The Surprising Computational Power of Nondeterministic Stack RNNs

Traditional recurrent neural networks (RNNs) have a fixed, finite number of memory cells. In theory (assuming bounded range and precision), this limits their formal language recognition power to regular languages, and in practice, RNNs have been shown to be unable to learn many context-free languages (CFLs). In order to expand the class of languages RNNs recognize, prior work has augmented RNNs with a nondeterministic stack data structure, putting them on par with pushdown automata and increasing their language recognition power to CFLs. Nondeterminism is needed for recognizing all CFLs (not just deterministic CFLs), but in this paper, we show that nondeterminism and the neural controller interact to produce two more unexpected abilities. First, the nondeterministic stack RNN can recognize not only CFLs, but also many non-context-free languages. Second, it can recognize languages with much larger alphabet sizes than one might expect given the size of its stack alphabet. Finally, to increase the information capacity in the stack and allow it to solve more complicated tasks with large alphabet sizes, we propose a new version of the nondeterministic stack that simulates stacks of vectors rather than discrete symbols. We demonstrate perplexity improvements with this new model on the Penn Treebank language modeling benchmark.Comment: 20 pages, 7 figures. Submitted to ICLR 202

Streaming algorithms for language recognition problems

We study the complexity of the following problems in the streaming model. Membership testing for \DLIN We show that every language in \DLIN\ can be recognised by a randomized one-pass $O(\log n)$ space algorithm with inverse polynomial one-sided error, and by a deterministic p-pass $O(n/p)$ space algorithm. We show that these algorithms are optimal. Membership testing for \LL$(k)$ For languages generated by \LL$(k)$ grammars with a bound of $r$ on the number of nonterminals at any stage in the left-most derivation, we show that membership can be tested by a randomized one-pass $O(r\log n)$ space algorithm with inverse polynomial (in $n$) one-sided error. Membership testing for \DCFL We show that randomized algorithms as efficient as the ones described above for \DLIN\ and \LL(k) (which are subclasses of \DCFL) cannot exist for all of \DCFL: there is a language in \VPL\ (a subclass of \DCFL) for which any randomized p-pass algorithm with error bounded by $\epsilon < 1/2$ must use $\Omega(n/p)$ space. Degree sequence problem We study the problem of determining, given a sequence $d_1, d_2,..., d_n$ and a graph $G$, whether the degree sequence of $G$ is precisely $d_1, d_2,..., d_n$. We give a randomized one-pass $O(\log n)$ space algorithm with inverse polynomial one-sided error probability. We show that our algorithms are optimal. Our randomized algorithms are based on the recent work of Magniez et al. \cite{MMN09}; our lower bounds are obtained by considering related communication complexity problems

Maintaining regularity and generalization in data using the minimum description length principle and genetic algorithm: case of grammatical inference

In this paper, a genetic algorithm with minimum description length (GAWMDL) is proposed for grammatical inference. The primary challenge of identifying a language of infinite cardinality from a finite set of examples should know when to generalize and specialize the training data. The minimum description length principle that has been incorporated addresses this issue is discussed in this paper. Previously, the e-GRIDS learning model was proposed, which enjoyed the merits of the minimum description length principle, but it is limited to positive examples only. The proposed GAWMDL, which incorporates a traditional genetic algorithm and has a powerful global exploration capability that can exploit an optimum offspring. This is an effective approach to handle a problem which has a large search space such the grammatical inference problem. The computational capability, the genetic algorithm poses is not questionable, but it still suffers from premature convergence mainly arising due to lack of population diversity. The proposed GAWMDL incorporates a bit mask oriented data structure that performs the reproduction operations, creating the mask, then Boolean based procedure is applied to create an offspring in a generative manner. The Boolean based procedure is capable of introducing diversity into the population, hence alleviating premature convergence. The proposed GAWMDL is applied in the context free as well as regular languages of varying complexities. The computational experiments show that the GAWMDL finds an optimal or close-to-optimal grammar. Two fold performance analysis have been performed. First, the GAWMDL has been evaluated against the elite mating pool genetic algorithm which was proposed to introduce diversity and to address premature convergence. GAWMDL is also tested against the improved tabular representation algorithm. In addition, the authors evaluate the performance of the GAWMDL against a genetic algorithm not using the minimum description length principle. Statistical tests demonstrate the superiority of the proposed algorithm. Overall, the proposed GAWMDL algorithm greatly improves the performance in three main aspects: maintains regularity of the data, alleviates premature convergence and is capable in grammatical inference from both positive and negative corpora

