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Given a grammar compressed string S, a pattern P, and \(d\ge 0\), we consider the problem of finding all occurrences of \(P'\) in S such that \(d(P,P')\le d\) with respect to Hamming distance. We propose an algorithm for this problem in \(O(\lg \lg n \lg ^* N(m+d\ occ_d \lg \frac{m}{d}\lg N))\) time, where \(N=|S|\), \(m=|P|\), n is the number of variables in the grammar compression, and \(occ_d\) is the frequency of an evidence of a substring of P. We implement this algorithm and compare with a naive filtering on the grammar compression.

We consider the relationship between grammars and formal languages, exploring the following idea: Normally, in the process of deriving a word in a language using a grammar, all structures remain fixed except for the intermediate strings which are changed only by the replacement of substrings. By introducing a more dynamic view of this process, we may allow the grammar to change in various ways as the derivation proceeds, or we may change the notion of application of a rule to a string, or the intermediate strings may be modified between application of rules. We call these more dynamic approaches to language generation ‘reactive grammars’ and explore, in this paper a range of such reactivities. Some of these are related to previously introduced notions of generative grammars, others appear to be new. We consider the expressivity of various reactive grammars and also their relationship to each other and to other formalisms including modal logic and deontic logic. Reactivity of computational structures has been explored in other areas, e.g. in Kripke structures and in the general areas of evolvable and adaptive systems.

Higher-order grammars have been extensively studied in 1980’s and interests in them have revived recently in the context of higher-order model checking and program verification, where higherorder grammars are used as models of higher-order functional programs. A lot of theoretical questions remain open, however, for unsafe higherorder grammars (grammars without the so-called safety condition). In this paper, we show that any tree languages generated by order-2 unsafe grammars are context-sensitive. This also implies that any unsafe order-3 word languages are context-sensitive. The proof involves novel technique based on typed lambda-calculus, such as type-based grammar transformation.

In this paper, we study the generative power of one-sided random context grammars working in a leftmost way. More specifically, by analogy with the three well-known types of leftmost derivations in regulated grammars, we introduce three types of leftmost derivations to one-sided random context grammars and prove the following three results. (I) One-sided random context grammars with type-1 leftmost derivations characterize the family of context-free languages. (II) One-sided random context grammars with type-2 and type-3 leftmost derivations characterize the family of recursively enumerable languages. (III) Propagating one-sided random context grammars with type-2 and type-3 leftmost derivations characterize the family of context-sensitive languages. In the conclusion, the generative power of random context grammars and one-sided random context grammars with leftmost derivations is compared.

Context Free Grammars (CFGs) are widely used in programming language descriptions, natural language processing, compilers, and other areas of software engineering where there is a need for describing the syntactic structures of programs. Grammar inference (GI) is the induction of CFGs from sample programs and is a challenging problem. We describe an unsupervised GI approach which uses simplicity as the criterion for directing the inference process and beam search for moving from a complex to a simpler grammar. We use several operators to modify a grammar and use the Minimum Description Length (MDL) Principle to favor simple and compact grammars. The effectiveness of this approach is shown by a case study of a domain specific language. The experimental results show that an accurate grammar can be inferred in a reasonable amount of time.

In DNA processing and RNA editing, gene insertion and deletion are considered as the basic operations. Based on the above evolutionary transformations, a computing model has been formulated in formal language theory known as insertion-deletion systems. In this paper we study about ambiguity and complexity measures of these systems. First, we define the various levels of ambiguity (i = 0,1,2,3,4,5) for insertion-deletion systems. Next, we show that there are inherently i-ambiguous insertion-deletion languages which are j-unambiguous for the combinations (i, j) ∈ {(5,4), (4,2), (3,1), (3,2), (2,1),(0,1)}. Further, We prove an important result that the ambiguity problem of insertion-deletion system is undecidable. Finally, we define three new complexity measures TLength − Con, TLength − Ins, TLength − Del for insertion-deletion systems and analyze the trade-off between the newly defined ambiguity levels and complexity measures.

A lazy-evaluation based top-down parsing algorithm has been implemented as a set of higher-order functions (combinators) which support directly-executable specifications of fully general attribute grammars. This approach extends aspects of previous approaches, and allows natural language processors to be constructed as modular and declarative specifications while accommodating ambiguous context-free grammars (including direct and indirect left-recursive rules), augmented with semantic rules with arbitrary attribute dependencies (including dependencies from right). This one-pass syntactic and semantic analysis method has polynomial time and space (w.r.t. the input length) for processing ambiguous input, and helps language developers build and test their models with little concern for the underlying computational methods.

