Turchin's Relation and Subsequence Relation in Loop Approximation

Abstract

The paper studies the subsequence relation through a notion of an intransitive binary relation on words in traces generated by prefix-rewriting systems. The relation was introduced in 1988 by V.F. Turchin for loop approximation in supercompilation. We study properties of this relation and introduce some refinements of the subsequence relation that inherit the useful features of Turchin's relation.

Turchin's Relation for Call-by-Name Computations: A Formal Approach

“…Turchin's relation can be used together with the homeomorphic embedding relation without the loss of well-binariness, and can be used not only for deciding when to terminate the computation path but also for deciding how to generalize the configurations. 15 If all the configurations in the computation tree contained no branching (i.e., were consisting only of unary function calls and constructors), Turchin's relation could be considered as a "one-gap version" of the homeomorphic embedding for the callstacks, and could be replaced by an annotated version of the homeomorphic embedding as shown in [12]. However, for terms having the tree structure (even with only one branching node), the "one-gap" (or even "n-gap") relation is not well-binary (that is also shown in [12]).…”

“…15 If all the configurations in the computation tree contained no branching (i.e., were consisting only of unary function calls and constructors), Turchin's relation could be considered as a "one-gap version" of the homeomorphic embedding for the callstacks, and could be replaced by an annotated version of the homeomorphic embedding as shown in [12]. However, for terms having the tree structure (even with only one branching node), the "one-gap" (or even "n-gap") relation is not well-binary (that is also shown in [12]). So, considering the call-stack configurations as words is somewhat essential for making Turchin's relation well-binary.…”

“…In this subsection we briefly recall notions from the paper [12] where a formal proof of the wellbinariness of Turchin's relation is given for the call-by-value semantics. Details are omitted.…”

“…Details are omitted. The full description of the formalization can be found in the original paper [12]. We use some notions from the paper as a basis to construct the formalization for the call-by-name semantics.…”

“…Then we delete all the nodes containing static data 12 . The remaining nodes are given the layer labels.…”