Greibach Normal Form Research Articles

Flap: A Deterministic Parser with Fused Lexing

Lexers and parsers are typically defined separately and connected by a token stream. This separate definition is important for modularity and reduces the potential for parsing ambiguity. However, materializing tokens as data structures and case-switching on tokens comes with a cost.We show how tofuseseparately-defined lexers and parsers, drastically improving performance without compromising modularity or increasing ambiguity. We propose a deterministic variant of Greibach Normal Form that ensures deterministic parsing with a single token of lookahead and makes fusion strikingly simple, and prove that normalizing context free expressions into the deterministic normal form is semantics-preserving. Our staged parser combinator library, flap, provides a standard interface, but generates specialized token-free code that runs two to six times faster than ocamlyacc on a range of benchmarks.

The Hardest LL(k) Language

This paper establishes an analogue of Greibach’s hardest language theorem (“The hardest context-free language”, SIAM J. Comp., 1973, http://dx.doi.org/10.1137/0202025 ) for the classical family of LL([Formula: see text]) languages. The first result is that there is a language [Formula: see text] defined by an LL(1) grammar in the Greibach normal form, to which every language [Formula: see text] defined by an LL(1) grammar in the Greibach normal form can be reduced by a homomorphism, that is, [Formula: see text] if and only if [Formula: see text]. Then it is shown that this statement does not hold for the full class of LL([Formula: see text]) languages. The other hardest language theorem is then established in the following form: there is a language [Formula: see text] defined by an LL(1) grammar in the Greibach normal form, such that, for every language [Formula: see text] defined by an LL([Formula: see text]) grammar, with [Formula: see text], there exists a homomorphism [Formula: see text], for which [Formula: see text] if and only if [Formula: see text] [Formula: see text] [Formula: see text], where [Formula: see text] is a new symbol. The results lead to two robust language families: the closures of the languages defined by LL(1) grammars in the Greibach normal form under inverse homomorphisms and under inverse finite transductions.

Building Greibach Normal Form Grammars Using Genetic Algorithms

Grammatical inference of context-free grammars using positive and negative language examples is among the most challenging task in modern artificial and natural language technology. Recently, several implementations combining various techniques, usually including the Backus–Naur form, have been proposed. In this paper, we explore a new implementation of grammatical inference using evolution methods focused on the Greibach normal form and exploiting its properties, and also propose new solutions both in the evolutionary processes and in the corresponding fitness estimation.

Greibach normal form for ω-algebraic systems and weighted simple ω-pushdown automata

In weighted automata theory, many classical results on formal languages have been extended into a quantitative setting. Here, we investigate weighted context-free languages of infinite words, a generalization of ω-context-free languages (as introduced by Cohen and Gold in 1977) and an extension of weighted context-free languages of finite words (that were already investigated by Chomsky and Schützenberger in 1963). As in the theory of formal grammars, these weighted context-free languages, or ω-algebraic series, can be represented as solutions of mixed ω-algebraic systems of equations and by weighted ω-pushdown automata.In our first main result, we show that (mixed) ω-algebraic systems can be transformed into Greibach normal form. We use the Greibach normal form in our second main result to prove that simple ω-reset pushdown automata recognize all ω-algebraic series. Simple ω-reset automata do not use ϵ-transitions and can change the stack only by at most one symbol. These results generalize fundamental properties of context-free languages to weighted context-free languages.

LL (1) Parser versus GNF inducted LL (1) Parser on Arithmetic Expressions Grammar: A Comparative Study

The prime objective of the proposed study is to determine the induction of Greibach Normal Form (GNF) in Arithmetic Expression Grammars to improve the processing speed of conventional LL(1) parser. Conventional arithmetic expression grammar and its equivalent LL(1) is used in the study which is converted. A transformation method is defined which converts the selected grammar into a Greibach normal form that is further converted into a GNF based parser through a method proposed in the study. These two parsers are analyzed by considering 399 cases of arithmetic expressions. During statistical analysis, the results are initially examined with the Kolmogorov-Smirnov and Shapiro-Wilk test. The statistical significance of the proposed method is evaluated with the Mann-Whitney U test. The study described that GNF based LL(1) parser for arithmetic take fewer steps than conventional LL(1) grammar. The ranks and asymptotic significance depict that the GNF based LL(1) method is significant than the conventional LL(1) approach. The study adds to the knowledge of parsers itself, parser expression grammars (PEG’s), LL(1) grammars, Greibach Normal Form (GNF) induced grammar structure, and the induction of Arithmetic PEG’s LL(1) to GNF based grammar.

Weak Greibach Normal Form for Hyperedge Replacement Grammars

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 every context-free hypergraph language (with nonsubstantial exceptions) can be generated by a grammar in this normal form. The proof presented in this paper generalizes a corresponding one for string grammars with a few more technicalities.

