigroup of functions (total, partial, or possibly general binary relations) that can be recombined using if–then–else. We show that this particular extension admits no finite complete axiomatization and instead focus on the case where composition of functions with predicates is also allowed (and we argue there is good reason to take this approach). In the case of total functions — modeling halting programs — we give a complete axiomatization for the theory in terms of a finite system of equations. We obtain a similar result when an operation of equality test and/or fixed point test is included

Identities in the Algebra of Partial Maps

We consider the identities of a variety of semigroup-related algebras modelling the algebra of partial maps. We show that the identities are intimately related to a weak semigroup deductive system and we show that the equational theory is decidable. We do this by giving a term rewriting system for the variety. We then show that this variety has many subvarieties whose equational theory interprets the full uniform word problem for semigroups and consequently are undecidable. As a corollary it is shown that the equational theory of Clifford semigroups whose natural order is a semilattice is undecidable

Monoids with tests and the algebra of possibly non-halting programs

We study the algebraic theory of computable functions, which can be viewed as arising from possibly non-halting computer programs or algorithms, acting on some state space, equipped with operations of composition, if-then-else and while-do defined in terms of a Boolean algebra of conditions. It has previously been shown that there is no finite axiomatisation of algebras of partial functions under these operations alone, and this holds even if one restricts attention to transformations (representing halting programs) rather than partial functions, and omits while-do from the signature. In the halting case, there is a natural “fix”, which is to allow composition of halting programs with conditions, and then the resulting algebras admit a finite axiomatisation. In the current setting such compositions are not possible, but by extending the notion of if-then-else, we are able to give finite axiomatisations of the resulting algebras of (partial) functions, with while-do in the signature if the state space is assumed finite. The axiomatisations are extended to consider the partial predicate of equality. All algebras considered turn out to be enrichments of the notion of a (one-sided) restriction semigrou

Partial maps with domain and range: extending Schein's representation

The semigroup of all partial maps on a set under the operation of composition admits a number of operations relating to the domain and range of a partial map. Of particular interest are the operations R and L returning the identity on the domain of a map and on the range of a map respectively. Schein [25] gave an axiomatic characterisation of the semigroups with R and L representable as systems of partial maps; the class is a finitely axiomatisable quasivariety closely related to ample semigroups (which were introduced—as type A semigroups—by Fountain, [7]). We provide an account of Schein's result (which until now appears only in Russian) and extend Schein's method to include the binary operations of intersection, of greatest common range restriction, and some unary operations relating to the set of fixed points of a partial map. Unlike the case of semigroups with R and L, a number of the possibilities can be equationally axiomatised

Value innovation modelling: Design thinking as a tool for business analysis and strategy

This paper explores the use of multiple perspective problem framing (English 2008) as a tool to reveal hidden value and commercial opportunity for business. Creative thinking involves the interrelationship of parameters held open and fluid within the cognitive span of the creative mind. The recognition of new associations can create new value that can lead to innovation in designed products, intellectual property and business strategy. The ‘Ideas-lab’ process is based on the proposition that a company’s capacity for innovation is dependent on the way the business is able to see its problems and opportunities. In this process the attributes of a company and the experience of the researchers are considered as the parameters of a design problem. It is therefore important to acknowledge the commercial experience of the project researchers, all of whom have a proven track record in helping businesses develop, exploit and protect their know how. Semi structured interviews were carried out with key individuals in 34 companies. The resulting data was assessed on a company-by-company basis through a process of multiple perspective problem framing, enabling key nodes, patterns and relationships to be identified and explored. A ‘Cornerstones of Innovation’ report was prepared to inform each company of the observations made by the researchers. The paper describes the methods adopted and summarises the feedback from participating companies. Case studies are highlighted to demonstrate ways in which the process influenced the actions of particular businesses, and the commercial outcomes that resulted. Finally the researchers reflect on the structure of the Ideas-lab process

Estimating direct and indirect rebound effects for UK households

Energy efficiency improvements by households lead to rebound effects that offset the potential energy and emissions savings. Direct rebound effects result from increased demand for cheaper energy services, while indirect rebound effects result from increased demand for other goods and services that also require energy to provide. Research to date has focused upon the former, but both are important for climate change. This study estimates the combined direct and indirect rebound effects from seven measures that improve the energy efficiency of UK dwellings. The methodology is based upon estimates of the income elasticity and greenhouse gas (GHG) intensity of 16 categories of household goods and services, and allows for the embodied emissions of the energy efficiency measures themselves. Rebound effects are measured in GHG terms and relate to the adoption of these measures by an average UK household. The study finds that the rebound effects from these measures are typically in the range 5-15% and arise mostly from indirect effects. This is largely because expenditure on gas and electricity is more GHG-intensive than expenditure on other goods and services. However, the anticipated shift towards a low carbon electricity system in the UK may lead to much larger rebound effects

Light Spectrum and its Implications on Milk Production

This information was part of the August 2013 issue of Eastern DairyBusiness Magazine. The Manager, a section within the Eastern DairyBusiness Magazine, is authored and organized by the PRO-DAIRY program in the College of Agriculture and Life Sciences at Cornell University