
Aschenbach, Martin; Kötzing, Timo; Seidel, Karen Learning from Informants: Relations between Learning Success Criteria. ArXiv 2018: 37
Learning from positive and negative information, socalled informants, being one of the models for human and machine learning introduced by Gold [1967], is investigated. Particularly, naturally arising questions about this learning setting, originating in results on learning from solely positive information, are answered. By a carefully arranged argument learners can be assumed to only change their hypothesis in case it is inconsistent with the data (such a learning behavior is called conservative). The deduced main theorem states the relations between the most important delayable learning success criteria, being the ones not ruined by a delayed in time hypothesis output. Additionally, our investigations concerning the nondelayable requirement of consistent learning underpin the claim for delayability being the right structural property to gain a deeper understanding concerning the nature of learning success criteria. Moreover, we obtain an anomalous hierarchy when allowing for an increasing finite number of anomalies of the hypothesized language by the learner compared with the language to be learned. In contrast to the vacillatory hierarchy for learning from solely positive information, we observe a duality depending on whether infinitely many vacillations between different (almost) correct hypotheses are still considered a successful learning behavior.

Göbel, Andreas; Lagodzinski, J. A. Gregor; Seidel, Karen Counting Homomorphisms to Trees Modulo a Prime. International Symposium on Mathematical Foundations of Computer Science (MFCS) 2018: 49:149:13
Many important graph theoretic notions can be encoded as counting graph homomorphism problems, such as partition functions in statistical physics, in particular independent sets and colourings. In this article we study the complexity of~\($\#_p\textsc{HomsTo}H$\), the problem of counting graph homomorphisms from an input graph to a graph \($H$\) modulo a prime number~\($p$\). Dyer and Greenhill proved a dichotomy stating that the tractability of nonmodular counting graph homomorphisms depends on the structure of the target graph. Many intractable cases in nonmodular counting become tractable in modular counting due to the common phenomenon of cancellation. In subsequent studies on counting modulo~\($2$\), however, the influence of the structure of~\($H$\) on the tractability was shown to persist, which yields similar dichotomies. Our main result states that for every tree~\($H$\) and every prime~\($p$\) the problem \($\#_p\textsc{HomsTo}H$\) is either polynomial time computable or \($\#_p\mathsf{P}$\)complete. This relates to the conjecture of Faben and Jerrum stating that this dichotomy holds for every graph \($H$\) when counting modulo~2. In contrast to previous results on modular counting, the tractable cases of \($\#_p\textsc{HomsTo}H$\) are essentially the same for all values of the modulo when \($H$\) is a tree. To prove this result, we study the structural properties of a homomorphism. As an important interim result, our study yields a dichotomy for the problem of counting weighted independent sets in a bipartite graph modulo some prime~\($p$\). These results are the first suggesting that such dichotomies hold not only for the onebit functions of the modulo~2 case but also for the modular counting functions of all primes~\($p$\).

Göbel, Andreas; Lagodzinski, J. A. Gregor; Seidel, Karen Counting Homomorphisms to Trees Modulo a Prime. arXiv 2018

Seidel, Karen Zu mathematischen Argumentationen eines Experten aus einer semiotischen Perspektive. Beiträge zum Mathematikunterricht 2017 2017: 897900
In Argumentationen unter in der Forschung tätigen Mathematikern kombiniert die erklärende Person Gesten, Sprache und Zeichen und verleiht damit unter anderem den concept images der verwendeten mathematischen Begriffe Ausdruck. Im Folgenden wird der Frage nachgegangen, inwiefern eine multimodale Analyse von erklärenden Phasen in Argumentationen von Experten mit dem Fokus auf Begriffsbildungsprozessen Aufschluss geben kann über operationale und strukturelle Vorstellungen von Begriffen (Sfard, 1991).

Hölzl, Rupert; Jain, Sanjay; Schlicht, Philipp; Seidel, Karen; Stephan, Frank Automatic Learning from Repetitive Texts. International Conference on Algorithmic Learning Theory (ALT) 2017: 129150
We study the connections between the learnability of automatic families of languages and the types of text used to present them to a learner. More precisely, we study how restrictions on the number of times that a correct datum appears in a text influence what classes of languages are automatically learnable. We show that an automatic family of languages is automatically learnable from fat text iff it is automatically learnable from thick text iff it is verifiable from balanced text iff it satisfies Angluin's telltale condition. Furthermore, many automatic families are automatically learnable from exponential text. We also study the relationship between automatic learnability and verifiability and show that all automatic families are automatically partially verifiable from exponential text and automatically learnable from thick text.

Kötzing, Timo; Schirneck, Martin; Seidel, Karen Normal Forms in Semantic Language Identification. International Conference on Algorithmic Learning Theory (ALT) 2017: 493516
We consider language learning in the limit from text where all learning restrictions are semantic, that is, where any conjecture may be replaced by a semantically equivalent conjecture. For different such learning criteria, starting with the wellknown TxtGBclearning, we consider three different normal forms: strongly locking learning, consistent learning and (partially) setdriven learning. These normal forms support and simplify proofs and give insight into what behaviors are necessary for successful learning (for example when consistency in conservative learning implies cautiousness and strong decisiveness). We show that strongly locking learning can be assumed for partially setdriven learners, even when learning restrictions apply. We give a very general proof relying only on a natural property of the learning restriction, namely, allowing for simulation on equivalent text. Furthermore, when no restrictions apply, also the converse is true: every strongly locking learner can be made partially setdriven. For several semantic learning criteria we show that learning can be done consistently. Finally, we deduce for which learning restrictions partial setdrivenness and setdrivenness coincide, including a general statement about classes of infinite languages. The latter again relies on a simulation argument.

Koepke, Peter; Räsch, Karen; Schlicht, Philipp A minimal Prikrytype forcing for singularizing a measurable cardinal. The Journal of Symbolic Logic 2013: 85100
Recently, Gitik, Kanovei and the first author proved that for a classical Prikry forcing extension the family of the intermediate models can be parametrized by \(\mathscr{P(\omega)/\mathrm{finite}\). By modifying the standard Prikry tree forcing we define a Prikrytype forcing which also singularizes a measurable cardinal but which is minimal, i.e. there are \emph{no} intermediate models properly between the ground model and the generic extension. The proof relies on combining the rigidity of the tree structure with indiscernibility arguments resulting from the normality of the associated measures.