Parsing Linear Context-Free Rewriting Systems with Fast Matrix Multiplication

We describe a matrix multiplication recognition algorithm for a subset of binary linear context-free rewriting systems (LCFRS) with running time $O(n^{\omega d})$ where $M(m) = O(m^{\omega})$ is the running time for $m \times m$ matrix multiplication and $d$ is the "contact rank" of the LCFRS -- the maximal number of combination and non-combination points that appear in the grammar rules. We also show that this algorithm can be used as a subroutine to get a recognition algorithm for general binary LCFRS with running time $O(n^{\omega d + 1})$. The currently best known $\omega$ is smaller than $2.38$. Our result provides another proof for the best known result for parsing mildly context sensitive formalisms such as combinatory categorial grammars, head grammars, linear indexed grammars, and tree adjoining grammars, which can be parsed in time $O(n^{4.76})$. It also shows that inversion transduction grammars can be parsed in time $O(n^{5.76})$. In addition, binary LCFRS subsumes many other formalisms and types of grammars, for some of which we also improve the asymptotic complexity of parsing

Parsing Unary Boolean Grammars Using Online Convolution

In contrast to context-free grammars, the extension of these grammars by explicit conjunction, the so-called conjunctive grammars can generate (quite complicated) non-regular languages over a single-letter alphabet (DLT 2007). Given these expressibility results, we study the parsability of Boolean grammars, an extension of context-free grammars by conjunction and negation, over a unary alphabet and show that they can be parsed in time O(|G| log^2(n) M(n)) where M(n) is the time to multiply two n-bit integers. This multiplication algorithm is transformed into a convolution algorithm which in turn is converted to an online convolution algorithm which is used for the parsing

Clique-Based Lower Bounds for Parsing Tree-Adjoining Grammars

up to lower order factors

Rewriting a Deep Generative Model

A deep generative model such as a GAN learns to model a rich set of semantic and physical rules about the target distribution, but up to now, it has been obscure how such rules are encoded in the network, or how a rule could be changed. In this paper, we introduce a new problem setting: manipulation of specific rules encoded by a deep generative model. To address the problem, we propose a formulation in which the desired rule is changed by manipulating a layer of a deep network as a linear associative memory. We derive an algorithm for modifying one entry of the associative memory, and we demonstrate that several interesting structural rules can be located and modified within the layers of state-of-the-art generative models. We present a user interface to enable users to interactively change the rules of a generative model to achieve desired effects, and we show several proof-of-concept applications. Finally, results on multiple datasets demonstrate the advantage of our method against standard fine-tuning methods and edit transfer algorithms.Comment: ECCV 2020 (oral). Code at https://github.com/davidbau/rewriting. For videos and demos see https://rewriting.csail.mit.edu

An Efficient Probabilistic Context-Free Parsing Algorithm that Computes Prefix Probabilities

We describe an extension of Earley's parser for stochastic context-free grammars that computes the following quantities given a stochastic context-free grammar and an input string: a) probabilities of successive prefixes being generated by the grammar; b) probabilities of substrings being generated by the nonterminals, including the entire string being generated by the grammar; c) most likely (Viterbi) parse of the string; d) posterior expected number of applications of each grammar production, as required for reestimating rule probabilities. (a) and (b) are computed incrementally in a single left-to-right pass over the input. Our algorithm compares favorably to standard bottom-up parsing methods for SCFGs in that it works efficiently on sparse grammars by making use of Earley's top-down control structure. It can process any context-free rule format without conversion to some normal form, and combines computations for (a) through (d) in a single algorithm. Finally, the algorithm has simple extensions for processing partially bracketed inputs, and for finding partial parses and their likelihoods on ungrammatical inputs.Comment: 45 pages. Slightly shortened version to appear in Computational Linguistics 2

