A game-theoretic characterization of Boolean grammars

We obtain a simple, purely game-theoretic characterization of Boolean grammars [A. Okhotin, Boolean grammars, Information and Computation, 194(1) (2004) 19–48]. In particular, we propose a two-player infinite game of perfect information for Boolean grammars, which is equivalent to their well-founded semantics. The game is directly applicable to the simpler classes of conjunctive and context-free grammars, and offers a promising new connection between game theory and formal languages.     

On the Q.R. algorithm with shift

Fuzzy grammars on Boolean lattices (B-fuzzy grammars) are newly defined and their basic properties are investigated. B-fuzzy grammars are defined as the extension of fuzzy grammars by Lee and Zadeh, where the grades of the application of rewriting rules of B-fuzzy grammars are the elements of Boolean lattice rather than the elements of unit interval [0,1].

It is shown that type 2 B-fuzzy grammars can generate type 1 languages though type 2 fuzzy grammars cannot generate type 2 languages. And the closure properties of B-fuzzy grammars are also studied.     

Case-factor diagrams for structured probabilistic modeling

We introduce a probabilistic formalism handling both Markov random fields of bounded tree width and probabilistic context-free grammars. Our models are based on case-factor diagrams (CFDs) which are similar to binary decision diagrams (BDDs) but are more concise for circuits of bounded tree width. A probabilistic model consists of a CFD defining a feasible set of Boolean assignments and a weight (or cost) for each individual Boolean variable. We give versions of the inside–outside algorithm and the Viterbi algorithm for these models.     

Die Verwandtschaft kontextfreier Grammatiken

We take up the question of relationship between grammars which has been initiated by G. Hotz. Two new theorems on factorization of arbitrary transformations into embeddings and reductions and on product constructions involving non-lengthpreserving homomorphisms are derived. With the help of these two theorems we can characterize the general relationship of unambiguous contextfree grammars by the relationship, defined by simulations. Simulations are wellknown in the case of normal-form-constructions. A necessary condition is given for the relationship of linear grammars, which is sufficient in the case of leftlinear grammars. A problem of G. Hotz, concerning the relationship of weakly equivalent linear grammars is answered negatively.     

Left context precedence grammars

Summary Left context precendence grammars which are defined in this paper are a proper subclass of the precedence grammars and contain properly the simple precedence grammars. Left context precedence grammars need not be uniquely invertible. The parsing algorithm developed for left context precedence languages works with linear time and has the viable prefix property which is a stronger property than the correct prefix property.     

Tree controlled grammars

Languages are studied which can be generated by context-free grammars under a single simple restriction which must be satisfied by its derivation trees. Using tree controlled grammars (TC grammars for short) all unambigous and some inherently ambigous context-free languages, and also some non context-free languages can be parsed in timeO(n ²). The classes of regular, linear, context-free, EOL, ETOL and type 0 languages can be characterized in a natural manner using TC grammars. A context-free generator for all type 0 languages is exhibited. Some normal forms for TC grammars are established but it is shown that many common normal forms (e. g. Greibach normal form) cannot be obtained for TC grammars in general.     

Multipass precedence analysis

Summary This paper defines a hierarchy of languages which is properly contained in the context sensitive languages and which starts with the context free family. The hierarchy is defined inductively by controlling labeled linear grammars with languages in one family to yield languages in the next larger family. The families of the hierarchy have properties analogous to those of the context free family, in particular, the new mechanism introduced is very suitable for parsing.A language in the n-th family is specified by a sequence of n — 1 labeled linear grammars and a context free grammar. By assuming that the reversals of the first n — 1 grammars and the last labeled linear grammar are precedence grammars, the concepts and parsing algorithm of Wirth and Weber extend to yield a parsing algorithm within the hierarchy. This considerably enhances the usefulness of the construction and allows much of the power of the context sensitive languages to become accessible in measured amounts for potential programming applications.     

Parameter reduction and automata evaluation for grammar-compressed trees

Trees can be conveniently compressed with linear straight-line context-free tree grammars. Such grammars generalize straight-line context-free string grammars which are widely used in the development of algorithms that execute directly on compressed structures (without prior decompression). It is shown that every linear straight-line context-free tree grammar can be transformed in polynomial time into a monadic (and linear) one. A tree grammar is monadic if each nonterminal uses at most one context parameter. Based on this result, polynomial time algorithms are presented for testing whether a given (i) nondeterministic tree automaton or (ii) nondeterministic tree automaton with sibling-constraints or (iii) nondeterministic tree walking automaton, accepts a tree represented by a linear straight-line context-free tree grammar. It is also shown that if tree grammars are nondeterministic or non-linear, then reducing their numbers of parameters cannot be done without an exponential blow-up in grammar size.     

LL(k)-PARSING OF COUPLED-CONTEXT-FREE GRAMMARS

Coupled-context-free grammars are a natural generalization of context-free grammars obtained by combining nonterminals to corresponding parentheses which can only be substituted simultaneously. Refering to the generative capacity of the grammars we obtain an infinite hierarchy of languages that comprises the context-free languages as the first and all the languages generated by TAGs as the second element. Here, we present a generalization of the context-free LL(k) -notion onto coupled-context-free grammars, which leads to a characterization of subclasses of coupled-context-free grammars–and in this way of TAGs as well–which can be parsed in linear time. The parsing procedure described works incrementally so that it can be used for on-line parsing of natural language. Examples show that important elements of the tree-adjoining languages can be generated by LL(k )-coupled-context-free grammars.     

