Programs Versus Finite Tree-Programs
Mikhail Moshkov
TL;DR
This work investigates when programs that implement total functions over a signature $\sigma$ (allowing cycles and countable node sets) are equivalent to finite tree-programs, by introducing total-relative-to-$K$ and program-saturated concepts. The authors establish a sharp criterion: a class $K$ is program-saturated if and only if it is compact; they further show that any axiomatizable class is program-saturated and extend these results to individual structures such as $\omega$-saturated models and $\alpha$-categorical theories, with complex examples like $\mathbb{C}$. They then demonstrate that one can always pass to a minimal, computation-program-saturated elementary extension of a given structure, quantified by a cardinal $\alpha(U)$, and transfer the saturation results to computation-tree contexts where finite-tree schemes suffice. The practical impact lies in identifying when complex program schemes collapse to finite-tree computations, and providing a model-theoretic toolkit (compactness, saturation, elementary extensions) to achieve this reduction, with implications for algorithmic representations and reasoning about computation trees in logic-based settings. ${S=(n,G)}$, ${\Pi(\tau)}$, ${\pi_\Gamma}$, ${\omega}$-saturation, and ${\alpha}$-categoricity appear as central symbols throughout the results.
Abstract
In this paper, we study classes of structures and individual structures for which programs implementing functions defined everywhere are equivalent to finite tree-programs. The programs under consideration may have cycles and at most countably many nodes. We start with programs in which arbitrary terms of a given signature may be used in function nodes and arbitrary formulas of this signature may be used in predicate nodes. We then extend our results to programs that are close in nature to computation trees: if such a program is a finite tree-program, then it is an ordinary computation tree.