Among the proposals for multidimensional grammars is a family of constraint-based grammatical frameworks, including Relational Grammars. In Relational languages, expressions are formally defined as a set of relations whose tuples are taken from an indexed set of symbols. Both bottom-up parsing and Earley-style parsing algorithms have previously been proposed for different classes of Relational languages. The Relational language class for Earley style parsing in Wittenburg (1992a) requires that each relation be a partial order. However, in some real-world domains, the relations do not naturally conform to these restrictions. In this paper I discuss motivations and methods for predictive, Earley-style parsing of multidimensional languages when the relations involved do not necessarily yield an ordering, e.g., when the relations are symmetric and/or nontransitive. The solution involves guaranteeing that a single initial start position for parsing can be associated with any member of the input set. The domains in which these issues are discussed involve incremental parsing in interfaces and off-line verification of multidimensional data.

In this paper we present a unification-based grammar formalism and parsing algorithm for the purposes of defining and processing non-concatenative languages. In order to encompass languages that are characterized by relations beyond simple string concatenation, we introduce relational constraints into a linguistically-based unification grammar formalism and extend bottom-up chart parsing methods. This work is currently being applied in the interpretation of hand-sketched mathematical expressions and structured flowcharts on notebook computers and interactive worksurfaces.

Extensions to Categorial Grammars proposed to account for nonconstitutent conjunction and long-distance dependencies introduce the problem of equivalent derivations, an issue we have characterized as spurious ambiguity from the parsing perspective. In Wittenburg (1987) a proposal was made for compiling Categorial Grammars into predictive forms in order to solve the spurious ambiguity problem. This paper investigates formal properties o f grammars that use predictive versions of function composition. Among our results are (1) that grammars with predictive composition are in general equivalent to the originals if and only if a restriction on predictive rules is applied, (2) that modulo this restriction, the predictive grammars have indeed eliminated the problem of spurious ambiguity, and (3) that the issue o f equivalence is decidable, i.e., for any particular grammar, whether one needs to apply the restriction or not to ensure equivalence is a decidable question.