Spinal-Formed Context-Free Tree Grammars

In this paper we introduce a restricted model of context-free tree grammars called spine grammars, and study their formal properties including considerably simple normal forms. Recent research on natural languages has suggested that formalisms for natural languages need to generate a slightly larger class of languages than context-free grammars, and for that reason tree adjoining grammars have been widely studied relating them to natural languages. It is shown that the class of string languages generated by spine grammars coincides with that of tree adjoining grammars. We also introduce acceptors called linear pushdown tree automata, and show that linear pushdown tree automata accept exactly the class of tree languages generated by spine grammars. Linear pushdown tree automata are obtained from pushdown tree automata with a restriction on duplicability for the pushdown stacks. Received May 29, 1998, and in revised form April 27, 1999, and in final form May 10, 1999.     

The covering problem for linear context-free grammars

Language equivalence, grammatical covering and structural equivalence are all notions of similarity defined on context-free grammars. We show that the problem of determining whether an arbitrary linear context-free grammar covers another is complete for the class of languages accepted by polynomially space bounded Turing machines. We then compare the complexity of this problem with the analogous problems for language equivalence and structural equivalence, not only for linear grammars, but also for regular grammars and unrestricted context-free grammars. As a step in obtaining the main result of this paper, we show that the equivalence problem for linear s-grammars is decidable in polynomial time.     

Extended linear macro grammars,iteration grammars,and register programs

Summary Extended macro grammars (of the linear basic type only) are introduced as a generalization of those in [5], and it is shown that they have the same language generating power as (parallel) iteration grammars. In particular the IO and OI versions of extended macro grammars correspond to the deterministic and the (usual) nondeterministic iteration grammars respectively. Hence iterated substitution can be formulated using extended macro grammars.A nondeterministic register program without tests may be viewed as a macro grammar. IO-extension of this macro grammar corresponds to the use of nonrecursive function procedures in the register program. OI-extended macro grammars correspond to register programs which compute on sets. Hence these features of register programs can be investigated by means of (extended) parallel rewriting systems.The work of the first author has been supported by the Netherlands Organization for the Advancement of Pure Research (Z.W.O.).     

On the generative capacity of compound string and array grammars

The operation of composition is a device introduced by Abraham [1] to increase the generative capacity of regular matrix grammars. But not all families of grammars have their generative capacity increased by this operation. P?un [5] has proved that composition increases the generative capacity of linear grammars but not of regular, CF, CS or PS grammars. In this note, we show that the generative power of k-linear (k ? 1) grammars is increased by composition. In fact, it is of interest to note that the families of compound linear and compound k-linear languages are equal. Furthermore, the family of compound linear languages is an AFL, properly contained in the family of CFLs. Certain other families of string and array grammars whose generative capacities are increased by the operation of composition are also provided.     

The Boolean closure of linear context-free languages

Closures of linear context-free languages under Boolean operations are investigated. The intersection closure and the complementation closure are incomparable. By closing these closures under further Boolean operations we obtain several new language families. The hierarchy obtained by such closures of closures is proper up to a certain level, where it collapses to the Boolean closure which, in turn, is incomparable with several closures of the family of context-free languages. The Boolean closure of the linear context-free languages is properly contained in the Boolean closure of the context-free languages. A characterization of a class of non-unary languages that cannot be expressed as a Boolean formula over the linear context-free languages is presented.     

On the Expressive Power of Abstract Categorial Grammars: Representing Context-Free Formalisms

We show how to encode context-free string grammars, linear context-free tree grammars, and linear context-free rewriting systems as Abstract Categorial Grammars. These three encodings share the same constructs, the only difference being the interpretation of the composition of the production rules. It is interpreted as a first-order operation in the case of context-free string grammars, as a second-order operation in the case of linear context-free tree grammars, and as a third-order operation in the case of linear context-free rewriting systems. This suggest the possibility of defining an Abstract Categorial Hierarchy.     

The Equivalence of Tree Adjoining Grammars and Monadic Linear Context-free Tree Grammars

The equivalence of leaf languages of tree adjoining grammars and monadic linear context-free grammars was shown about a decade ago. This paper presents a proof of the strong equivalence of these grammar formalisms. Non-strict tree adjoining grammars and monadic linear context-free grammars define the same class of tree languages. We also present a logical characterisation of this tree language class showing that a tree language is a member of this class iff it is the two-dimensional yield of an MSO-definable three-dimensional tree language.     

BRNGLR: a cubic Tomita-style GLR parsing algorithm

Tomita-style generalised LR (GLR) algorithms extend the standard LR algorithm to non-deterministic grammars by performing all possible choices of action. Cubic complexity is achieved if all rules are of length at most two. In this paper we shall show how to achieve cubic time bounds for all grammars by binarising the search performed whilst executing reduce actions in a GLR-style parser. We call the resulting algorithm Binary Right Nulled GLR (BRNGLR) parsing. The binarisation process generates run-time behaviour that is related to that shown by a parser which pre-processes its grammar or parse table into a binary form, but without the increase in table size and with a reduced run-time space overhead. BRNGLR parsers have worst-case cubic run time on all grammars, linear behaviour on LR(1) grammars and produce, in worst-case cubic time, a cubic size binary SPPF representation of all the derivations of a given sentence.     

Constructing LR parsers for regular right part grammars

Summary Regular right part (RRP) grammars differ from context free (CF) grammars by virtue of the fact that the production right parts are nondeterministic finite automatons (FAs). LR(k) parsers for RRP grammars are linear time parsers which can determine the right end of each handle by considering at most k terminal symbols to its right and the left end (after the right end has been found) by considering at most one parse stack state to its left. This paper is concerned with the construction of a class of LR(k) parsers for RRP grammars which makes use of FAs for determining both the right and left ends of the handle.This work has been supported by a Carleton University grant and National Research Council of Canada grant no. A3585