Maintaining regularity and generalization in data using the minimum description length principle and genetic algorithm: Case of grammatical inference

In this paper, a genetic algorithm with minimum description length (GAWMDL) is proposed for grammatical inference. The primary challenge of identifying a language of infinite cardinality from a finite set of examples should know when to generalize and specialize the training data. The minimum description length principle that has been incorporated addresses this issue is discussed in this paper. Previously, the e-GRIDS learning model was proposed, which enjoyed the merits of the minimum description length principle, but it is limited to positive examples only. The proposed GAWMDL, which incorporates a traditional genetic algorithm and has a powerful global exploration capability that can exploit an optimum offspring. This is an effective approach to handle a problem which has a large search space such the grammatical inference problem. The computational capability, the genetic algorithm poses is not questionable, but it still suffers from premature convergence mainly arising due to lack of population diversity. The proposed GAWMDL incorporates a bit mask oriented data structure that performs the reproduction operations, creating the mask, then Boolean based procedure is applied to create an offspring in a generative manner. The Boolean based procedure is capable of introducing diversity into the population, hence alleviating premature convergence. The proposed GAWMDL is applied in the context free as well as regular languages of varying complexities. The computational experiments show that the GAWMDL finds an optimal or close-to-optimal grammar. Two fold performance analysis have been performed. First, the GAWMDL has been evaluated against the elite mating pool genetic algorithm which was proposed to introduce diversity and to address premature convergence. GAWMDL is also tested against the improved tabular representation algorithm. In addition, the authors evaluate the performance of the GAWMDL against a genetic algorithm not using the minimum description length principle. Statistical tests demonstrate the superiority of the proposed algorithm. Overall, the proposed GAWMDL algorithm greatly improves the performance in three main aspects: maintains regularity of the data, alleviates premature convergence and is capable in grammatical inference from both positive and negative corpora

Maintaining regularity and generalization in data using the minimum description length principle and genetic algorithm: case of grammatical inference

In this paper, a genetic algorithm with minimum description length (GAWMDL) is proposed for grammatical inference. The primary challenge of identifying a language of infinite cardinality from a finite set of examples should know when to generalize and specialize the training data. The minimum description length principle that has been incorporated addresses this issue is discussed in this paper. Previously, the e-GRIDS learning model was proposed, which enjoyed the merits of the minimum description length principle, but it is limited to positive examples only. The proposed GAWMDL, which incorporates a traditional genetic algorithm and has a powerful global exploration capability that can exploit an optimum offspring. This is an effective approach to handle a problem which has a large search space such the grammatical inference problem. The computational capability, the genetic algorithm poses is not questionable, but it still suffers from premature convergence mainly arising due to lack of population diversity. The proposed GAWMDL incorporates a bit mask oriented data structure that performs the reproduction operations, creating the mask, then Boolean based procedure is applied to create an offspring in a generative manner. The Boolean based procedure is capable of introducing diversity into the population, hence alleviating premature convergence. The proposed GAWMDL is applied in the context free as well as regular languages of varying complexities. The computational experiments show that the GAWMDL finds an optimal or close-to-optimal grammar. Two fold performance analysis have been performed. First, the GAWMDL has been evaluated against the elite mating pool genetic algorithm which was proposed to introduce diversity and to address premature convergence. GAWMDL is also tested against the improved tabular representation algorithm. In addition, the authors evaluate the performance of the GAWMDL against a genetic algorithm not using the minimum description length principle. Statistical tests demonstrate the superiority of the proposed algorithm. Overall, the proposed GAWMDL algorithm greatly improves the performance in three main aspects: maintains regularity of the data, alleviates premature convergence and is capable in grammatical inference from both positive and negative corpora

