J. Joao Almeida, Eliana Grande, Georgi Smirnov
In this paper we consider the problem of context-free grammars comparison from the analysis point of view. We show that the problem can be reduced to numerical solution of systems of nonlinear matrix equations. The approach presented here forms a basis for probabilistic comparison algorithms oriented to automatic assessment of of student's answers in computer science.
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The L(PSCG)=L(CS) problem asks whether propagating scattered context grammars and context sensitive grammars are equivalent. The presented paper reformulates and answers this problem in terms of CD grammar systems. More specifically, it characterizes the family of context sensitive languages by two-component CD grammar systems with propagating scattered context rules.
In this paper, we prove the semidecidability of the problem of saying whether or not a context-free grammar generates a regular language. We introduce the notion of context-free grammar in Marciani Normal Form. We prove that a context-free grammar in Marciani Normal Form always generates a regular language.
Context-free S grammars are introduced, for arbitrary (storage) type S, as a uniform framework for recursion-based grammars, automata, and transducers, viewed as programs. To each occurrence of a nonterminal of a context-free S grammar an object of type S is associated, that can be acted upon by tests and operations, as indicated in the rules of the grammar. Taking particular storage types gives particular formalisms, such as indexed grammars, top-down tree transducers, attribute grammars, etc. Context-free S grammars are equivalent to pushdown S automata. The context-free S languages can be o…
Context-dependent fusion grammars were recently introduced as devices for the generation of hypergraph languages. In this paper, we show that this new type of hypergraph grammars, where the application of fusion rules is restricted by positive and negative context conditions, is a universal computation model. Our main result is that Turing machines can be transformed into these grammars such that the recognized language of the Turing machine and the generated language of the corresponding context-dependent fusion grammar coincide up to representation of strings as graphs. As a corollary we get…
Context-free grammar simplification is a subject of high importance in computer language processing technology as well as in formal language theory. This paper presents a formalization, using the Coq proof assistant, of the fact that general context-free grammars generate languages that can be also generated by simpler and equivalent context-free grammars. Namely, useless symbol elimination, inaccessible symbol elimination, unit rules elimination and empty rules elimination operations were described and proven correct with respect to the preservation of the language generated by the original g…
Ordered Context-Free Grammars Revisited
本文重新研究有序上下文无关文法，简化了字符串推导的定义，并提出基于共享压缩解析森林的 O(n^4) 时间解析算法。
Context-Free Grammars (CFGs) and Parsing Expression Grammars (PEGs) have several similarities and a few differences in both their syntax and semantics, but they are usually presented through formalisms that hinder a proper comparison. In this paper we present a new formalism for CFGs that highlights the similarities and differences between them. The new formalism borrows from PEGs the use of parsing expressions and the recognition-based semantics. We show how one way of removing non-determinism from this formalism yields a formalism with the semantics of PEGs. We also prove, based on these new…
In a recent paper (M. Barash, A. Okhotin, "Defining contexts in context-free grammars", LATA 2012), the authors introduced an extension of the context-free grammars equipped with an operator for referring to the left context of the substring being defined. This paper proposes a more general model, in which context specifications may be two-sided, that is, both the left and the right contexts can be specified by the corresponding operators. The paper gives the definitions and establishes the basic theory of such grammars, leading to a normal form and a parsing algorithm working in time O(n^4), …
Grammatical inference is a machine learning area, whose fundamentals are built around learning sets. At present, real-life data and examples from manually crafted grammars are used to test their learning performance. This paper aims to present a method of generating artificial context-free grammars with their optimal learning sets, which could be successfully applied as a benchmarking tool for empirical grammar inference methods.
In 1973, Greibach ("The hardest context-free language", SIAM J. Comp., 1973) constructed a context-free language $L_0$ with the property that every context-free language can be reduced to $L_0$ by a homomorphism, thus representing it as an inverse homomorphic image $h^{-1}(L_0)$. In this paper, a similar characterization is established for a family of grammars equipped with operators for referring to the left context of any substring, recently defined by Barash and Okhotin ("An extension of context-free grammars with one-sided context specifications", Inform. Comput., 2014). An essential step …
In this paper we prove several results on normal forms for linear displacement context-free grammars. The results themselves are rather simple and use well-known techniques, but they are extensively used in more complex constructions. Therefore this article mostly serves educational and referential purposes.