The Syntactic Concept Lattice is a residuated lattice based on the distributional structure of a language; the natural representation based on this is a context sensitive formalism. Here we examine the possibility of basing a context free grammar (CFG) on the structure of this lattice; in particular by choosing non-terminals to correspond to concepts in this lattice. We present a learning algorithm for context free grammars which uses positive data and membership queries, and prove its correctness under the identification in the limit paradigm. Since the lattice itself may be infinite, we consider only a polynomially bounded subset of the set of concepts, in order to get an efficient algorithm. We compare this on the one hand to learning algorithms for context free grammars, where the non-terminals correspond to congruence classes, and on the other hand to the use of context sensitive techniques such as Binary Feature Grammars and Distributional Lattice Grammars. The class of CFGs that can be learned in this way includes inherently ambiguous and thus non-deterministic languages; this approach therefore breaks through an important barrier in CFG inference.

Program inversion has many applications such as in the implementation of serialization/deserialization and in providing support for redo/undo, and has been studied by many researchers. However, little attention has been paid to two problems: how to characterize programs that are easy or hard to invert and whether, for each class of programs, efficient inverses can be obtained. In this paper, we propose an inversion framework that we call grammar-based inversion, where a program is associated with an unambiguous grammar describing the range of the program. The complexity of the grammar indicates how hard it is to invert the program, while the complexity is related to how efficient an inverse can be obtained.

We define context-free grammars with Muller acceptance condition that generate languages of countable words. We establish several elementary properties of the class of Muller context-free languages including closure properties and others. We show that every Muller context-free grammar can be transformed into a normal form grammar in polynomial space without increasing the size of the grammar, and then we show that many decision problems can be solved in polynomial time for Muller context-free grammars in normal form. These problems include deciding whether the language generated by a normal form grammar contains only well-ordered, scattered, or dense words. In a further result we establish a limitedness property of Muller context-free grammars: If the language generated by a grammar contains only scattered words, then either there is an integer n such that each word of the language has Hausdorff rank at most n, or the language contains scattered words of arbitrarily large Hausdorff rank. We also show that it is decidable which of the two cases applies.

International audience; Functional graph grammars are finite devices which generate the class of regular automata. We recall the notion of synchronization by grammars, and for any given grammar we consider the class of languages recognized by automata generated by all its synchronized grammars. The synchronization is an automaton-related notion: all grammars generating the same automaton synchronize the same languages. When the synchronizing automaton is unambiguous, the class of its synchronized languages forms an effective boolean algebra lying between the classes of regular languages and unambiguous context-free languages. We additionally provide sufficient conditions for such classes to be closed under concatenation and its iteration.

The final publication is available at Springer via http://dx.doi.org/10.1007/978-3-642-02478-8_59, Proceedings of 10th International Work-Conference on Artificial Neural Networks, IWANN 2009, Salamanca, Spain., This work tests the suitability of NEPs to parse languages. We propose PNEP, a simple extension to NEP, and a procedure to translate a grammar into a PNEP that recognizes the same language. These parsers based on NEPs do not impose any additional constrain to the structure of the grammar, which can contain all kinds of recursive, lambda or ambiguous rules. This flexibility makes this procedure specially suited for Natural Languge Processing (NLP). In a first proof with a simplified English grammar, we got a performance (a linear time complexity) similar to that of the most popular syntactic parsers in the NLP area (Early and its derivatives). All the possible derivations for ambiguous grammars were generated, This work was partially supported by MEC, project TIN2008-02081/TIN and by DGUI CAM/UAM, project CCG08-UAM/TIC-4425.

We define context-free grammars with Buchi acceptance condition generating languages of countable words. We establish several closure properties and decidability results for the class of Buchi context-free languages generated by these grammars. We also define context-free grammars with Muller acceptance condition and show that there is a language generated by a grammar with Muller acceptance condition which is not a Buchi context-free language.

A top-down parsing algorithm has been constructed to accommodate any form of ambiguous context-free grammar, augmented with semantic rules with arbitrary attribute dependencies. A memoization technique is used with this non-strict method for efficiently processing ambiguous input. This one-pass approach allows Natural Language (NL) processors to be constructed as executable, modular and declarative specifications of Attribute Grammars.

