Corrigendum to “Shifting : one-inclusion mistake bounds and sample compression” [J. Comput. System Sci. 75 (1) (2009) 37–59]

H. Simon and B. Szörényi have found an error in the proof of Theorem 52 of “Shifting: One-inclusion mistake bounds and sample compression”, Rubinstein et al. (2009). In this note we provide a corrected proof of a slightly weakened version of this theorem. Our new bound on the density of one-inclusion hypergraphs is again in terms of the capacity of the multilabel concept class. Simon and Szörényi have recently proved an alternate result in Simon and Szörényi (2009).

Hypergraph-based modeling of ad-hoc business processes

Process models are usually depicted as directed graphs, with nodes representing activities and directed edges control flow. While structured processes with pre-defined control flow have been studied in detail, flexible processes including ad-hoc activities need further investigation. This paper presents flexible process graph, a novel approach to model processes in the context of dynamic environment and adaptive process participants’ behavior. The approach allows defining execution constraints, which are more restrictive than traditional ad-hoc processes and less restrictive than traditional control flow, thereby balancing structured control flow with unstructured ad-hoc activities. Flexible process graph focuses on what can be done to perform a process. Process participants’ routing decisions are based on the current process state. As a formal grounding, the approach uses hypergraphs, where each edge can associate any number of nodes. Hypergraphs are used to define execution semantics of processes formally. We provide a process scenario to motivate and illustrate the approach.

On algebraic immunity and annihilators

Algebraic immunity AI(f) defined for a boolean function f measures the resistance of the function against algebraic attacks. Currently known algorithms for computing the optimal annihilator of f and AI(f) are inefficient. This work consists of two parts. In the first part, we extend the concept of algebraic immunity. In particular, we argue that a function f may be replaced by another boolean function f^c called the algebraic complement of f. This motivates us to examine AI(f ^c ). We define the extended algebraic immunity of f as AI *(f)= min {AI(f), AI(f^c )}. We prove that 0≤AI(f)–AI *(f)≤1. Since AI(f)–AI *(f)= 1 holds for a large number of cases, the difference between AI(f) and AI *(f) cannot be ignored in algebraic attacks. In the second part, we link boolean functions to hypergraphs so that we can apply known results in hypergraph theory to boolean functions. This not only allows us to find annihilators in a fast and simple way but also provides a good estimation of the upper bound on AI *(f).