Parsing multi-ordered grammars with the Gray algorithm

Background. Context-free grammars (CFGs) and Parsing-expression Grammars (PEGs) are the two main formalisms used by formal specifications and parsing frameworks to describe programming languages. They mainly differ in the definition of the choice operator, describing language alternatives. CFGs support the use of non-deterministic choice (i.e., unordered choice), where all alternatives are equally explored. PEGs support a deterministic choice (i.e., ordered choice), where alternatives are explored in strict succession. In practice the two formalisms, are used through concrete classes of parsing algorithms (such as Left-to-right, rightmost derivation (LR) for CFGs and Packrat parsing for PEGs), that follow the semantics of the formal operators. Problem Statement. Neither the two formalisms, nor the accompanying algorithms are sufficient for a complete description of common cases arising in language design. In order to properly handle ambiguity, recursion, precedence or associativity, parsing frameworks either introduce implementation specific directives or ask users to refactor their grammars to fit the needs of the framework/algorithm/formalism combo. This introduces significant complexity even in simple cases and results in incompatible grammar specifications. Our Proposal. We introduce Multi-Ordered Grammars (MOGs) as an alternative to the CFG and PEG formalisms. MOGs aim for a better exploration of ambiguity, ordering, recursion and associativity during language design. This is achieved by (a) allowing both deterministic and non-deterministic choices to co-exist, and (b) introducing a form of recursive and scoped ordering. The formalism is accompanied by a new parsing algorithm (Gray) that extends chart parsing (normally used for Natural Language Processing) with the proposed MOG operators. Results. We conduct two case-studies to assess the expressiveness of MOGs, compared to CFGs and PEGs. The first consists of two idealized examples from literature (an expression grammar and a simple procedural language). The second examines a real-world case (the entire Smalltalk grammar and eleven new Smalltalk extensions) probing the complexities of practical needs. We show that in comparison, MOGs are able to reduce complexity and naturally express language constructs, without resorting to implementation specific directives. Conclusion. We conclude that combining deterministic and non-deterministic choices in a single grammar specification is indeed not only possible but also beneficial. Moreover, augmented by operators for recursive and scoped ordering the resulting multi-ordered formalism presents a viable alternative to both CFGs and PEGs. Concrete implementations of MOGs can be constructed by extending chart parsing with MOG operators for recursive and scoped ordering.