Recent work by Atserias and Dawar [Albert Atserias and Anuj Dawar, 2019] and Tucker-Foltz [Jamie Tucker-Foltz, 2024] has established undefinability results in fixed-point logic with counting (FPC) corresponding to many classical complexity results from the hardness of approximation. In this line of work, NP-hardness results are turned into unconditional FPC undefinability results. We extend this work by showing the FPC undefinability of any constant factor approximation of weighted 2-to-2 games, based on the NP-hardness results of Khot, Minzer and Safra. Our result shows that the completely satisfiable 2-to-2 games are not FPC-separable from those that are not ε-satisfiable, for arbitrarily small ε. The perfect completeness of our inseparability is an improvement on the complexity result, as the NP-hardness of such a separation is still only conjectured. This perfect completeness enables us to show the FPC undefinability of other problems whose NP-hardness is conjectured. In particular, we are able to show that no FPC formula can separate the 3-colourable graphs from those that are not t-colourable, for any constant t.

Many companies look for novel ways to trace their operational sustainability. The application of AI to analyze and make sense of the big data the company holds represents one promising approach for this aim. The authors study how one can set and design an AI-based solution for improving the sustainability of complex business processes and decision-making in companies of different types. First, they provide a general analysis of current frameworks for measuring adherence to sustainability goals for companies, then they present a conceptual framework and architecture design for an AI-enabled sustainability service for companies. The implications of our research suggest that AI can provide distinct functions: (a) automation: taking big data from different departments and analyzing them with the aim of tracing the sustainability of the company; (b) support: to help decision-making and create relevant insights for stakeholders that are coherent with defined sustainability decision criteria. To the authors' knowledge, no previous research has provided analysis and design of such AI solution for companies.

Teaching innovation and entrepreneurship is a challenging task. It requires students to gain an understanding of key terminology and theoretical knowledge as well as practical experience to build transversal skills. This is what is needed to develop a problem-solving and enterprising attitude for the jobs of tomorrow. At doctoral level, educating STEM graduates to become entrepreneurs and digital leaders in the digital transformation era calls for new learning formats and contents. In this paper we present a new design for the EIT Digital Doctoral School Innovation and Entrepreneurship (I&E) education and we specifically focus on the online educational components developed to cover key I&E content and topics. We present the first results of an evaluation conducted with the students participating in all the courses of the EIT Digital Doctoral School in 2019 and the first impressions and learnings from the new design. We found that the initial feedback was satisfying and that our students appreciated the approach and said that they recommend that instructors use this approach in future teaching. This study contributes to the ongoing debate on how to design and implement I&E education leveraging European universities networks.