Decomposition and Descriptional Complexity of Shuffle on Words and Finite Languages

We investigate various questions related to the shuffle operation on words and finite languages. First we investigate a special variant of the shuffle decomposition problem for regular languages, namely, when the given regular language is the shuffle of finite languages. The shuffle decomposition into finite languages is, in general not unique. Thatis,therearelanguagesL^,L2,L3,L4withLiluL2= £3luT4but{L\,L2}^ {I/3, L4}. However, if all four languages are singletons (with at least two combined letters), it follows by a result of Berstel and Boasson [6], that the solution is unique; that is {L\,L2} = {L3,L4}. We extend this result to show that if L\ and L2 are arbitrary finite sets and Lz and Z-4 are singletons (with at least two letters in each), the solution is unique. This is as strong as it can be, since we provide examples showing that the solution can be non-unique already when (1) both L\ and L2 are singleton sets over different unary alphabets; or (2) L\ contains two words and L2 is singleton. We furthermore investigate the size of shuffle automata for words. It was shown by Campeanu, K. Salomaa and Yu in [11] that the minimal shuffle automaton of two regular languages requires 2mn states in the worst case (where the minimal automata of the two component languages had m and n states, respectively). It was also recently shown that there exist words u and v such that the minimal shuffle iii DFA for u and v requires an exponential number of states. We study the size of shuffle DFAs for restricted cases of words, namely when the words u and v are both periods of a common underlying word. We show that, when the underlying word obeys certain conditions, then the size of the minimal shuffle DFA for u and v is at most quadratic. Moreover we provide an efficient algorithm, which decides for a given DFA A and two words u and v, whether u lu u C L(A)

An infinite hierarchy induced by depth synchronization

AbstractDepth-synchronization measures the number of parallel derivation steps in a synchronized context-free (SCF) grammar. When not bounded by a constant the depth-synchronization measure of an SCF grammar is at least logarithmic and at most linear with respect to the word length. Languages with linear depth-synchronization measure and languages with a depth-synchronization measure in between logarithmic and linear are proven to exist. This gives rise to a strict infinite hierarchy within the family of SCF (and ET0L) languages

Algorithmic Decomposition of Shuffle on Words

We investigate shuffle-decomposability into two words. We give an algorithm which takes as input a DFA M (under certain conditions) and determines the unique candidate decomposition into words u and v such that L(M) = u v ifM is shuffle decomposable, in time O(|u| + |v|). Even though this algorithm does not determine whether or not the DFA is shuffle decomposable, the sublinear time complexity of only determining the two words under the assumption of decomposability is surprising given the complexity of shuffle, and demonstrates an interesting property of the operation. We also show that for given words u and v and a DFA M we can determine whether u v ⊆ L(M) in polynomial time

Automating the search for a patent's prior art with a full text similarity search

More than ever, technical inventions are the symbol of our society's advance. Patents guarantee their creators protection against infringement. For an invention being patentable, its novelty and inventiveness have to be assessed. Therefore, a search for published work that describes similar inventions to a given patent application needs to be performed. Currently, this so-called search for prior art is executed with semi-automatically composed keyword queries, which is not only time consuming, but also prone to errors. In particular, errors may systematically arise by the fact that different keywords for the same technical concepts may exist across disciplines. In this paper, a novel approach is proposed, where the full text of a given patent application is compared to existing patents using machine learning and natural language processing techniques to automatically detect inventions that are similar to the one described in the submitted document. Various state-of-the-art approaches for feature extraction and document comparison are evaluated. In addition to that, the quality of the current search process is assessed based on ratings of a domain expert. The evaluation results show that our automated approach, besides accelerating the search process, also improves the search results for prior art with respect to their quality

On the Shuffle Automaton Size for Words

We investigate the state size of DFAs accepting the shuffle of two words. We provide words u and v, such that the minimal DFA for u shuffled with v requires an exponential number of states. We also show some conditions for the words u and v which ensure a quadratic upper bound on the state size of u shuffled with v. Moreover, switching only two letters within one of u or v is enough to trigger the change from quadratic to exponential

Graph-Controlled Insertion-Deletion Systems

In this article, we consider the operations of insertion and deletion working in a graph-controlled manner. We show that like in the case of context-free productions, the computational power is strictly increased when using a control graph: computational completeness can be obtained by systems with insertion or deletion rules involving at most two symbols in a contextual or in a context-free manner and with the control graph having only four nodes.Comment: In Proceedings DCFS 2010, arXiv:1008.127