garbage collection and the use of pushdown systems each enhance the precision of control-flow analysis (CFA). However, their respective needs conflict: abstract garbage collection requires the stack but pushdown systems obscure it. Though several existing techniques address this conflict, none take full advantage of the underlying interplay. In this paper, we dissolve this conflict with a technique which exploits the precision of pushdown systems to decompose the heap across the continuation. This technique liberates abstract garbage collection from the stack, increasing its effectiveness and the compositionality of its host analysis. We generalize our approach to apply compositional treatment to abstract timestamps which induces the context abstraction of m-CFA, an abstraction more precise than k-CFA's for many common programming patterns.
Okasaki introduced the canonical formulation of functional red-black trees when he gave a concise, elegant method of persistent element insertion. Persistent element deletion, on the other hand, has not enjoyed the same treatment. For this reason, many functional implementations simply omit persistent deletion. Those that include deletion typically take one of two approaches. The more-common approach is a superficial translation of the standard imperative algorithm. The resulting algorithm has functional airs but remains clumsy and verbose, characteristic of its imperative heritage. (Indeed, even the term insertion is a holdover from imperative origins, but is now established in functional contexts. Accordingly, we use the term deletion which has the same connotation.) The less-common approach leverages the features of advanced type systems, which obscures the essence of the algorithm. Nevertheless, foreign-language implementors reference such implementations and, apparently unable to tease apart the algorithm and its type specification, transliterate the entirety unnecessarily. Our goal is to provide for persistent deletion what Okasaki did for insertion: a succinct, comprehensible method that will liberate implementors. We conceptually simplify deletion by temporarily introducing a "double-black" color into Okasaki's tree type. This third color, with its natural interpretation, significantly simplifies the preservation of invariants during deletion.
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