Defining Contexts in Context-Free Grammars

This thesis introduces an extension of Chomsky’s context-free grammars equipped with operators for referring to left and right contexts of strings.The new model is called grammar with contexts. The semantics of these grammars are given in two equivalent ways — by language equations and by logical deduction, where a grammar is understood as a logic for the recursive definition of syntax. The motivation for grammars with contexts comes from an extensive example that completely defines the syntax and static semantics of a simple typed programming language. Grammars with contexts maintain most important practical properties of context-free grammars, including a variant of the Chomsky normal form. For grammars with one-sided contexts (that is, either left or right), there is a cubic-time tabular parsing algorithm, applicable to an arbitrary grammar. The time complexity of this algorithm can be improved to quadratic,provided that the grammar is unambiguous, that is, it only allows one parsefor every string it defines. A tabular parsing algorithm for grammars withtwo-sided contexts has fourth power time complexity. For these grammarsthere is a recognition algorithm that uses a linear amount of space. For certain subclasses of grammars with contexts there are low-degree polynomial parsing algorithms. One of them is an extension of the classical recursive descent for context-free grammars; the version for grammars with contexts still works in linear time like its prototype. Another algorithm, with time complexity varying from linear to cubic depending on the particular grammar, adapts deterministic LR parsing to the new model. If all context operators in a grammar define regular languages, then such a grammar can be transformed to an equivalent grammar without context operators at all. This allows one to represent the syntax of languages in a more succinct way by utilizing context specifications. Linear grammars with contexts turned out to be non-trivial already over a one-letter alphabet. This fact leads to some undecidability results for this family of grammars