In this paper we associate permitting symbols with rules of Grammars in the components of cooperating distributed context free hexagonal array grammar systems as a control mechanism and investigating the generative power of the resulting systems in the terminal mode. This feature of associating permitting symbols with rules when extended to patterns in the form of connected arrays also requires checking of symbols, but this is simpler than usual pattern matching. The benefit of allowing permitting symbols is that it enables us to reduce the number of components required in a cooperating distri…
For any context-free grammar, we build a transition diagram, that is, a finite directed graph with labeled arcs, which describes the work of the grammar. This approach is new, and it is different from previously known graph models. We define the concept of proper walk in this transition diagram and we prove that a word belongs to a given context-free language if and only if this word can be obtained with the help of a proper walk.
We study the computational complexity of reachability, coverability and inclusion for extensions of context-free commutative grammars with integer counters and reset operations on them. Those grammars can alternatively be viewed as an extension of communication-free Petri nets. Our main results are that reachability and coverability are inter-reducible and both NP-complete. In particular, this class of commutative grammars enjoys semi-linear reachability sets. We also show that the inclusion problem is, in general, coNEXP-complete and already $Π_2^\text{P}$-complete for grammars with only one …
Context-free language theory is a well-established area of mathematics, relevant to computer science foundations and technology. This paper presents the preliminary results of an ongoing formalization project using context-free grammars and the Coq proof assistant. The results obtained so far include the representation of context-free grammars, the description of algorithms for some operations on them (union, concatenation and closure) and the proof of related theorems (e.g. the correctness of these algorithms). A brief survey of related works is presented, as well as plans for further develop…
The pumping lemma and Ogden lemma offer a powerful method to prove that a particular language is not context-free. In 2008 Kanazawa proved an analogue of pumping lemma for well-nested multiple-context free languages. However, the statement of lemma is too weak for practical usage. We prove a stronger variant of pumping lemma and an analogue of Ogden lemma for this language family. We also use these statements to prove that some natural context-sensitive languages cannot be generated by tree-adjoining grammars.
We study the computational complexity of universality and inclusion problems for unambiguous finite automata and context-free grammars. We observe that several such problems can be reduced to the universality problem for unambiguous context-free grammars. The latter problem has long been known to be decidable and we propose a PSPACE algorithm that works by reduction to the zeroness problem of recurrence equations with convolution. We are not aware of any non-trivial complexity lower bounds. However, we show that computing the coin-flip measure of an unambiguous context-free language, a quantit…
Parikh's Theorem states that every context-free grammar (CFG) is equivalent to some regular CFG when the ordering of symbols in the words is ignored. The same is not true for the so-called weighted CFGs, which additionally assign a weight to each grammar rule. If the result holds for a given weighted CFG $G$, we say that $G$ satisfies the Parikh property. We prove constructively that the Parikh property holds for every weighted nonexpansive CFG. We also give a decision procedure for the property when the weights are over the rationals.
A formal-linguistic approach for solving an entertaining task is made in this paper. The well-known task of the Hanoi towers is discussed in relation to some concepts of discrete mathematics. A context-free grammar which generate an algorithm for solving this task is described. A deterministic pushdown automation which in its work imitates the work of monks in solving the task of the Hanoi towers is built.
We consider commutative regular and context-free grammars, or, in other words, Parikh images of regular and context-free languages. By using linear algebra and a branching analog of the classic Euler theorem, we show that, under an assumption that the terminal alphabet is fixed, the membership problem for regular grammars (given v in binary and a regular commutative grammar G, does G generate v?) is P, and that the equivalence problem for context free grammars (do G_1 and G_2 generate the same language?) is in $\mathrm{Π_2^P}$.