Logic for ω-pushdown automata

Context-free languages of infinite words have recently found increasing interest. Here, we will present a second-order logic with the same expressive power as Büchi or Muller pushdown automata for infinite words. This extends fundamental logical characterizations of Büchi, Elgot, Trakhtenbrot for regular languages of finite and infinite words and a more recent logical characterization of Lautemann, Schwentick and Thérien for context-free languages of finite words to ω-context-free languages. For our argument, we will investigate Greibach normal forms of ω-context-free grammars as well as a new type of Büchi pushdown automata which can alter their stack by at most one element and without ϵ-transitions. We show that they suffice to accept all ω-context-free languages. This enables us to use similar results recently developed for infinite nested words.

Non-erasing Chomsky-Schützenberger theorem with grammar-independent alphabet

The famous theorem by Chomsky and Schützenberger (CST) says that every context-free language L over an alphabet Σ is representable as h(D∩R), where D is a Dyck language over a set Ω of brackets, R is a local language and h is an alphabetic homomorphism that erases unboundedly many symbols. Berstel found that the number of erasures can be linearly limited if the grammar is in Greibach normal form; Berstel and Boasson (and later, independently, Okhotin) proved a non-erasing variant of CST for grammars in Double Greibach Normal Form. In all these CST statements, however, the size of the Dyck alphabet Ω depends on the grammar size for L. In the Stanley variant of the CST, |Ω| only depends on |Σ| and not on the grammar, but the homomorphism erases many more symbols than in the other versions of CST; also, the regular language R is strictly locally testable but not local. We prove a new version of CST which combines both features of being non-erasing and of using a grammar-independent alphabet. In our construction, |Ω| is polynomial in |Σ|, namely O(|Σ|46), and the regular language R is strictly locally testable. Using a recent generalization of Medvedev's homomorphic characterization of regular languages, we prove that the degree in the polynomial dependence of |Ω| on |Σ| may be reduced to just 2 in the case of linear grammars in Double Greibach Normal Form.

Generating context-free languages using spiking neural P systems with structural plasticity

Spiking neural P system (SNP system) is a model of computation inspired by networks of spiking neurons. An SNP system is a network of neurons that can send an object, known as a spike, to each other. Spiking neural P system with structural plasticity (SNPSP system) is a variant of the classical SNP system. SNPSP system that incorporates the ideas of synaptogenesis (creating new synapses) and synaptic pruning (deletion of existing synapses), collectively known as structural plasticity, as features of the model. This gives SNPSP systems the ability to change their own structure/topology. In this work, we use SNPSP systems to generate context-free languages. We create a procedure for constructing an SNPSP system given a context-free grammar in Greibach normal form (GNF). The resulting SNPSP system essentially simulates the way in which a context-free grammar in GNF is used to generate languages. We use modules known as arithmetic-memory modules, also created using SNPSP systems, to perform arithmetic operations which are needed for the simulation.

Categorial dependency grammars

Categorial Dependency Grammars (CDGs) are classical categorial grammars extended by oriented polarized valencies. At the same time, CDGs represent a class of completely lexicalized dependency grammars. They express both projective and non-projective dependencies. CDGs generate non-context-free languages. At that, they are parsed in polynomial time under realistic conditions. CDGs possess a normal form that is analogous to Greibach normal form for cf-grammars. CDG-languages are closed under almost all AFL operations and are accepted by a special class of push-down automata with independent counters.

On Intuitionistic Fuzzy Context-Free Languages

Taking intuitionistic fuzzy sets as the structures of truth values, we propose the notions of intuitionistic fuzzy context-free grammars (IFCFGs, for short) and pushdown automata with final states (IFPDAs). Then we investigate algebraic characterization of intuitionistic fuzzy recognizable languages including decomposition form and representation theorem. By introducing the generalized subset construction method, we show that IFPDAs are equivalent to their simple form, called intuitionistic fuzzy simple pushdown automata (IF-SPDAs), and then prove that intuitionistic fuzzy recognizable step functions are the same as those accepted by IFPDAs. It follows that intuitionistic fuzzy pushdown automata with empty stack and IFPDAs are equivalent by classical automata theory. Additionally, we introduce the concepts of Chomsky normal form grammar (IFCNF) and Greibach normal form grammar (IFGNF) based on intuitionistic fuzzy sets. The results of our study indicate that intuitionistic fuzzy context-free languages generated by IFCFGs are equivalent to those generated by IFGNFs and IFCNFs, respectively, and they are also equivalent to intuitionistic fuzzy recognizable step functions. Then some operations on the family of intuitionistic fuzzy context-free languages are discussed. Finally, pumping lemma for intuitionistic fuzzy context-free languages is investigated.

Alternative Vidhi to Conversion of Cyclic CNF->GNF

In automata theory Greibach Normal Form shows that A->aV
 n*, where ‘a’ is terminal symbol and Vn is nonterminal symbol where * shows zero or more rates of Vn [1]. Most popular questions, conversion of following cyclic CNF into GNF are:
 Question 1 S->AA | a, A->SS | b
 Question 2 S->AB, A->BS | b, B->SA | a
 Question 3 S->AB, A->BS | b, B->AS | a [1]
 To solve these questions, we need two technical lemmas and required one or more another variable like Z1. In these questions, we have cyclic nature of production called cyclic CNF. We have modified the same rule by which we get the more reliable answer with less number of productions in right hand side without using lemmas and any another variable. This above method can be applied on all problems by which we produce the GNF.
 