Grammar convergence is a lightweight verification method for establishing and maintaining the correspondence between grammar knowledge ingrained in all kinds of software artifacts, e.g., object models, XML schemas, parser descriptions, or language documents. The central idea is to extract grammars from diverse software artifacts, and to transform the grammars until they become syntactically identical. The present paper introduces and illustrates the basics of grammar convergence.

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 REGULAR constraint.

It is well known that certain language families generated by cooperating distributed (CD) grammar systems can be characterized in terms of context-free random context grammars. In particular, the language families generated by CD grammar systems working in the t- and -modes of derivation obey a characterization in terms of ET0L systems, or equivalently by context-free disjoint forbidding random context grammars, and of context-free random context grammars with appearance checking, respectively. Now the question arises whether or not other random context like language families can be characterized in terms of CD grammar systems. We positively answer this question, proving that there are derivation modes for CD grammar systems, namely the negated versions of the aforementioned modes, which precisely characterize the family of context-free disjoint forbidding random context languages and that of languages generated by context-free random context grammars without appearance checking. In passing we show that every language generated by a context-free random context grammar without appearance checking can also be generated by a context-free recurrent programmed grammar without appearance checking, and vice versa.

Non-terminally separated (NTS) languages are a subclass of deterministic context free languages where there is a stable relationship between the substrings of the language and the non-terminals of the grammar. We show that when the distribution of samples is generated by a PCFG, based on the same grammar as the target language, the class of unambiguous NTS languages is PAC-learnable from positive data alone, with polynomial bounds on data and computation.

The Minimalist Grammars (MGs) proposed by Stabler(1997) have tree-shaped derivations (Harkema, 2001b; Michaelis, 2001a). As in categorial grammars, each lexical item is an association between a vocabulary element and complex of features, and so the ”yields” or ”fringes” of the derivation trees are sequences of these lexical items, and the string parts of these lexical items are reordered in the course of the derivation. This paper shows that while the derived string languages can be ambiguous and non-context-free, the set of yields of the derivation trees is always context-free and unambiguous. In fact, the derivation yield languages are strictly deterministic context-free languages, which implies that they are LR(0), and that the generation of derivation trees from a yield language string can be computed in linear time. This result suggests that the work of MG parsing consists essentially of guessing the lexical entries associated with words and empty categories.

The final publication is available at Springer via http://dx.doi.org/10.1007/11499305_19, Proceedings of First International Work-Conference on the Interplay Between Natural and Artificial Computation, IWINAC 2005, Las Palmas, Canary Islands, Spain, June 15-18, 2005, This paper describes Attribute Grammar Evolution (AGE), a new Automatic Evolutionary Programming algorithm that extends standard Grammar Evolution (GE) by replacing context-free grammars by attribute grammars. GE only takes into account syntactic restrictions to generate valid individuals. AGE adds semantics to ensure that both semantically and syntactically valid individuals are generated. Attribute grammars make it possible to semantically describe the solution. The paper shows empirically that AGE is as good as GE for a classical problem, and proves that including semantics in the grammar can improve GE performance. An important conclusion is that adding too much semantics can make the search difficult.

This paper discusses on the learnability of context free grammars(CFGs) within the framework of identification in the limit from positive structural examples. We will introduce a new subclass of CFGs, called a class of k-reversiblecontext-free grammars and propose an O(kdn 3) algorithm for learning this new subclass of CFGs, where d is the maximum of the ranks of given skeletal trees, and n is the total size of the given examples.

This paper describes recent improvements and extensions of Synapse system for learning of context free grammars (CFGs) from sample strings. The system uses rule generation based on bottom-up parsing, incremental learning, and search for rule sets. By the recent improvements, Synapse more rapidly synthesizes several nontrivial CFGs including an unambiguous grammar for the set of strings containing twice as many a’s as b’s and an ambiguous grammar for strings not of the form of ww. By employing a novel rule generation method called inverse derivation, the form of grammars is extended to definite clause grammars (DCGs) and broader classes of grammars such as graph and hypergraph grammars.

