There Exist some Omega-Powers of Any Borel Rank

Omega-powers of finitary languages are languages of infinite words (omega-languages) in the form V^omega, where V is a finitary language over a finite alphabet X. They appear very naturally in the characterizaton of regular or context-free omega-languages. Since the set of infinite words over a finite alphabet X can be equipped with the usual Cantor topology, the question of the topological complexity of omega-powers of finitary languages naturally arises and has been posed by Niwinski (1990), Simonnet (1992) and Staiger (1997). It has been recently proved that for each integer n > 0, there exist some omega-powers of context free languages which are Pi^0_n-complete Borel sets, that there exists a context free language L such that L^omega is analytic but not Borel, and that there exists a finitary language V such that V^omega is a Borel set of infinite rank. But it was still unknown which could be the possible infinite Borel ranks of omega-powers. We fill this gap here, proving the following very surprising result which shows that omega-powers exhibit a great topological complexity: for each non-null countable ordinal alpha, there exist some Sigma^0_alpha-complete omega-powers, and some Pi^0_alpha-complete omega-powers.Comment: To appear in the Proceedings of the 16th EACSL Annual Conference on Computer Science and Logic, CSL 2007, Lausanne, Switzerland, September 11-15, 2007, Lecture Notes in Computer Science, (c) Springer, 200

Finite Automata for the Sub- and Superword Closure of CFLs: Descriptional and Computational Complexity

We answer two open questions by (Gruber, Holzer, Kutrib, 2009) on the state-complexity of representing sub- or superword closures of context-free grammars (CFGs): (1) We prove a (tight) upper bound of $2^{\mathcal{O}(n)}$ on the size of nondeterministic finite automata (NFAs) representing the subword closure of a CFG of size $n$. (2) We present a family of CFGs for which the minimal deterministic finite automata representing their subword closure matches the upper-bound of $2^{2^{\mathcal{O}(n)}}$ following from (1). Furthermore, we prove that the inequivalence problem for NFAs representing sub- or superword-closed languages is only NP-complete as opposed to PSPACE-complete for general NFAs. Finally, we extend our results into an approximation method to attack inequivalence problems for CFGs

Adaptive Intelligent Tutoring System for learning Computer Theory

In this paper, we present an intelligent tutoring system developed to help students in learning Computer Theory. The Intelligent tutoring system was built using ITSB authoring tool. The system helps students to learn finite automata, pushdown automata, Turing machines and examines the relationship between these automata and formal languages, deterministic and nondeterministic machines, regular expressions, context free grammars, undecidability, and complexity. During the process the intelligent tutoring system gives assistance and feedback of many types in an intelligent manner according to the behavior of the student. An evaluation of the intelligent tutoring system has revealed reasonably acceptable results in terms of its usability and learning abilities are concerned

Descriptional Complexity of Finite Automata -- Selected Highlights

The state complexity, respectively, nondeterministic state complexity of a regular language $L$ is the number of states of the minimal deterministic, respectively, of a minimal nondeterministic finite automaton for $L$. Some of the most studied state complexity questions deal with size comparisons of nondeterministic finite automata of differing degree of ambiguity. More generally, if for a regular language we compare the size of description by a finite automaton and by a more powerful language definition mechanism, such as a context-free grammar, we encounter non-recursive trade-offs. Operational state complexity studies the state complexity of the language resulting from a regularity preserving operation as a function of the complexity of the argument languages. Determining the state complexity of combined operations is generally challenging and for general combinations of operations that include intersection and marked concatenation it is uncomputable