Symmetric Weighted First-Order Model Counting

The FO Model Counting problem (FOMC) is the following: given a sentence $\Phi$ in FO and a number $n$, compute the number of models of $\Phi$ over a domain of size $n$; the Weighted variant (WFOMC) generalizes the problem by associating a weight to each tuple and defining the weight of a model to be the product of weights of its tuples. In this paper we study the complexity of the symmetric WFOMC, where all tuples of a given relation have the same weight. Our motivation comes from an important application, inference in Knowledge Bases with soft constraints, like Markov Logic Networks, but the problem is also of independent theoretical interest. We study both the data complexity, and the combined complexity of FOMC and WFOMC. For the data complexity we prove the existence of an FO$^{3}$ formula for which FOMC is #P$_1$-complete, and the existence of a Conjunctive Query for which WFOMC is #P$_1$-complete. We also prove that all $\gamma$-acyclic queries have polynomial time data complexity. For the combined complexity, we prove that, for every fragment FO$^{k}$, $k\geq 2$, the combined complexity of FOMC (or WFOMC) is #P-complete.Comment: To appear at PODS'1

Students´ language in computer-assisted tutoring of mathematical proofs

Truth and proof are central to mathematics. Proving (or disproving) seemingly simple statements often turns out to be one of the hardest mathematical tasks. Yet, doing proofs is rarely taught in the classroom. Studies on cognitive difficulties in learning to do proofs have shown that pupils and students not only often do not understand or cannot apply basic formal reasoning techniques and do not know how to use formal mathematical language, but, at a far more fundamental level, they also do not understand what it means to prove a statement or even do not see the purpose of proof at all. Since insight into the importance of proof and doing proofs as such cannot be learnt other than by practice, learning support through individualised tutoring is in demand. This volume presents a part of an interdisciplinary project, set at the intersection of pedagogical science, artificial intelligence, and (computational) linguistics, which investigated issues involved in provisioning computer-based tutoring of mathematical proofs through dialogue in natural language. The ultimate goal in this context, addressing the above-mentioned need for learning support, is to build intelligent automated tutoring systems for mathematical proofs. The research presented here has been focused on the language that students use while interacting with such a system: its linguistic propeties and computational modelling. Contribution is made at three levels: first, an analysis of language phenomena found in students´ input to a (simulated) proof tutoring system is conducted and the variety of students´ verbalisations is quantitatively assessed, second, a general computational processing strategy for informal mathematical language and methods of modelling prominent language phenomena are proposed, and third, the prospects for natural language as an input modality for proof tutoring systems is evaluated based on collected corpora

Q(sqrt(-3))-Integral Points on a Mordell Curve

We use an extension of quadratic Chabauty to number fields,recently developed by the author with Balakrishnan, Besser and M ̈uller,combined with a sieving technique, to determine the integral points overQ(√−3) on the Mordell curve y2 = x3 − 4

The 1974 Bilingual Education Amendments: Revolution, Reaction or Reform

Purpose: The study examined in detail the legislative history of the 1974 Bilingual Education Act, Section 105 of the Education Amendments of 1974, Public Law 93-380. The study examined the roles of Representatives, Senators, lobbyists, judicial decisions, minority groups and Administration officials in developing the 1974 Bilingual Education Act

Logic and lattices for a statistics advisor

The work partially reported here concerned the development ot a prototype Expert System for giving advice about Statistics experiments, called ASA, and an inference engine to support ASA, called ABASE.This involved discovering what knowledge was necessary for performing the task at a satis¬ factory level of competence, working out how to represent this knowledge in a computer, and how to process the representations efficiently.Two areas of Statistical knowledge are described in detail: the classification of measure¬ ments and statistical variables, and the structure of elementary statistical experiments. A knowledge representation system based on lattices is proposed, and it is shown that such representations are learnable by computer programs, and lend themselves to particularly efficient implementation.ABASE was influenced by MBASE, the inference engine of MECHO [Bundy et al 79a]. Both are theorem provers working on typed function-free Horn clauses, with controlled creation of new entities. Their type systems and proof procedures are radically different, though, and ABASE is "conversational" while MBASE is not

LIPIcs, Volume 261, ICALP 2023, Complete Volume

LIPIcs, Volume 261, ICALP 2023, Complete Volum