Subclasses of Presburger Arithmetic and the Weak EXP Hierarchy

It is shown that for any fixed $i>0$, the $\Sigma_{i+1}$-fragment of Presburger arithmetic, i.e., its restriction to $i+1$ quantifier alternations beginning with an existential quantifier, is complete for $\mathsf{\Sigma}^{\mathsf{EXP}}_{i}$, the $i$-th level of the weak EXP hierarchy, an analogue to the polynomial-time hierarchy residing between $\mathsf{NEXP}$ and $\mathsf{EXPSPACE}$. This result completes the computational complexity landscape for Presburger arithmetic, a line of research which dates back to the seminal work by Fischer & Rabin in 1974. Moreover, we apply some of the techniques developed in the proof of the lower bound in order to establish bounds on sets of naturals definable in the $\Sigma_1$-fragment of Presburger arithmetic: given a $\Sigma_1$-formula $\Phi(x)$, it is shown that the set of non-negative solutions is an ultimately periodic set whose period is at most doubly-exponential and that this bound is tight.Comment: 10 pages, 2 figure

Can learning organisations survive in the newer NHS?

BACKGROUND: This paper outlines the principal characteristics of a learning organisation and the organisational features that define it. It then compares these features with the organisational conditions that currently obtain, or are being created, within the British NHS. The contradictory development of recent British health policy, resulting in the NHS becoming both more marketised and more bureaucratised has correspondingly ambiguous implications for attempts to implement a 'learning organisation' model. METHODS: Texts that define and debate the characteristics of a learning organisation were found by snowballing references from the founding learning organisation books and published papers, and then by searching a database specifically devised for a literature review on organisational structures and processes in health care. COPAC and ABI-Info databases for subsequent peer-reviewed publications that also appeared relevant to the present study were searched. RESULTS: The outcomes of the above search are summarised and mapped onto the current constituent organisations of the NHS to identify the extent to which they achieve or approximate to a learning organisation status. CONCLUSION: Because of the complexity of the NHS and the contradictory processes of marketisation and bureaucratisation characterising it, it cannot, as a whole system, become a learning organisation. However, it is possible that its constituent organisations may achieve this status to varying degrees. Constraints upon NHS managers to speak their minds freely place an ultimate limit on learning organisation development. This limitation suggests that current British health service policy encourages organisational learning-but not too openly and not too much