The paper demonstrates the non-closure of the family of unambiguous linear languages (that is, those defined by unambiguous linear context-free grammars) under complementation. To be precise, a particular unambiguous linear grammar is presented, and it is proved that the complement of this language is not defined by any context-free grammar. This also constitutes an alternative proof for the result of Hibbard and Ullian ("The independence of inherent ambiguity from complementedness among context-free languages", J.ACM, 1966) on the non-closure of the unambiguous languages under complementation…
We consider the problem of learning an unknown context-free grammar when the only knowledge available and of interest to the learner is about its structural descriptions with depth at most $\ell.$ The goal is to learn a cover context-free grammar (CCFG) with respect to $\ell$, that is, a CFG whose structural descriptions with depth at most $\ell$ agree with those of the unknown CFG. We propose an algorithm, called $LA^\ell$, that efficiently learns a CCFG using two types of queries: structural equivalence and structural membership. We show that $LA^\ell$ runs in time polynomial in the number o…
We study the computational complexity of universality and inclusion problems for unambiguous finite automata and context-free grammars. We observe that several such problems can be reduced to the universality problem for unambiguous context-free grammars. The latter problem has long been known to be decidable and we propose a PSPACE algorithm that works by reduction to the zeroness problem of recurrence equations with convolution. We are not aware of any non-trivial complexity lower bounds. However, we show that computing the coin-flip measure of an unambiguous context-free language, a quantit…
On Quantum Context-Free Grammars
本文针对量子上下文无关文法（QCFGs）的合式性问题，在已有研究基础上提出了若干确保推导词概率不超1的充要条件（幺正条件），完善了量子文法领域的相关理论。
Recently, it has been shown that every recursively enumerable language can be generated by a scattered context grammar with no more than three nonterminals. However, in that construction, the maximal number of nonterminals simultaneously rewritten during a derivation step depends on many factors, such as the cardinality of the alphabet of the generated language and the structure of the generated language itself. This paper improves the result by showing that the maximal number of nonterminals simultaneously rewritten during any derivation step can be limited by a small constant regardless of o…
Forbidden-Context & Ordered Grammar Systems
本文研究将禁止随机上下文文法与有序文法结合到协作分布式文法系统中，探讨其生成能力，提出四种场景，解决部分开放性问题。
Probabilistic context-free grammars (PCFGs) are used to define distributions over strings, and are powerful modelling tools in a number of areas, including natural language processing, software engineering, model checking, bio-informatics, and pattern recognition. A common important question is that of comparing the distributions generated or modelled by these grammars: this is done through checking language equivalence and computing distances. Two PCFGs are language equivalent if every string has identical probability with both grammars. This also means that the distance (whichever norm is us…
This paper solves an open problem concerning the generative power of nonerasing context-free rewriting systems using a simple mechanism for checking for context dependencies, in the literature known as semi-conditional grammars of degree (1,1). In these grammars, two nonterminal symbols are attached to each context-free production, and such a production is applicable if one of the two attached symbols occurs in the current sentential form, while the other does not. Specifically, this paper demonstrates that the family of languages generated by semi-conditional grammars of degree (1,1) coincide…
Parsing Hypergraphs using Context-Free Positional Grammars
本文提出了一种基于位置语法规则的超图解析方法，通过转换超边替换为带约束的位置语法规则，利用排列操作确定超边顺序，并揭示了生成与识别歧义的区别。
Passive Model Learning of Visibly Deterministic Context-free Grammars
本文提出PAPNI算法，将RPNI推广到可见确定性下推自动机，在已知字母表分解的前提下实现确定性上下文无关文法的被动学习，并通过实验验证其预测准确率优于RPNI。
Around Context-Free Grammars -- a Normal Form, a Representation Theorem, and a Regular Approximation
本文提出上下文无关文法的 Dyck 范式，证明任意 CFL 可表示为单侧 Dyck 语言子集的同态像，并据此构造其正则超集逼近。
It is well-known that: (i) every context-free language over a singleton terminal alphabet is regular, and (ii) the class of languages that satisfy the Pumping Lemma is a proper super-class of the context-free languages. We show that any language in this superclass over a singleton terminal alphabet is regular. Our proof is based on a transformational approach and does not rely on Parikh's Theorem. Our result extends previously known results because there are languages that are not context-free, do satisfy the Pumping Lemma, and do not satisfy the hypotheses of Parikh's Theorem.
It is known that hyperedge replacement grammars are similar to string context-free grammars in the sense of definitions and properties. Therefore, we expect that there is a generalization of the well-known Greibach normal form from string grammars to hypergraph grammars. Such generalized normal forms are presented in several papers; however, they do not cover a large class of hypergraph languages (e.g. languages consisting of star graphs). In this paper, we introduce a weak Greibach normal form, whose definition corresponds to the lexicalized normal form for string grammars, and prove that eve…