We propose and describe a minimal cooperative problem that captures essential features of cooperative behavior and permits detailed study of the mechanisms involved. We characterize this problem as one of language generation by cooperating grammars, and present initial results for language induction by pairs of right-linear grammars using grammatically based genetic programming. Populations of cooperating grammar systems were found to induce grammars for regular languages more rapidly than non-cooperating controls. Cooperation also resulted in greater absolute accuracy in the steady state, even though the control performance exceeded that of prior results for the induction of regular languages by a genetic algorithm.

A string generating hypergraph grammar is a hyperedge replacement grammar where the resulting language consists of string graphs i.e. hypergraphs modeling strings. With the help of these grammars, string languages like a n b n c n can be modeled that can not be generated by context-free grammars for strings. They are well suited to model discontinuous constituents in natural languages, i.e. constituents that are interrupted by other constituents. For parsing context-free Chomsky grammars, the Earley parser is well known. In this paper, an Earley parser for string generating hypergraph grammars is presented, leading to a parser for natural languages that is able to handle discontinuities.

This paper describes inductive inference for synthesizing context free grammars from positive and negative sample strings, implemented in Synapse system. For effective inference of grammars, Synapse employs the following mechanisms. 1. A rule generating method called "inductive CYK algorithm," which generates minimum production rules required for parsing positive samples. 2. Incremental learning for adding newly generated rules to previously obtained rules.Synapse can synthesize both ambiguous grammars and unambiguous grammars. Experimental results show recent improvement of Synapse system to synthesize context free grammars.

This paper describes a method of synthesizing context free grammars from positive and negative sample strings, which is implemented in a grammatical inference system called Synapse. The method is based on incremental learning for positive samples and a rule generation method by “inductive CYK algorithm,” which generates minimal production rules required for parsing positive samples. Synapse can generate unambiguous grammars as well as ambiguous grammars. Some experiments showed that Synapse can synthesize several simple context free grammars in considerably short time.

Determinism plays an important role in grammatical inference. However, in practice, ambiguous grammars (and non determinism grammars in particular) are more used than determinism grammars. Computing the probability of parsing a given string or its most probable parse with stochastic regular grammars can be performed in linear time. However, the problem of finding the most probable string has yet not given any satisfactory answer. In this paper we prove that the problem is NP-hard and does not allow for a polynomial time approximation scheme. The result extends to stochastic regular syntax-directed translation schemes.

Context-Free Grammars are the object of increasing interest in the pattern recognition research community in an attempt to overcome the limited modeling capabilities of the simpler regular grammars, and have application in a variety of fields such as language modeling, speech recognition, optical character recognition, computational biology, etc. This paper proposes an efficient algorithm to solve one of the problems associated to the use of weighted and stochastic Context-Free Grammars: the problem of computing the N best parse trees of a given string. After the best parse tree has been computed using the CYK algorithm, a large number of alternative parse trees are obtained, in order by weight (or probability), in a small fraction of the time required by the CYK algorithm to find the best parse tree. This is confirmed by experimental results using grammars from two different domains: a chromosome grammar, and a grammar modeling natural language sentences from the Wall Street Journal corpus.

We consider parallel communicating grammar systems which consist of several grammars and perform derivation steps, where each of the grammars works in a parallel and synchronized manner on its own sentential form, and communication steps, where a transfer of sentential forms is done. We discuss accepting and analyzing versions of such grammar systems with context-free productions and present characterizations of the family of recursively enumerable languages by them. In accepting parallel communicating grammar systems rules of the form α → A with a word α and a nonterminal A are applied as in the generating case, and the language consists of all terminal words which can derive the axiom. We prove that all types of these accepting grammar systems describe the family of recursively enumerable languages, even if λ-rules are forbidden. Moreover, we study analyzing parallel communicating grammar systems, the derivations of which perform the generating counterparts backwards. This requires a modification of the generating derivation concept to strong-returning parallel communicating grammar systems which also generate the family of recursively enumerable languages.

In this paper, Apical growth Pure context-free grammars (AGPCFG) which are a variation of the pure grammars of Maurer et al are introduced. These grammars allow rewriting of active symbols at the ends of a string simultaneously. They also provide a variation of another kind of grammars called filamentous systems with apical growth, which are motivated by biological considerations. The family of languages generated by AGPCFGs is a subclass of context-free languages. An algorithm for learning this language subclass, in the framework of identification in the limit from positive examples, is provided.

