Algorithms for learning regular expressions from positive data

We describe algorithms that directly infer very simple forms of 1-unambiguous regular expressions from positive data. Thus, we characterize the regular language classes that can be learned this way, both in terms of regular expressions and in terms of (not necessarily minimal) deterministic finite automata.     

Valuations,regular expressions,and fractal geometry

Valuations — morphisms from ( _n ^* ,·,) to ((0, ), ·, 1) — are a simple generalization of so-called Bernoulli morphisms. In this paper, we show a characterization of strongly unambiguous regular expressions with the help of valuations and formal power series. We apply these algebraic results to the determination of Hausdorff dimensions of fractals described by regular expressions.     

A refined search tree technique for Dominating Set on planar graphs

We establish a refined search tree technique for the parameterized DOMINATING SET problem on planar graphs. Here, we are given an undirected graph and we ask for a set of at most k vertices such that every other vertex has at least one neighbor in this set. We describe algorithms with running times O(8^kn) and O(8^kk+n³), where n is the number of vertices in the graph, based on bounded search trees. We describe a set of polynomial time data-reduction rules for a more general “annotated” problem on black/white graphs that asks for a set of k vertices (black or white) that dominate all the black vertices. An intricate argument based on the Euler formula then establishes an efficient branching strategy for reduced inputs to this problem. In addition, we give a family examples showing that the bound of the branching theorem is optimal with respect to our reduction rules. Our final search tree algorithm is easy to implement; its analysis, however, is involved.     

Iterated Function Systems and Control Languages

Valuations—morphisms from (Σ*, ·, e) to ((0, ∞), ·, 1)—are a generalization of Bernoulli morphisms introduced by Eilenberg [“Automata, Languages, and Machines”, Academic Press, New York, 1974]. Here, we show how to generalize the notion of entropy (of a language) in order to obtain new formulas to determine the Hausdorff dimension of fractal sets (also in Euclidean spaces), especially defined via regular (ω-)languages. By doing this, we can sharpen and generalize earlier results in two ways: first, we treat the case where the underlying basic iterated function system contains noncontractive mappings and, second, we obtain results valid for nonregular languages as well.     

Computational completeness of simple semi-conditional insertion–deletion systems of degree (2,1)

Natural Computing - Insertion–deletion (or ins–del for short) systems are simple models of bio-inspired computing. They are well studied in formal language theory, especially regarding...     

Parallel communicating grammar systems with terminal transmission

We introduce a new variant of PC grammar systems, called PC grammar systems with terminal transmission, PCGSTT for short. We show that right-linear centralized PCGSTT have nice formal language theoretic properties: they are closed under gsm mappings (in particular, under intersection with regular sets and under homomorphisms) and union; a slight variant is, in addition, closed under concatenation and star; their power lies between that of n-parallel grammars introduced by Wood and that of matrix languages of index n, and their relation to equal matrix grammars of degree n is discussed. We show that membership for these language classes is complete for NL. In a second part of the paper, we discuss questions concerning grammatical inference of these systems. More precisely, we show that PCGSTT whose component grammars are terminal distinguishable right-linear, a notion introduced by Radhakrishnan and Nagaraja in [33,34], are identifiable in the limit if certain data communication information is supplied in addition.     

Identification of function distinguishable languages

We show how appropriately chosen functions f which we call distinguishing can be used to make deterministic finite automata backward deterministic. This idea can be exploited to design regular language classes called f-distinguishable which are identifiable in the limit from positive samples. Special cases of this approach are the k-reversible and terminal distinguishable languages, as discussed in Angluin (J. Assoc. Comput. Mach. 29 (3) (1982) 741), Fernau (Technical Report WSI-99-23, Universität Tübingen (Germany), Wilhelm-Schickard-Institut für Informatik, 1999, Short version published in the proceedings of AMAI 2000, see \tt http://rutcor.rutgers.edu/˜amai/aimath00/AcceptedCont.htm, Proc. 15th Internat. Conf. on Pattern Recognition (ICPR 2000), Vol. 2, IEEE Press, New York, 2000, pp. 125–128), Radhakrishnan (Ph.D. Thesis, Department of Computer Science and Engineering, Indian Institute of Technology, Bombay, India, 1987), Radhakrishnan and Nagaraja (IEEE Trans. Systems, Man Cybernet. 17 (6) (1987) 982). Moreover, we show that all regular languages may be approximated in the setting introduced by Kobayashi and Yokomori (in: K. P. Jantke, T. Shinohara, Th. Zeugmann (Eds.), Proc. Sixth Internat. Conf. Algorithmic Learning Theory (ALT’95), Lecture Notes in Computer Science/Lecture Notes in Artificial Intelligence, Vol. 997, Springer, Berlin, 1995, pp. 298–312), (Theoret. Comput. Sci. 174 (1997) 251–257) by any class of f-distinguishable languages. Observe that the class of all function-distinguishable languages is equal to the class of regular languages.     

Fixed Parameter Algorithms for DOMINATING SET and Related Problems on Planar Graphs

Abstract. We present an algorithm that constructively produces a solution to the k -DOMINATING SET problem for planar graphs in time O(c^ \sqrt k n) , where c=4^ 6\sqrt 34 . To obtain this result, we show that the treewidth of a planar graph with domination number γ (G) is O(\sqrt \rule 0pt 4pt \smash γ (G) ) , and that such a tree decomposition can be found in O(\sqrt \rule 0pt 4pt \smash γ (G) n) time. The same technique can be used to show that the k -FACE COVER problem (find a size k set of faces that cover all vertices of a given plane graph) can be solved in O(c ₁ ^ \sqrt k n) time, where c ₁ =3^ 36\sqrt 34 and k is the size of the face cover set. Similar results can be obtained in the planar case for some variants of k -DOMINATING SET, e.g., k -INDEPENDENT DOMINATING SET and k -WEIGHTED DOMINATING SET.