 

A comment on “Automata theory based on complete residuated lattice-valued logic: Pushdown automata”

Automata theory based on complete residuated lattice-valued logic has been first established by Qiu, and then has been systematically studied by Qiu and others. The definition of L-valued Chomsky Normal Form in Xing and Qiu [Automata theory based on complete residuated lattice-valued logic: pushdown automata, Fuzzy Sets and Systems 160 (2009) 1125–1140] is somewhat different from that in Xing and Qiu [Pumping lemma in context-free grammar theory based on complete residuated lattice-valued logic, Fuzzy Sets and Systems 160 (2009) 1141–1151]. In this note, we give a more general L-valued Chomsky Normal Form to unify the two definitions. We mainly show that, for an L-valued context-free grammar, an L-valued Greibach Normal Form can be equivalently constructed.

An elementary proof of a generalization of double Greibach normal form

An elementary proof of a generalization of double Greibach normal form

Generating all permutations by context-free grammars in Greibach normal form

We consider context-free grammars G n in Greibach normal form and, particularly, in Greibach m -form ( m = 1 , 2 ) which generates the finite language L n of all n ! strings that are permutations of n different symbols ( n ≥ 1 ). These grammars are investigated with respect to their descriptional complexity, i.e., we determine the number of nonterminal symbols and the number of production rules of G n as functions of n . As in the case of Chomsky normal form, these descriptional complexity measures grow faster than any polynomial function.

GENERATING ALL CIRCULAR SHIFTS BY CONTEXT-FREE GRAMMARS IN GREIBACH NORMAL FORM

For each alphabet Σn = {a1,a2,…,an}, linearly ordered by a1 < a2 < ⋯ < an, let Cn be the language of circular or cyclic shifts over Σn, i.e., Cn = {a1a2 ⋯ an-1an, a2a3 ⋯ ana1,…,ana1 ⋯ an-2an-1}. We study a few families of context-free grammars Gn (n ≥1) in Greibach normal form such that Gn generates Cn. The members of these grammar families are investigated with respect to the following descriptional complexity measures: the number of nonterminals ν(n), the number of rules π(n) and the number of leftmost derivations δ(n) of Gn. As in the case of Chomsky normal form, these ν, π and δ are functions bounded by low-degree polynomials. However, the question whether there exists a family of grammars that is minimal w. r. t. all these measures remains open.

Analogical Conception of Chomsky Normal Form and Greibach Normal Form for Linear, Monadic Context-Free Tree Grammars

This paper presents the analogical conception of Chomsky normal form and Greibach normal form for linear, monadic context-free tree grammars (LM-CFTGs). LM-CFTGs generate the same class of languages as four well-known mildly context-sensitive grammars. It will be shown that any LM-CFTG can be transformed into equivalent ones in both normal forms. As Chomsky normal form and Greibach normal form for context-free grammars (CFGs) play a very important role in the study of formal properties of CFGs, it is expected that the Chomsky-like normal form and the Greibach-like normal form for LM-CFTGs will provide deeper analyses of the class of languages generated by mildly context-sensitive grammars.

A descriptive characterisation of linear languages

Lautemann et al. (1995) gave a descriptive characterisation of the class of context-free languages, showing that a language is context-free iff it is definable as the set of words satisfying some sentence of a particular logic (fragment) over words. The present notes discuss how to specialise this result to the class of linear languages. Somewhat surprisingly, what would seem the most straightforward specialisation actually fails, due to the fact that linear grammars fail to admit a Greibach normal form. We identify an alternative specialisation, based on an alternative characterisation of context-free languages, also noted by Lautemann et al. (1995).

Algebraically complete semirings and Greibach normal form

We give inequational and equational axioms for semirings with a fixed-point operator and formally develop a fragment of the theory of context-free languages. In particular, we show that Greibach’s normal form theorem depends only on a few equational properties of least pre-fixed points in semirings, and eliminations of chain and deletion rules depend on their inequational properties (and the idempotence of addition). It follows that these normal form theorems also hold in non-continuous semirings having enough fixed points.

Solving Algebraic Equations Using Coalgebra

Algebraic systems of equations define functions using recursion where parameter passing is permitted. This generalizes the notion of a rational system of equations where parameter passing is prohibited. It has been known for some time that algebraic systems in Greibach Normal Form have unique solutions. This paper presents a categorical approach to algebraic systems of equations which generalizes the traditional approach in two ways i) we define algebraic equations for locally finitely presentable categories rather than just Set; and ii) we define algebraic equations to allow right-hand sides which need not consist of finite terms. We show these generalized algebraic systems of equations have unique solutions by replacing the traditional metric-theoretic arguments with coalgebraic arguments.