A Classification Approach for Automated Reasoning Systems--A Case Study in Graph Theory

Reasoning systems which create classifications of structured objects face the problem of how object descriptions can be used to reflect their components as well as relations among these components. Current reasoning systems on graph theory do not adequately provide models to discover complex relations among mathematical concepts (eg: relations involving subgraphs) mainly due to the inability to solve this problem. This thesis presents an approach to construct a knowledge-based system, GC (Graph Classification), which overcomes this difficulty in performing automated reasoning in graph theory. We describe graph concepts based on an attribute called Linear Recursive Constructivity (LRC). LRC defines classes by an algebraic formula supported by background knowledge of graph types. We use subsumption checking on decomposed algebraic expressions of graph classes as a major proof method. The search is guided by case-split-based inferencing. Using the approach GC has generated proofs for many theorems such as any two distinct cycles (closed paths) having a common edge e contain a cycle not traversing e , if cycle C1 contains edges e1, e2, and cycle C2 contains edges e2, e3, then there exists a cycle that contains e1 and e3 and the union of a tree and a path is a tree if they have only a single common vertex. The main contributions of this thesis are: (1) Development of a classification-based knowledge representation and a reasoning approach for graph concepts, thus providing a simple model for structured mathematical objects. (2) Development of an algebraic theory for simplifying and decomposing graph concepts. (3) Development of a proof search and a case-splitting technique with the guidance of graph type knowledge. (4) Development of a proving mechanism that can be generate constructive proofs by manipulating only simple linear formalization of theorems

Equivalence-Invariant Algebraic Provenance for Hyperplane Update Queries

The algebraic approach for provenance tracking, originating in the semiring model of Green et. al, has proven useful as an abstract way of handling metadata. Commutative Semirings were shown to be the "correct" algebraic structure for Union of Conjunctive Queries, in the sense that its use allows provenance to be invariant under certain expected query equivalence axioms. In this paper we present the first (to our knowledge) algebraic provenance model, for a fragment of update queries, that is invariant under set equivalence. The fragment that we focus on is that of hyperplane queries, previously studied in multiple lines of work. Our algebraic provenance structure and corresponding provenance-aware semantics are based on the sound and complete axiomatization of Karabeg and Vianu. We demonstrate that our construction can guide the design of concrete provenance model instances for different applications. We further study the efficient generation and storage of provenance for hyperplane update queries. We show that a naive algorithm can lead to an exponentially large provenance expression, but remedy this by presenting a normal form which we show may be efficiently computed alongside query evaluation. We experimentally study the performance of our solution and demonstrate its scalability and usefulness, and in particular the effectiveness of our normal form representation

Algebraic thinking of grade 8 students in solving word problems with a spreadsheet

This paper describes and discusses the activity of grade 8 students on two word problems, using a spreadsheet. We look at particular uses of the spreadsheet, namely at the students’ representations, as ways of eliciting forms of algebraic thinking involved in solving the problems. We aim to see how the spreadsheet allows the solution of formally impracticable problems at students’ level of algebra knowledge, by making them treatable through the computational logic that is intrinsic to the operating modes of the spreadsheet. The protocols of the problem solving sessions provided ways to describe and interpret the relationships that students established between the variables in the problems and their representations in the spreadsheet

Preservice Teachers’ Algebraic Reasoning and Symbol Use on a Multistep Fraction Word Problem

Previous research on preservice teachers’ understanding of fractions and algebra has focused on one or the other. To extend this research, we examined 85 undergraduate elementary education majors and middle school mathematics education majors’ solutions and solution paths (i.e., the ways or methods in which preservice teachers solve word problems) when combining fractions with algebra on a multistep word problem. In this article, we identify and describe common strategy clusters and approaches present in the preservice teachers’ written work. Our results indicate that preservice teachers’ understanding of algebra include arithmetic methods, proportions, and is related to their understanding of a whole

Graphical representation and generalization in sequences problems

In this paper we present different ways used by Secondary students to generalize when they try to solve problems involving sequences. 359 Spanish students solved generalization problems in a written test. These problems were posed through particular terms expressed in different representations. We present examples that illustrate different ways of achieving various types of generalization and how students express generalization. We identify graphical representation of generalization as a useful tool of getting other ways of expressing generalization, and we analyze its connection with other ways of expressing it