Formal models of the extension activity of DNA polymerase enzymes

The study of formal language operations inspired by enzymatic actions on DNA is part of ongoing efforts to provide a formal framework and rigorous treatment of DNA-based information and DNA-based computation. Other studies along these lines include theoretical explorations of splicing systems, insertion-deletion systems, substitution, hairpin extension, hairpin reduction, superposition, overlapping concatenation, conditional concatenation, contextual intra- and intermolecular recombinations, as well as template-guided recombination. First, a formal language operation is proposed and investigated, inspired by the naturally occurring phenomenon of DNA primer extension by a DNA-template-directed DNA polymerase enzyme. Given two DNA strings u and v, where the shorter string v (called the primer) is Watson-Crick complementary and can thus bind to a substring of the longer string u (called the template) the result of the primer extension is a DNA string that is complementary to a suffix of the template which starts at the binding position of the primer. The operation of DNA primer extension can be abstracted as a binary operation on two formal languages: a template language L1 and a primer language L2. This language operation is called L1-directed extension of L2 and the closure properties of various language classes, including the classes in the Chomsky hierarchy, are studied under directed extension. Furthermore, the question of finding necessary and sufficient conditions for a given language of target strings to be generated from a given template language when the primer language is unknown is answered. The canonic inverse of directed extension is used in order to obtain the optimal solution (the minimal primer language) to this question. The second research project investigates properties of the binary string and language operation overlap assembly as defined by Csuhaj-Varju, Petre and Vaszil as a formal model of the linear self-assembly of DNA strands: The overlap assembly of two strings, xy and yz, which share an overlap y, results in the string xyz. In this context, we investigate overlap assembly and its properties: closure properties of various language families under this operation, and related decision problems. A theoretical analysis of the possible use of iterated overlap assembly to generate combinatorial DNA libraries is also given. The third research project continues the exploration of the properties of the overlap assembly operation by investigating closure properties of various language classes under iterated overlap assembly, and the decidability of the completeness of a language. The problem of deciding whether a given string is terminal with respect to a language, and the problem of deciding if a given language can be generated by an overlap assembly operation of two other given languages are also investigated

If the Current Clique Algorithms are Optimal, so is Valiant's Parser

The CFG recognition problem is: given a context-free grammar $\mathcal{G}$ and a string $w$ of length $n$, decide if $w$ can be obtained from $\mathcal{G}$. This is the most basic parsing question and is a core computer science problem. Valiant's parser from 1975 solves the problem in $O(n^{\omega})$ time, where $\omega<2.373$ is the matrix multiplication exponent. Dozens of parsing algorithms have been proposed over the years, yet Valiant's upper bound remains unbeaten. The best combinatorial algorithms have mildly subcubic $O(n^3/\log^3{n})$ complexity. Lee (JACM'01) provided evidence that fast matrix multiplication is needed for CFG parsing, and that very efficient and practical algorithms might be hard or even impossible to obtain. Lee showed that any algorithm for a more general parsing problem with running time $O(|\mathcal{G}|\cdot n^{3-\varepsilon})$ can be converted into a surprising subcubic algorithm for Boolean Matrix Multiplication. Unfortunately, Lee's hardness result required that the grammar size be $|\mathcal{G}|=\Omega(n^6)$. Nothing was known for the more relevant case of constant size grammars. In this work, we prove that any improvement on Valiant's algorithm, even for constant size grammars, either in terms of runtime or by avoiding the inefficiencies of fast matrix multiplication, would imply a breakthrough algorithm for the $k$-Clique problem: given a graph on $n$ nodes, decide if there are $k$ that form a clique. Besides classifying the complexity of a fundamental problem, our reduction has led us to similar lower bounds for more modern and well-studied cubic time problems for which faster algorithms are highly desirable in practice: RNA Folding, a central problem in computational biology, and Dyck Language Edit Distance, answering an open question of Saha (FOCS'14)