(An) LR(k) and LL(k) parsing framework: abstract interpretation approach

Abstract

Abstract interpretation was introduced as a framework for construction of static analysis of computer programs. Abstract interpretation can be regarded as a partial execution of programs with information about its semantics and without performing all the computation. Originally abstract interpretation was applied to static analysis of programs. However it turned out that abstract interpretation also can be used in representing the relationships between various formal semantics which are expressed by fixpoint semantics. On the other hand, it is well known that a context free grammar can be regarded as a kind of deductive system. Moreover it is also known that a deductive system can expresses a fixpoint semantics. Based on these two ideas, it is possible to represent a semantics of a given context free grammar by a fixpoint semantics and express a parser of the grammar with an abstract interpretation of the semantics of the grammar. Cousot and Cousot represented Earley``s parsing algorithm as an abstract interpretation in their research, and they expected that it would be possible to express other parsing algorithms with similar method. On the other hand, LR(k) parsing is the strongest deterministic parsing algorithm, and LL(k) parsing is a top-down dual of LR(k) parsing. These are among the strongest deterministic parsing algorithms and used widely in practical applications. Nonetheless the theory of construction of these parsing algorithms is not understood widely. While most textbooks of compiler construction or automata theory mention LR parsing, the correctness proofs of the construction algorithms of LR parsers and LR parsing algorithms are omitted usually. Even in the texts which describe the theory and proofs in detail, developments of the theory are mainly procedural and it is not easy to understand them intuitively. In this research, after the line of Cousot and Cousot``s works, we represented LR(k) and LL(k) parsing algorithms as abs...