Seurat games on Stockmeyer graphs

We define a family of vertex colouring games played over a pair of graphs or digraphs (G, H) by players ∀ and ∃. These games arise from work on a longstanding open problem in algebraic logic. It is conjectured that there is a natural number n such that ∀ always has a winning strategy in the game with n colours whenever G 6∼= H. This is related to the reconstruction conjecture for graphs and the degree-associated reconstruction conjecture for digraphs. We show that the reconstruction conjecture implies our game conjecture with n = 3 for graphs, and the same is true for the degree-associated reconstruction conjecture and our conjecture for digraphs. We show (for any k < ω) that the 2-colour game can distinguish certain non-isomorphic pairs of graphs that cannot be distinguished by the k-dimensional Weisfeiler-Leman algorithm. We also show that the 2-colour game can distinguish the non-isomorphic pairs of graphs in the families defined by Stockmeyer as counterexamples to the original digraph reconstruction conjecture

Premise Selection for Mathematics by Corpus Analysis and Kernel Methods

Smart premise selection is essential when using automated reasoning as a tool for large-theory formal proof development. A good method for premise selection in complex mathematical libraries is the application of machine learning to large corpora of proofs. This work develops learning-based premise selection in two ways. First, a newly available minimal dependency analysis of existing high-level formal mathematical proofs is used to build a large knowledge base of proof dependencies, providing precise data for ATP-based re-verification and for training premise selection algorithms. Second, a new machine learning algorithm for premise selection based on kernel methods is proposed and implemented. To evaluate the impact of both techniques, a benchmark consisting of 2078 large-theory mathematical problems is constructed,extending the older MPTP Challenge benchmark. The combined effect of the techniques results in a 50% improvement on the benchmark over the Vampire/SInE state-of-the-art system for automated reasoning in large theories.Comment: 26 page

Enacting Reasoning-and-Proving in Secondary Mathematics Classrooms through Tasks

Proof is the mathematical way of convincing oneself and others of the truth of a claim for all cases in the domain under consideration. As such, reasoning-and-proving is a crucial, formative practice for all students in kindergarten through twelfth grade, which is reflected in the Common Core State Standards in Mathematics. However, students and teachers exhibit many difficulties employing, writing, and understanding reasoning-and-proving. In particular, teachers are challenged by their knowledge base, insufficient resources, and unsupportive pedagogy. The Cases of Reasoning and Proving (CORP) materials were designed to offer teachers opportunities to engage in reasoning-and-proving tasks, discuss samples of authentic practice, examine research-based frameworks, and develop criteria for evaluating reasoning-and-proving products based on the core elements of proof. A six-week graduate level course was taught with the CORP materials with the goal of developing teachers’ understanding of what constitutes reasoning-and-proving, how secondary students benefit from reasoning-and-proving, and how they can support the development of students’ capacities to reason-and-prove. Research was conducted on four participants of the course during either their first or second year of teaching. The purpose of the research was to study the extent to which the participants selected, implemented, and evaluated students’ work on reasoning-and-proving tasks. The participants’ abilities were examined through an analysis of answers to interview questions, tasks used in class, and samples of student work, and scoring criteria. The results suggest that: 1.) participants were able to overcome some of the limitations of their insufficient resource by modifying and creating some reasoning-and-proving exercises; 2.) participants were able to maintain the level of cognitive demand of proof tasks during implementation; and 3) participants included some if not all of the core elements of proof in their definition of proof and in their evaluation criteria for student products of reasoning-and-proving products

¬¬¬¬Modeling: Neutral, Null, and Baseline

This paper distinguishes two reasoning strategies for using a model as a “null”. Null modeling evaluates whether a process is causally responsible for a pattern by testing it against a null model. Baseline modeling measures the relative significance of various processes responsible for a pattern by detecting deviations from a baseline model. Scientists sometimes conflate these strategies because their formal similarities, but they must distinguish them lest they privilege null models as accepted until disproved. I illustrate this problem with the neutral theory of ecology and use this as a case study to draw general lessons. First, scientists cannot draw certain kinds of causal conclusions using null modeling. Second, scientists can draw these kinds of causal conclusions using baseline modeling, but this requires more evidence than does null modeling

Learning-Assisted Automated Reasoning with Flyspeck

The considerable mathematical knowledge encoded by the Flyspeck project is combined with external automated theorem provers (ATPs) and machine-learning premise selection methods trained on the proofs, producing an AI system capable of answering a wide range of mathematical queries automatically. The performance of this architecture is evaluated in a bootstrapping scenario emulating the development of Flyspeck from axioms to the last theorem, each time using only the previous theorems and proofs. It is shown that 39% of the 14185 theorems could be proved in a push-button mode (without any high-level advice and user interaction) in 30 seconds of real time on a fourteen-CPU workstation. The necessary work involves: (i) an implementation of sound translations of the HOL Light logic to ATP formalisms: untyped first-order, polymorphic typed first-order, and typed higher-order, (ii) export of the dependency information from HOL Light and ATP proofs for the machine learners, and (iii) choice of suitable representations and methods for learning from previous proofs, and their integration as advisors with HOL Light. This work is described and discussed here, and an initial analysis of the body of proofs that were found fully automatically is provided