Arithmetic Classification of Perfect Models of Stratified Programs (Addendum)

RECURSION-FREE PROGRAMS The following section completes the analysis of arithmetic complexity of perfect models and has been inadvertently omitted in the previous version of the paper. We say that a general program P is recursion-free if in its dependency graph Dp there is no cycle. Clearly recursion-free programs form a subclass of stratified programs. Recursion-free programs form a very simple generalization of the class of hierarchical programs introduced in [C78]. Hierarchical programs satisfy an additional condition on variable occurrences in clauses that prevents floundering, i.e. a forced selection of a non-ground negative literal in an SLDNF- derivation. In this section we study the complexity of perfect models of recursion-free programs

The Expressiveness of Locally Stratified Programs

This paper completes an investigation of the logical expressibility of finite, locally stratified, general logic programs. We show that every hyperarithmetic set can be computed by a suitably chosen locally stratified logic program (as a set of values of a predicate over its perfect model). This is an optimal result, since the perfect model of a locally stratified program is itself an implicitly definable hyperarithmetic set (under a recursive coding of the Herbrand base); hence to obtain all hyperarithmetic sets requires something new, in this case selecting one predicate from the model. We find that the expressive power of programs does not increase when one considers the programs which have a unique stable model or a total well-founded model. This shows that all these classes of structures (perfect models of locally stratified logic programs, well-founded models which turn out to be total, and stable models of programs possessing a unique stable model) are all closely connected with Kleene\u27s hyperarithmetical hierarchy. Thus, for general logic programming, negation with respect to two-valued logic is related to the hyperarithmetic hierarchy in the same way as Horn logic is to the class of recursively enumerable sets

Infinite time Turing machines and an application to the hierarchy of equivalence relations on the reals

We describe the basic theory of infinite time Turing machines and some recent developments, including the infinite time degree theory, infinite time complexity theory, and infinite time computable model theory. We focus particularly on the application of infinite time Turing machines to the analysis of the hierarchy of equivalence relations on the reals, in analogy with the theory arising from Borel reducibility. We define a notion of infinite time reducibility, which lifts much of the Borel theory into the class $\bm{\Delta}^1_2$ in a satisfying way.Comment: Submitted to the Effective Mathematics of the Uncountable Conference, 200

Application of Volcano Plots in Analyses of mRNA Differential Expressions with Microarrays

Volcano plot displays unstandardized signal (e.g. log-fold-change) against noise-adjusted/standardized signal (e.g. t-statistic or -log10(p-value) from the t test). We review the basic and an interactive use of the volcano plot, and its crucial role in understanding the regularized t-statistic. The joint filtering gene selection criterion based on regularized statistics has a curved discriminant line in the volcano plot, as compared to the two perpendicular lines for the "double filtering" criterion. This review attempts to provide an unifying framework for discussions on alternative measures of differential expression, improved methods for estimating variance, and visual display of a microarray analysis result. We also discuss the possibility to apply volcano plots to other fields beyond microarray.Comment: 8 figure