David Harel
Abstract
Elementary translations between various kinds of recursive trees are presented. It is shown that trees of either finite or countably infinite branching can be effectively put into one-one correspondence with infinitely branching trees in such a way that the infinite paths of the latter correspond to the “P-abiding” infinite paths of the former. Here P can be any member of a very wide class of properties of infinite paths. For many properties ??, the converse holds too. Two of the applications involve (a) the formulation of large classes of highly undecidable variants of classical computational problems, and in particular, easily describable domino problems that are III11-complete, and (b) the existence of a general method for proving termination of nondeterministic or concurrent programs under any reasonable notion of fairness.
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Index Terms
- Effective transformations on infinite trees, with applications to high undecidability, dominoes, and fairness
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Reviews
Kurt Sieber
This paper presents a theorem relating recursive &ohgr;-trees (with possibly infinite branching) and recursive marked b-trees (with finitely bounded branching only). Undecidability results for Turing machines, domino problems, various kinds of logics, and fair termination problems are obtained as corollaries. The theorem comes in two steps, both establishing (different) recursive isomorphisms. In the first step (called infinite tree recurrence lemma), infinite paths of &ohgr;-trees correspond to recurrent paths of b-trees (i.e., paths with infinitely many marked nodes). Hence, the class of well-founded &ohgr;-trees is isomorphic to the class of recurrent free b-trees and the &Pgr;1-1--completeness of the former transfers to the latter. This result translates to Turing machines and to domino problems, recurrence being understood as infinitely many occurrences of a particular machine state in one computation or of a particular domino in one tiling. Finally, the importance of domino problems is illustrated by hardness proofs for various logics, where, in particular, recurrent dominoes play a role in the satisfiability problem of dynamic logic. The second step (i.e., the theorem itself) is a considerable generalization of the lemma. Recurrent paths are now replaced by 4 -abiding paths where 4 is an arbitrary formula of a (very powerful) path language. Hence, undecidability results for infinitely many classes of trees are obtained. Applied to formal computation trees of nondeterministic programs, the results translate to fair termination problems with respect to any imaginable notion of fairness. Moreover, by working through the proof of the theorem, explicit schedulers for nondeterministic programs, i.e., programs which simulate fair termination by conventional termination, can be obtained. Working through the paper requires much effort, but most of the proofs can be finally understood. An exception is the construction of the schedulers, which remains mysterious for the reader. Moreover, the scheduler for strong fairness seems to be wrong; hence, the examples are of no help in this case. Throughout the paper, the author provides a lot of motivation for the reader, but sometimes at the expense of the overall structure. That's why I've given a rather long (hopefully more structured) summary. The paper appears to be really important to me because it provides very general tools for proving undecidability results. However, I'm not a specialist in this subject, so the reader himself should decide about this issue, possibly with the aid of my summary.
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