We investigate the conversion of one-way nondeterministic finite automata and context-free grammars into Parikh equivalent one-way and two-way deterministic finite automata, from a descriptional complexity point of view. We prove that for each one-way nondeterministic automaton with $n$ states there exist Parikh equivalent one-way and two-way deterministic automata with $e^{O(\sqrt{n \ln n})}$ and $p(n)$ states, respectively, where $p(n)$ is a polynomial. Furthermore, these costs are tight. In contrast, if all the words accepted by the given automaton contain at least two different letters, th…
Higher-order grammars are extensions of regular and context-free grammars, where non-terminals may take parameters. They have been extensively studied in 1980's, and restudied recently in the context of model checking and program verification. We show that the class of unsafe order-(n+1) word languages coincides with the class of frontier languages of unsafe order-n tree languages. We use intersection types for transforming an order-(n+1) word grammar to a corresponding order-n tree grammar. The result has been proved for safe languages by Damm in 1982, but it has been open for unsafe language…
Multiple (simple) context-free tree grammars are investigated, where "simple" means "linear and nondeleting". Every multiple context-free tree grammar that is finitely ambiguous can be lexicalized; i.e., it can be transformed into an equivalent one (generating the same tree language) in which each rule of the grammar contains a lexical symbol. Due to this transformation, the rank of the nonterminals increases at most by 1, and the multiplicity (or fan-out) of the grammar increases at most by the maximal rank of the lexical symbols; in particular, the multiplicity does not increase when all lex…
Probabilistic context-free grammars have a long-term record of use as generative models in machine learning and symbolic regression. When used for symbolic regression, they generate algebraic expressions. We define the latter as equivalence classes of strings derived by grammar and address the problem of calculating the probability of deriving a given expression with a given grammar. We show that the problem is undecidable in general. We then present specific grammars for generating linear, polynomial, and rational expressions, where algorithms for calculating the probability of a given expres…
Idefix-Free Languages and Their Application in External Contextual Grammars
本文研究从正则语言子族中选择语言的外部上下文语法的能力，引入并比较了infix-free、prefix-free和suffix-free（统称idefix-free）语言与其他子正则语言族，扩展了现有层次结构。
Context-free language theory is a subject of high importance in computer language processing technology as well as in formal language theory. This paper presents a formalization, using the Coq proof assistant, of fundamental results related to context-free grammars and languages. These include closure properties (union, concatenation and Kleene star), grammar simplification (elimination of useless symbols inaccessible symbols, empty rules and unit rules) and the existence of a Chomsky Normal Form for context-free grammars.
This paper presents an extension of the GLL parsing algorithm for context-free grammars which also supports parsing expression grammars with ordered choice and lookahead. The new PEGLL algorithm retains support for unordered choice, and thus parses a common superset of context-free grammars and parsing expression grammars. As part of this work, the authors have modified an existing GLL parser-generator to support parsing expression grammars, adding operators for common parsing expressions and modifying the lexer algorithm to better support ordered choice.
A Uniform Framework for Problems on Context-Free Grammars
本文改进Bouajjani等人基于自动机的统一方法，提出空间复杂度更优、更易理解的新算法，并表明其在多数上下文无关文法标准问题上可与知名方案竞争。
Classifying sequences by combining context-free grammars and OWL ontologies
本文提出一种在OWL中形式化上下文无关文法的模式用于序列分类，并在计算复杂度与实际适用性上与现有方法比较，展示了音乐领域的应用示例。
The problem of identifying a probabilistic context free grammar has two aspects: the first is determining the grammar's topology (the rules of the grammar) and the second is estimating probabilistic weights for each rule. Given the hardness results for learning context-free grammars in general, and probabilistic grammars in particular, most of the literature has concentrated on the second problem. In this work we address the first problem. We restrict attention to structurally unambiguous weighted context-free grammars (SUWCFG) and provide a query learning algorithm for structurally unambiguou…
We consider Parikh images of languages accepted by non-deterministic finite automata and context-free grammars; in other words, we treat the languages in a commutative way --- we do not care about the order of letters in the accepted word, but rather how many times each one of them appears. In most cases we assume that the alphabet is of fixed size. We show tight complexity bounds for problems like membership, equivalence, and disjointness. In particular, we show polynomial algorithms for membership and disjointness for Parikh images of non-deterministic finite automata over fixed alphabet, an…
How to Demonstrate Metalinearness and Regularity by Tree-Restricted General Grammars
本文引入通用文法的推导树与上下文相关节点对概念，证明了满足相邻路径上下文相关节点对数量有界条件时，线性核心通用文法生成k-线性语言、正则核心通用文法生成正则语言，为语言性质证明提供工具。
We investigate the global GRAMMAR constraint over restricted classes of context free grammars like deterministic and unambiguous context-free grammars. We show that detecting disentailment for the GRAMMAR constraint in these cases is as hard as parsing an unrestricted context free grammar.We also consider the class of linear grammars and give a propagator that runs in quadratic time. Finally, to demonstrate the use of linear grammars, we show that a weighted linear GRAMMAR constraint can efficiently encode the EDITDISTANCE constraint, and a conjunction of the EDITDISTANCE constraint and the RE…