Multiple context-free path querying by matrix multiplication

Many graph analysis problems can be formulated as formal language-constrained path querying problems where the formal languages are used as constraints for navigational path queries. Recently, the context-free language (CFL) reachability formulation has become very popular and can be used in many areas, for example, querying graph databases, Resource Description Framework (RDF) analysis. However, the generative capacity of context-free grammars (CFGs) is too weak to generate some complex queries, for example, from natural languages, and the various extensions of CFGs have been proposed. Multiple context-free grammar (MCFG) is one of such extensions of CFGs. Despite the fact that, to the best of our knowledge, there is no algorithm for MCFL-reachability, this problem is known to be decidable. This paper is devoted to developing the first such algorithm for the MCFL-reachability problem. The essence of the proposed algorithm is to use a set of Boolean matrices and operations on them to find paths in a graph that satisfy the given constraints. The main operation here is Boolean matrix multiplication. As a result, the algorithm returns a set of matrices containing all information needed to solve the MCFL-reachability problem. The presented algorithm is implemented in Python using GraphBLAS API. An analysis of real RDF data and synthetic graphs for some MCFLs is performed. The study showed that using a sparse format for matrix storage and parallel computing for graphs with tens of thousands of edges the analysis time can be 10–20 minutes. The result of the analysis provides tens of millions of reachable vertex pairs. The proposed algorithm can be applied in problems of static code analysis, bioinformatics, network analysis, as well as in graph databases when a path query cannot be expressed using context-free grammars. The provided algorithm is linear algebra-based, hence, it allows one to use high-performance libraries and utilize modern parallel hardware

Survey on Instruction Selection: An Extensive and Modern Literature Review

Instruction selection is one of three optimisation problems involved in the code generator backend of a compiler. The instruction selector is responsible of transforming an input program from its target-independent representation into a target-specific form by making best use of the available machine instructions. Hence instruction selection is a crucial part of efficient code generation. Despite on-going research since the late 1960s, the last, comprehensive survey on the field was written more than 30 years ago. As new approaches and techniques have appeared since its publication, this brings forth a need for a new, up-to-date review of the current body of literature. This report addresses that need by performing an extensive review and categorisation of existing research. The report therefore supersedes and extends the previous surveys, and also attempts to identify where future research should be directed.Comment: Major changes: - Merged simulation chapter with macro expansion chapter - Addressed misunderstandings of several approaches - Completely rewrote many parts of the chapters; strengthened the discussion of many approaches - Revised the drawing of all trees and graphs to put the root at the top instead of at the bottom - Added appendix for listing the approaches in a table See doc for more inf

Improving the Representation and Conversion of Mathematical Formulae by Considering their Textual Context

Mathematical formulae represent complex semantic information in a concise form. Especially in Science, Technology, Engineering, and Mathematics, mathematical formulae are crucial to communicate information, e.g., in scientific papers, and to perform computations using computer algebra systems. Enabling computers to access the information encoded in mathematical formulae requires machine-readable formats that can represent both the presentation and content, i.e., the semantics, of formulae. Exchanging such information between systems additionally requires conversion methods for mathematical representation formats. We analyze how the semantic enrichment of formulae improves the format conversion process and show that considering the textual context of formulae reduces the error rate of such conversions. Our main contributions are: (1) providing an openly available benchmark dataset for the mathematical format conversion task consisting of a newly created test collection, an extensive, manually curated gold standard and task-specific evaluation metrics; (2) performing a quantitative evaluation of state-of-the-art tools for mathematical format conversions; (3) presenting a new approach that considers the textual context of formulae to reduce the error rate for mathematical format conversions. Our benchmark dataset facilitates future research on mathematical format conversions as well as research on many problems in mathematical information retrieval. Because we annotated and linked all components of formulae, e.g., identifiers, operators and other entities, to Wikidata entries, the gold standard can, for instance, be used to train methods for formula concept discovery and recognition. Such methods can then be applied to improve mathematical information retrieval systems, e.g., for semantic formula search, recommendation of mathematical content, or detection of mathematical plagiarism.Comment: 10 pages, 4 figure