Conjunctive Grammars over a Unary Alphabet: Undecidability and Unbounded Growth

It has recently been proved (Je?, DLT 2007) that conjunctive grammars (that is, context-free grammars augmented by conjunction) generate some non-regular languages over a one-letter alphabet. The present paper improves this result by constructing conjunctive grammars for a larger class of unary languages. The results imply undecidability of a number of decision problems of unary conjunctive grammars, as well as non-existence of a recursive function bounding the growth rate of the generated languages. An essential step of the argument is a simulation of a cellular automaton recognizing positional notation of numbers using language equations.     

Conjunctive Grammars and Systems of Language Equations

This paper studies systems of language equations that are resolved with respect to variables and contain the operations of concatenation, union and intersection. Every system of this kind is proved to have a least fixed point, and the equivalence of these systems to conjunctive grammars is established. This allows us to obtain an algebraic characterization of the language family generated by conjunctive grammars.     

Second-Order Abstract Categorial Grammars as Hyperedge Replacement Grammars

Second-order abstract categorial grammars (de Groote in Association for computational linguistics, 39th annual meeting and 10th conference of the European chapter, proceedings of the conference, pp. 148–155, 2001) and hyperedge replacement grammars (Bauderon and Courcelle in Math Syst Theory 20:83–127, 1987; Habel and Kreowski in STACS 87: 4th Annual symposium on theoretical aspects of computer science. Lecture notes in computer science, vol 247, Springer, Berlin, pp 207–219, 1987) are two natural ways of generalizing “context-free” grammar formalisms for string and tree languages. It is known that the string generating power of both formalisms is equivalent to (non-erasing) multiple context-free grammars (Seki et al. in Theor Comput Sci 88:191–229, 1991) or linear context-free rewriting systems (Weir in Characterizing mildly context-sensitive grammar formalisms, University of Pennsylvania, 1988). In this paper, we give a simple, direct proof of the fact that second-order ACGs are simulated by hyperedge replacement grammars, which implies that the string and tree generating power of the former is included in that of the latter. The normal form for tree-generating hyperedge replacement grammars given by Engelfriet and Maneth (Graph transformation. Lecture notes in computer science, vol 1764. Springer, Berlin, pp 15–29, 2000) can then be used to show that the tree generating power of second-order ACGs is exactly the same as that of hyperedge replacement grammars.     

A Special Case of a Unary Regular Language Containment

Given an n-state unary finite automaton accepting a language T, and an ultimately periodic set S given as a union of arithmetic progressions that can be represented using n^O(1) bits, and whose periods are mutually coprime, deciding whether T ⫅ S is in n^{O(log n)} time. Dropping the mutual coprimality condition, this containment problem becomes NP-hard.     

GPU Shape Grammars

GPU Shape Grammars provide a solution for interactive procedural generation, tuning and visualization of massive environment elements for both video games and production rendering. Our technique generates detailed models without explicit geometry storage. To this end we reformulate the grammar expansion for generation of detailed models at the tesselation control and geometry shader stages. Using the geometry generation capabilities of modern graphics hardware, our technique generated massive, highly detailed models. GPU Shape Grammars integrate within a scalable framework by introducing automatic generation of levels of detail at reduced cost. We apply our solution for interactive generation and rendering of scenes containing thousands of buildings and trees.     

Unary Quantifiers on Finite Models

In this paper (except in Section 5) all quantifiers are assumedto be so called simple unaryquantifiers, and all models are assumedto be finite. We give a necessary and sufficientcondition for a quantifier to be definablein terms of monotone quantifiers. For amonotone quantifier we give a necessaryand sufficient condition for beingdefinable in terms of a given set of bounded monotonequantifiers. Finally, we give a necessaryand sufficient condition for a monotonequantifier to be definable in terms of agiven monotone quantifier.Our analysis shows that the quantifierat least one half and its relatives behavedifferently than other monotone quantifiers.     

