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

authors: Jackson, Marcel

subjects: Science & Technology

abstract: 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 semigroup

text: 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 semigroup

year: 2015-03

title: Do we expect too much? Reflection on chemistry content in higher education

authors: Coll, Richard K.

subjects: chemistry

abstract: Education research in the 1970s, like other related areas, was dominated by quantitative work¹ during an era for which social sciences sought to draw upon the successful scientific approach typically used in the physical sciences (in particular) to investigate teaching and learning.¹’² So if we felt a cohort of students did not understand some con¬cept, we tried to find out whether or not a different teach¬ing approach could fix their misconceptions.³ But how to do this? Well, drawing on a scientific approach, we would divide the class or classes up, teach one cohort the same way we always had, and the other cohort in our new way, and evaluate any differences in conceptual understanding using, e.g. a standardized topic test. Differences would be examined for statistical significance of evidence that our new approach to teaching had worked. And this is the way much research was done at the time. Control of variables, randomized sampling, and so on, were all embedded in such an approach to educational research.

text: Do we expect too much? Reflection on chemistry content in higher education Education research in the 1970s, like other related areas, was dominated by quantitative work¹ during an era for which social sciences sought to draw upon the successful scientific approach typically used in the physical sciences (in particular) to investigate teaching and learning.¹’² So if we felt a cohort of students did not understand some con¬cept, we tried to find out whether or not a different teach¬ing approach could fix their misconceptions.³ But how to do this? Well, drawing on a scientific approach, we would divide the class or classes up, teach one cohort the same way we always had, and the other cohort in our new way, and evaluate any differences in conceptual understanding using, e.g. a standardized topic test. Differences would be examined for statistical significance of evidence that our new approach to teaching had worked. And this is the way much research was done at the time. Control of variables, randomized sampling, and so on, were all embedded in such an approach to educational research.

year: 2008

title: Approaches to benefit sharing

authors: Sterling, Rogena

abstract: This information sheet provides successful examples of different approaches to benefit sharing across a range of industries, from around the world and within Aotearoa. It highlights what works and where challenges and gaps exist in current approaches. Finally, a spectrum of Indigenous sharing approaches is provided, based on the OECD framework of benefit sharing and type of benefit (direct and collective) to Indigenous peoples.

text: Approaches to benefit sharing This information sheet provides successful examples of different approaches to benefit sharing across a range of industries, from around the world and within Aotearoa. It highlights what works and where challenges and gaps exist in current approaches. Finally, a spectrum of Indigenous sharing approaches is provided, based on the OECD framework of benefit sharing and type of benefit (direct and collective) to Indigenous peoples.

year: 2024-06-01