Relative Predicativity and dependent recursion in second-order set theory and higher-orders theories

This article reports that some robustness of the notions of predicativity and of autonomous progression is broken down if as the given infinite total entity we choose some mathematical entities other than the traditional ω. Namely, the equivalence between normal transfinite recursion scheme and new dependent transfinite recursion scheme, which does hold in the context of subsystems of second order number theory, does not hold in the context of subsystems of second order set theory where the universe V of sets is treated as the given totality (nor in the contexts of those of n+3-th order number or set theories, where the class of all n+2-th order objects is treated as the given totality).

The Meta in Meta-object Architectures

Behavioral reflection is crucial to support for example functional upgrades, on-the-fly debugging, or monitoring critical applications. However the use of reflective features can lead to severe problems due to infinite metacall recursion even in simple cases. This is especially a problem when reflecting on core language features since there is a high chance that such features are used to implement the reflective behavior itself. In this paper we analyze the problem of infinite meta-object call recursion and solve it by providing a first class representation of meta-level execution: at any point in the execution of a system it can be determined if we are operating on a meta-level or base level so that we can prevent infinite recursion. We present how meta-level execution can be represented by a meta-context and how reflection becomes context-aware. Our solution makes it possible to freely apply behavioral reflection even on system classes: the meta-context brings stability to behavioral reflection. We validate the concept with a robust implementation and we present benchmarks.

Justifying induction on modal μ-formulae

We define a rank function for formulae of the propositional modal μ-calculus such that the rank of a fixed point is strictly bigger than the rank of any of its finite approximations. A rank function of this kind is needed, for instance, to establish the collapse of the modal μ-hierarchy over transitive transition systems. We show that the range of the rank function is ωω. Further we establish that the rank is computable by primitive recursion, which gives us a uniform method to generate formulae of arbitrary rank below ωω.

Applicative theories for logarithmic complexity classes

We present applicative theories of words corresponding to weak, and especially logarithmic, complexity classes. The theories for the logarithmic hierarchy and alternating logarithmic time formalise function algebras with concatenation recursion as main principle. We present two theories for logarithmic space where the first formalises a new two-sorted algebra which is very similar to Cook and Bellantoni's famous two-sorted algebra B for polynomial time [4]. The second theory describes logarithmic space by formalising concatenation- and sharply bounded recursion. All theories contain the predicates WW representing words, and VV representing temporary inaccessible words. They are inspired by Cantini's theories [6] formalising B.