String rewriting for Double Coset Systems

In this paper we show how string rewriting methods can be applied to give a new method of computing double cosets. Previous methods for double cosets were enumerative and thus restricted to finite examples. Our rewriting methods do not suffer this restriction and we present some examples of infinite double coset systems which can now easily be solved using our approach. Even when both enumerative and rewriting techniques are present, our rewriting methods will be competitive because they i) do not require the preliminary calculation of cosets; and ii) as with single coset problems, there are many examples for which rewriting is more effective than enumeration. Automata provide the means for identifying expressions for normal forms in infinite situations and we show how they may be constructed in this setting. Further, related results on logged string rewriting for monoid presentations are exploited to show how witnesses for the computations can be provided and how information about the subgroups and the relations between them can be extracted. Finally, we discuss how the double coset problem is a special case of the problem of computing induced actions of categories which demonstrates that our rewriting methods are applicable to a much wider class of problems than just the double coset problem.Comment: accepted for publication by the Journal of Symbolic Computatio

Using Natural Language as Knowledge Representation in an Intelligent Tutoring System

Knowledge used in an intelligent tutoring system to teach students is usually acquired from authors who are experts in the domain. A problem is that they cannot directly add and update knowledge if they don’t learn formal language used in the system. Using natural language to represent knowledge can allow authors to update knowledge easily. This thesis presents a new approach to use unconstrained natural language as knowledge representation for a physics tutoring system so that non-programmers can add knowledge without learning a new knowledge representation. This approach allows domain experts to add not only problem statements, but also background knowledge such as commonsense and domain knowledge including principles in natural language. Rather than translating into a formal language, natural language representation is directly used in inference so that domain experts can understand the internal process, detect knowledge bugs, and revise the knowledgebase easily. In authoring task studies with the new system based on this approach, it was shown that the size of added knowledge was small enough for a domain expert to add, and converged to near zero as more problems were added in one mental model test. After entering the no-new-knowledge state in the test, 5 out of 13 problems (38 percent) were automatically solved by the system without adding new knowledge

A closed formula for the number of convex permutominoes

In this paper we determine a closed formula for the number of convex permutominoes of size n. We reach this goal by providing a recursive generation of all convex permutominoes of size n+1 from the objects of size n, according to the ECO method, and then translating this construction into a system of functional equations satisfied by the generating function of convex permutominoes. As a consequence we easily obtain also the enumeration of some classes of convex polyominoes, including stack and directed convex permutominoes