In this paper, we investigate various concepts of leftmost derivation in grammars controlled by bicoloured digraphs, especially regarding their descriptive capacity. This approach allows us to unify the presentation of known results regarding especially programmed grammars and matrix grammars, and to obtain new results concerning grammars with regular control, and periodically time-variant grammars. Moreover, we get new results on leftmost derivations in conditional grammars.

We describe a language model employing a new headed-disjuncts formulation of Lafferty et al.'s (1992) probabilistic link grammar, together with (1) an EM training method for estimating the probabilities, and (2) a procedure for learning some simple lexicalized grammar structures. The model in its simplest form is a generalization of n-gram models, but in its general form possesses context-free expressiveness. Unlike the original experiments on probabilistic link grammars, we assume that no hand-coded grammar is initially available (as with n-gram models). We employ untyped links to concentrate the learning on lexical dependencies, and our formulation uses the lexical identities of heads to influence the structure of the parse graph. After learning, the language model consists of grammatical rules in the form of a set of simple disjuncts for each word, plus several sets of probability parameters. The formulation extends cleanly toward learning more powerful context-free grammars. Several issues relating to generalization bias, linguistic constraints, and parameter smoothing are considered. Preliminary experimental results on small artificial corpora are supportive of our approach.

Although Attribute Grammars were introduced long ago, their lack of expressiveness has resulted in limited use outside the domain of static language processing. With the new notion of Dynamic Attribute Grammars defined on top of Grammar Couples, we show that it is possible to extend this expressiveness and to describe computations on structures that are not just trees, but also on abstractions allowing for infinite structures. The result is a language that is comparable in power to most first-order functional languages, with a distinctive declarative character.

In contrast to the usual depth-first derivations of context-free (CF) grammars, breadth-first derivations (also in combination with depth-first ones) yield a class of augmented context-free grammars (ACF) (also termed multi-breadth-depth grammars) endowed with greater generative capacity, yet manageable. The inadequacy of CF grammars to treat distant dependencies is overcome by the new model. ACF grammars can be classified with respect to their disposition, a concept related to the data structure needed to parse their strings. For such augmented CF grammars we consider the LL(k) condition, that ensures top-down deterministic parsing. We restate the condition as an adjacency problem and we prove that it is decidable for any disposition. The deterministic linear-time parser differs from a recursive descent parser by using instead of a LIFO stack a more general data structure, involving FIFO queues and LIFO stacks in accordance with the disposition. ACF grammars can be also viewed as a formalized version of ATN (Augmented Transition Networks).

Every context-free hypergraph grammar that generates a language of bounded degree can be transformed into an equivalent one that has the apex property, i.e., that cannot “pass” nodes from nonterminal to nonterminal. This generalizes Double Greibach Normal Form of context-free grammars. Moreover, it provides a natural grammatical characterization of the context-free hypergraph languages of bounded degree. For grammars with the apex property it is not possible to put a bound on the number of nonterminals in the right-hand sides of the productions.

Affix Grammars are members of the family of Two-Level Grammars, along with W-grammars, Metamorphosis Grammars and Attribute Grammars. In this tutorial we shall be concerned with the nature and rationale of Affix Grammars and their application in describing programming languages. Some parsing and affix evaluation methods for deterministic and nondeterministic Affix Grammars are discussed. By means of an example, a comparison is made with W-grammars and Attribute Grammars.

A method is presented for obtaining a simple deterministic pushdown transducer which acts as a parser for simple chain grammars. It is shown that a simple deterministic grammar can be constructed which covers the simple chain grammar. To obtain both the simple deterministic pushdown transducer and the cover result, a new type of parse is introduced which differs from the left and right parses which are common for the usual one pass no back-tracking parsing algorithms. For the simple chain grammars this parse, the so-called left part parse, follows from a simple left part property which is satisfied by the grammatical trees of simple chain grammars.

Bracketed two-level grammars are not a variation of two-level grammars (van Wijngaarden grammars) but consitute a restriction within the general scheme. The resulting grammars give rise to an effective top-down analysis, where the replacement of metanotions is governed by rules similar to the evaluation dependencies in attribute grammars. Supplemented by the formalized concept of predicates, the class of languages is shown to be decidable and includes EXSPACE. Moreover it has been demonstrated by a description of PASCAL-S, that the grammars are versatile enough to yield quite readable formal definitions of programming languages. To allow a critical comparison, a grammar for the syntax of ASPLE is given in an Appendix.

