Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 2608.20424

Abstract computation over first-order structures. Extras: From programs to decision trees I

Christine Gaßner
View original source ↗Read source document ↗Export metadata ↓

Abstract

Decisions and their consequences can be described and analyzed by means of decision trees. The decisions themselves depend on questions that, whenever possible, should be answered with yes or no. The original BSS machines over the real numbers are graphs with computation nodes and branching nodes for decisions. The evaluation of BSS machines and algebraic decision trees in computer-aided geometry which have been introduced for various types of numbers generally involves the evaluation of systems of literals of first-order logic. Flowchart-like representations and decision trees are also helpful for analyzing decisions made by BSS RAMs over first-order structures. All algorithms defined by machine-oriented programs of BSS RAMs can be illustrated using flowcharts, walks, and paths in program trees. The basic structures of these tools are graphs and their visual representations can help to characterize the behavior of individual BSS RAMs and other first-order machines. We offer a theoretical framework for linking various models. Here, we introduce program paths and transition systems for transforming partial configurations which are defined syntactically and can be extended and refined later. Finally, we define standard orders for program paths and present algorithms for enumerating finite program paths.

Explore related subjects

math.LOcs.LO

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Christine Gaßner. 2026-08-19. Abstract computation over first-order structures. Extras: From programs to decision trees I. https://arxiv.org/abs/2608.20424

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Bluebirds and mockingbirds cannot produce a fixed-point combinator

Let $B$ be the bluebird combinator with reduction rule $Bxyz \to_{w} x\left(yz\right)$, let $M$ be the mockingbird combinator with reduction rule $Mx \to_{w} xx$, and let $I$ be the identity bird combinator with reduction rule $Ix \to_{w} x$. A fixed-point combinator, called a sage bird by Smullyan, is a closed term $Y$ such that, for a fresh variable $x$, $Yx$ is equivalent to $x\left(Yx\right)$ under these reduction rules. For a fixed variable $x$, we construct an invariant $\mathrm{Tr}_{x}\left(u\right)$ of a $BMI$-term $u$ with respect to $\to_{w}$. This invariant traces the occurrences of $x$ in the leftmost-innermost reduction sequence of $u$. We then prove that $\mathrm{Tr}_{x}\left(Yx\right) \neq \mathrm{Tr}_{x}\left(x^{r}\left( Yx \right)\right)$ for every $x$-free $BMI$-term $Y$ and every $r\geq 1$. Consequently, there exists no fixed-point combinator in $BMI$-combinatory logic. This provides a negative answer to the problem posed by Smullyan in 1985.

math.LO↗
arXiv2026

Pointwise provable equality and the failure of composition

Montagna (1989) and Di Paola--Montagna (1991) claim that the algebraic systems $S'$ and $S'_T$, respectively, are categories. We show that the proposed composition is not independent of the choice of representatives. For every consistent recursively enumerable extension $T$ of Peano arithmetic ($\mathrm{PA}$), we exhibit two program indices that are pointwise provably equal in $T$ but yield inequivalent composites when each is run after the same program. Montagna's $S'$ is the case $T=\mathrm{PA}$. The failure already occurs for partial maps from $ω$ to itself. Weak totality and the proposed range assignment also depend on the choice of representatives. More generally, for consistent $T\supseteq\mathrm{PA}$, pointwise provable equality is a composition congruence exactly when $T$ proves every true $Π^0_1$ sentence, in which case it is extensional equality. This completeness condition fails for every consistent recursively enumerable $T\supseteq\mathrm{PA}$ by Gödel's second incompleteness theorem. For every extension $T\supseteq\mathrm{PA}$, the least composition congruence containing pointwise provable equality is extensional equality if $T$ is $Σ^0_1$-sound and the universal relation otherwise.

math.LO↗
arXiv2026

Compactness via Consistency Properties

We will use consistency properties to characterize strongly compact cardinals, first showing an adequate Model Existence Theorem for larger fragments.

math.LO↗