Conjunctive Visual Forms

Visual exploration of multidimensional data is a process of isolating and extracting relationships within and between dimensions. Coordinated multiple view approaches are particularly effective for visual exploration because they support precise expression of heterogeneous multidimensional queries using simple interactions. Recent visual analytics research has made significant progress in identifying and understanding patterns of composed views and coordinations that support fast, flexible, and open-ended data exploration. What is missing is formalization of the space of expressible queries in terms of visual representation and interaction. This paper introduces the conjunctive visual form model in which visual exploration consists of interactively-driven sequences of transitions between visual states that correspond to conjunctive normal forms in boolean logic. The model predicts several new and useful ways to extend the space of rapidly expressible queries through addition of simple interactive capabilities to existing compositional patterns. Two recent related visual tools offer a subset of these capabilities, providing a basis for conjecturing about such extensions.     

Creating Specific Grammars with Generic Grammars: Towards Flexible Urban Design

The aim of the City Induction project is to develop an urban design tool consisting of 3 parts: an urban programme formulation module, a generation module and an evaluation module. The generation module relies on a very generic Urban Grammar composed of several generic grammars called Urban Induction Patterns (UIPs) corresponding to typical urban design moves. Specific grammars, such as the analytical grammars inferred from our case studies, can be obtained by defining specific arrangements of Urban Induction Patterns and specific constraints on the rule parameters. We show that variations on the UIP arrangements or rule parameters can provide design variations and specific grammars to be synthesised through design exploration. It is therefore seen as a process for synthesizing a specific design grammar within the field of urban design and has two main features: (1) it allows for the synthesis of specific grammars during the design process and (2) it allows for the customization of a personal design language within the broad scope of the generic grammar.     

A Faithful Representation of Non-Associative Lambek Grammars in Abstract Categorial Grammars

This paper solves a natural but still open question: can abstract categorial grammars (ACGs) respresent usual categorial grammars? Despite their name and their claim to be a unifying framework, up to now there was no faithful representation of usual categorial grammars in ACGs. This paper shows that Non-Associative Lambek grammars as well as their derivations can be defined using ACGs of order two. To conclude, the outcome of such a representation are discussed.     

Verifying Object-based Graph Grammars

The development of concurrent and reactive systems is gaining importance since they are well-suited to modern computing platforms, such as the Internet. However, the development of correct concurrent and reactive systems is a non-trivial task. Object-based graph grammar (OBGG) is a visual formal language suitable for the specification of this class of systems. In previous work, a translation from OBGG to PROMELA (the input language of the SPIN model checker) was defined, enabling the verification of OBGG models using SPIN. In this paper we extend this approach in two different ways: (1) the approach for property specification is improved, enabling to prove properties not only about possible OBGG derivations, but also about the internal state of involved objects; (2) an approach is defined to interpret PROMELA races as OBGG derivations, generating graphical counter-examples for properties that are not true for a given OBGG model. Another contribution of this paper is (3) the definition of a method for model checking partial systems (isolated objects or a set of objects) using an assume-guarantee approach. A gas station system modeled with OBGGs is used to illustrate the contributions.This work is partially sponsored by projects IQ-MObile (CNPq-Brazil/CNR-Italy) and PLATUS (CNPq).Osmar Marchi dos Santos is partially sponsored by CAPES-Brazil.     

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.     

Probabilistic Grammars and Languages

Using an asymptotic characterization of probabilistic finite state languages over a one-letter alphabet we construct a probabilistic language with regular support that cannot be generated by probabilistic CFGs. Since all probability values used in the example are rational, our work is immune to the criticism leveled by Suppes (Synthese 22:95–116, 1970) against the work of Ellis (1969) who first constructed probabilistic FSLs that admit no probabilistic FSGs. Some implications for probabilistic language modeling by HMMs are discussed.     

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.     

Grammars on partial graphs

Summary The concept of Chomsky-grammars is generalized to graph-grammars; the gluing of graphs is defined by a pushout-construction. In the present paper, we allow the left-hand and right-hand side of a production to be partial graphs, i.e. graphs in which there may be edges without a source or target node. A necessary and sufficient condition for applicability of productions is given. Furthermore, convex graph-grammars are studied.