In a recent paper, Girard (2012) proposed to use his recent construction of a geometry of interaction in the hyperfinite factor (Girard 2011) in an innovative way to characterize complexity classes. We begin by giving a detailed explanation of both the choices and the motivations of Girard's definitions. We then provide a complete proof that the complexity class co-NL can be characterized using this new approach. We introduce the non-deterministic pointer machine as a technical tool, a concrete model to compute algorithms.

What does it mean that an encoding is fully abstract? What does it not mean? In this position paper, we want to help the reader to evaluate the real benefits of using such a notion when studying the expressiveness of programming languages. Several examples and counterexamples are given. In some cases, we work at a very abstract level; in other cases, we give concrete samples taken from the field of process calculi, where the theory of expressiveness has been mostly developed in the last years.

Full abstraction, i.e. that a function preserves equivalence from a source to a target, has been used extensively as a correctness criterion for mappings between models of computation. I here show that with fixed equivalences, fully abstract functions almost always exist. Also, with the function and one of the equivalences fixed the other equivalence can almost always be found.

Completeness is a key feature of abstract interpretation. It corresponds to exactness of the abstraction of fix-points and relies upon the need of absence of false alarms in static program analysis. Making abstract interpretation complete is therefore a major problem in approximating the semantics of programming languages. In this paper, we consider the problem of making abstract interpretations complete by minimally modifying the predicate transformer, i.e. the semantics, of a program. We study the mathematical properties of complete functions on complete lattices and prove the existence of minimal transformations of monotone functions to achieve completeness. We then apply minimal complete transformers to prove the minimality of standard program transformations in security, such as static program monitoring.

We investigate the relationships between the dynamical properties of Boolean networks and properties of their Jacobian matrices, in particular the existence of local cycles in the associated interaction graphs. We define the notion of hereditarily bijective maps, and we use it to strengthen the property of unicity of a fixed point and to provide simplified proofs and generalizations of theorems relating attractors to the existence of local cycles, in particular local positive cycles. We then argue that this notion may not suffice to prove, under a suitable hypothesis such as the existence of a cyclic attractor or a stronger hypothesis, the existence of local negative cycles. We then consider a class of Boolean networks called and-or-nets, and for this class, we prove that the hypothesis of an antipodal attractive cycle implies the existence of a local negative cycle.