Two Extensions of Cover Automata

Deterministic Finite Cover Automata (DFCA) are compact representations of finite languages. Deterministic Finite Automata with “do not care” symbols and Multiple Entry Deterministic Finite Automata are both compact representations of regular languages. This paper studies the benefits of combining these representations to get even more compact representations of finite languages. DFCAs are extended by accepting either “do not care” symbols or considering multiple entry DFCAs. We study for each of the two models the existence of the minimization or simplification algorithms and their computational complexity, the state complexity of these representations compared with other representations of the same language, and the bounds for state complexity in case we perform a representation transformation. Minimization for both models proves to be NP-hard. A method is presented to transform minimization algorithms for deterministic automata into simplification algorithms applicable to these extended models. DFCAs with “do not care” symbols prove to have comparable state complexity as Nondeterministic Finite Cover Automata. Furthermore, for multiple entry DFCAs, we can have a tight estimate of the state complexity of the transformation into equivalent DFCA.

Preservice Mathematics Teachers' Representation Transformation Competence Levels in the Process of Solving Limit Problems

In this study, representations used by preservice mathematics teachers in the process of solving limit problems were determined, the inter-representation transformation competence levels were investigated and the relationship between them was examined. In this context, “Limit Representation Transformation Test” with a reliability of .908 was administered to 50 preservice teachers attending to a state university in the Central of Turkey. Preservice teachers had most difficulty in solving problems that had verbal representation inputs, especially they achieved low performances in transformation from verbal to numerical representation. Although, in general, they achieved the highest performance in the problem that had numerical representation input, they also achieved very high performances in the problems that had graphical and algebraic representation inputs. Specifically, they performed very well in the problems that required transformation from an algebraic representation to a verbal representation. Moreover, significant positive correlations were found among preservice teachers’ representation transformation competence levels.