🤖 AI Summary
This study addresses the problem of elementarily embedding trees with unbounded paths into bounded trees to support the formal modeling of infinite computational processes. To this end, the authors propose a set of sufficient conditions guaranteeing the existence of such embeddings and introduce several tree operations that yield a compositional semantic framework grounded in the Feferman–Vaught theorem. This framework employs first-order logic to characterize path structures and establishes that the defined operations preserve logical properties. The main contribution lies in developing an extendable method for bounded trees that enables tractable modeling of infinite computation sequences, thereby providing a model-theoretic foundation for finite representations of infinite behaviors.
📝 Abstract
A tree is a partially ordered set that is downwards linear and downwards connected. A tree is called bounded when each of its paths (i.e. maximal linearly ordered subsets) contains a greatest element. In a bounded tree, each path can be defined by a first-order formula using the leaf of the path as parameter. Bounded trees can be used to model computational systems such as Zeno machines whereby the leaf of a path represents the state to which an infinitely long sequence of computations converges, or a state that is assigned to a computational sequence that loops. We identify a sufficient condition under which certain trees that are not bounded, can be elementarily embedded in trees that are bounded. Several tree operations are also given, and Feferman-Vaught style preservation properties for these operations are proved.