The use of control strings to direct derivations in context-free grammars is generalized in this paper. Recording grammars (rgs) are introduced. Rgs don't have a pregiven set of control strings, but generate these during the course of derivations. The generative capacity for several models of rgs is studied. The control mechanism inherent to rgs establishes relationships between substrings in words. The nature of these relationships is investigated too. Applicability of rgs within both compiler theory and programming language description methods is demonstrated. New characterizations of a-transducer mappings and Turing-transductions are displayed. It is shown further how rgs can be used to formalize non-contextfree features in programming languages.

This paper deals with the question of canonical forms, or canonical types, for both grammar forms and context-free grammars. Certain canonical types are position restricted if they depend only upon the positions of terminals and nonterminals. We show that with a few very minor restrictions every possible production-type may be specified as a canonical type for arbitrary context free grammars. The result does not hold for grammar forms where a certain restriction must be made.

An abstract notion of context-free grammar is introduced. It deals with abstract objects that can be words, trees, graphs or other combinatorial objects. It is applied to NLC graph grammars introduced by Rozenberg and Janssens. The monadic second-order theory of a context-free NLC set of graphs is decidable.

Several results on the computational complexity of general context-free language parsing and recognition are given. In particular we show that parsing strings of length n is harder than recognizing such strings by a factor of only 0(log n), at most. The same is true for linear and/or unambiguous context-free languages. We also show that the time to multiply \(\sqrt n \times \sqrt n\) Boolean Matrices is a lower bound on the time to recognize all prefixes of a string (or do on-line recognition), which in turn is a lower bound on the time to generate a particular convenient representation of all parses of a string (in an ambiguous grammar). Thus these problems are solvable in linear time only if n×n Boolean matrix multiplication can be done in 0(n2).

The problem of covering context-free grammars by grammars in some normal forms is considered. It is shown that certain example grammars cannot be covered by grammars which are in e-free form or in Greibach normal form or in Chomsky normal form. The results are generalized using a concept called the structural similarity of context-free grammars. Finally, a grammatical transformation method for constructing e-free covers is given.

This paper is a report on an ongoing work which started in 1981 and is aiming at a general method which would help to considerably reduce the time necessary to develop a syntax-directed editor for any given diagram technique. The main idea behind the approach is to represent diagrams by (formal) graphs whose nodes are enriched with attributes. Then, any manipulation of a diagram (typically the insertion of an arrow, a box, text, coloring, etc.) can be expressed in terms of the manipulation of its underlying attributed representation graph. The formal description of the manipulation is done by programmed attributed graph grammars.

Strict interpretations of grammar forms are studied with respect to parsing, ambiguity, and decidability for intersection and containment. In particular, a parsing procedure for an arbitrary strict interpretation grammar is given which is based on a given parsing method for the master grammar. Time and space bounds on the new procedure are then obtained. Each ambiguous interpretation grammar of an unambiguous grammar form can be converted to an equivalent unambiguous interpretation of the same grammar form. Unambiguity is always decidable for strict interpretation grammars of unambiguous grammar forms. Also, for languages obtained from “compatible” strict interpretations of an unambiguous grammar form, the following problems are solvable: empty intersection, finite intersection, containment, and equality.

The paper gives another understanding of the two-level grammar formalism: any two-level grammar can be splitted into two parts — a context-free "syntax" and an equational "semantics". It has been shown that in the case of a repetition-free and regular based two-level grammar one can always solve the equations assigned to each derivation of the resulting CF grammar. This suggests an approach to the parsing problem of two-level grammars based on well known methods for CF grammars and the algorithm presented. The approach may occur efficient however, only if some restrictions are imposed on two-level grammars. One sort of restrictions we have discussed in the paper (repetition-free and regular based grammars). Others should result from the requirements of programming languages. For example, one obvious requirement is that two-level grammars should be (semanticaly) unambiguous, i.e., here — in terms of the corresponding CF grammars and equations — that each set of equations has at most one solution.

From a given context free grammar, it is possible in a variety of ways to generate automatically a program that acts as a recogniser for the language of that grammar. Under a number of conditions, depending on the particular technique used, this program is an “exact recogniser” of that language, accepting only sentences of the language and rejecting all other strings of symbols.