Learning language of protein sequences, which captures non-local interactions between amino acids close in the spatial structure, is a long-standing bioinformatics challenge, which requires at least context-free grammars. However, complex character of protein interactions impedes unsupervised learning of context-free grammars. Using structural information to constrain the syntactic trees proved effective in learning probabilistic natural and RNA languages. In this work, we establish a framework for learning probabilistic context-free grammars for protein sequences from syntactic trees partiall…
A practical tool for natural language modeling and development of human-machine interaction is developed in the context of formal grammars and languages. A new type of formal grammars, called grammars with prohibition, is introduced. Grammars with prohibition provide more powerful tools for natural language generation and better describe processes of language learning than the conventional formal grammars. Here we study relations between languages generated by different grammars with prohibition based on conventional types of formal grammars such as context-free or context sensitive grammars. …
This paper presents a restricted form of linear indexed grammars, called even linear indexed grammars, which yield the even linear indexed languages. These languages properly contain the context-free languages and are contained in the set of linear indexed languages. We show that several patterns found in natural languages are also generated by these grammars, including crossing dependencies, copying, and multiple agreements. We discuss the learning problem for even linear indexed languages and show that it is reducible to that of the context-free languages. The closure properties for this cla…
Detecting and Explaining (In-)equivalence of Context-Free Grammars
本文提出了一种可扩展框架，用于决定、证明和解释上下文无关文法的等价性。通过实现该框架并在教育支持系统中收集的大型数据集上进行评估，展示了其在处理大量数据时的有效性。
On Languages Generated by Signed Grammars
本文研究带符号文法生成的语言，利用派生树产出的形式级数来指定形式语言，并分析所得语言族的性质。
A test set for a formal language (set of strings) L is a subset T of L such that for any two string homomorphisms f and g defined on L, if the restrictions of f and g on T are identical functions, then f and g are identical on the entire L. Previously, it was shown that there are context-free grammars for which smallest test sets are cubic in the size of the grammar, which gives a lower bound on tests set size. Existing upper bounds were higher degree polynomials; we here give the first algorithm to compute test sets of cubic size for all context-free grammars, settling the gap between the upp…
Synchronous Context-Free Grammars (SCFGs), also known as syntax-directed translation schemata, are unlike context-free grammars in that they do not have a binary normal form. In general, parsing with SCFGs takes space and time polynomial in the length of the input strings, but with the degree of the polynomial depending on the permutations of the SCFG rules. We consider linear parsing strategies, which add one nonterminal at a time. We show that for a given input permutation, the problems of finding the linear parsing strategy with the minimum space and time complexity are both NP-hard.
Improving the previously known best bound, we show that any recursively enumerable language can be generated with a non-returning parallel communicating (PC) grammar system having six context-free components. We also present a non-returning universal PC grammar system generating unary languages, that is, a system where not only the number of components, but also the number of productions and the number of nonterminals are limited by certain constants, and these size parameters do not depend on the generated language.
This technical report presents a general framework for parsing a variety of grammar formalisms. We develop a grammar formalism, called an Abstract Grammar, which is general enough to represent grammars at many levels of the hierarchy, including Context Free Grammars, Minimalist Grammars, and Generalized Context-free Grammars. We then develop a single parsing framework which is capable of parsing grammars which are at least up to GCFGs on the hierarchy. Our parsing framework exposes a grammar interface, so that it can parse any particular grammar formalism that can be reduced to an Abstract Gra…
The primary use of any probabilistic model involving a set of random variables is to run inference and sampling queries on it. Inference queries in classical probabilistic models is concerned by the computation of marginal or conditional probabilities of events given as an input. When the probabilistic model is sequential, more sophisticated marginal inference queries involving complex grammars may be of interest in fields such as computational linguistics and NLP. In this work, we address the question of computing the likelihood of context-free grammars (CFGs) in Hidden Markov Models (HMMs). …
Random Graph Generation in Context-Free Graph Languages
本文将Mairson的上下文无关字符串文法生成算法适配到超边替换文法，提出一种在无歧义条件下以均匀随机方式、二次方时间生成随机超图的方法，并用生成项图的示例加以验证。
We describe a matrix multiplication recognition algorithm for a subset of binary linear context-free rewriting systems (LCFRS) with running time $O(n^{ωd})$ where $M(m) = O(m^ω)$ 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 $O(n^{ωd + 1})$. The currently best known $ω$ is smaller than $2.38$. Our result provi…
Multiplicity Problems on Algebraic Series and Context-Free Grammars
本文研究多交换变量代数幂级数的三类计算问题，通过归约为算术电路相关问题证明其属于计数谱系，改进了代数级数零性判定的复杂度界，并得到上下文无关文法相关语言问题的新复杂度推论。
A GLR-like Parsing Algorithm for Three-Valued Interpretations of Boolean Grammars with Strong Negation
针对带否定的布尔文法在经典二值逻辑下规则矛盾难处理的问题，本文定义三值解释并借鉴逻辑编程方法处理矛盾文法，提出类GLR算法判定字符串在对应三值语言中的成员资格。
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