arXiv subjects

Heer Tern Koh

Publications and source records attributed to Heer Tern Koh.

4 recordsLinked to original sources

arXiv2026

Almost free algebras: from the word problem to elimination of quantifiers

Term algebras are important objects in computer science and are correspondingly well-studied. A natural generalization is to quotient these algebras by finitely many ground term equations, obtaining what we call almost free algebras. One of the earliest results on almost free algebras is that their word problem is polynomial time decidable. In this paper, we show that other natural problems: finding canonical representatives; computing the cardinality of a congruence class; checking if all congruence classes are infinite; checking if the algebra is finite; checking if two algebras are isomorphic, are all polynomial time decidable. Another famous result regarding term algebras is that they admit quantifier elimination in a suitably expanded language. Following this pattern, we also show that almost free algebras admit quantifier elimination by expanding the language with the standard tester predicates. While this is implied by existing results, we view our main contribution here as providing a different approach, which we posit can be easily extended to a larger class that is not covered by existing works. Finally, we provide an application to the quantifier elimination procedure, constructing examples of non-initial algebras over arbitrary signatures with a polynomial time word problem.

math.LO↗

arXiv2026

Primitive recursive categoricity spectra

We study the primitive recursive analogue of computable categoricity spectra for various natural classes of structures. We show that these notions coincide for all relatively $Δ_{2}^{0}$-categorical equivalence structures and linear orders, relatively $Δ_{3}^{0}$-categorical Boolean algebras, and computably categorical tree as partial orders.

math.LO↗

arXiv2026

Primitive recursive categoricity spectra of functional structures

For the notion of degree of categoricity, we study an analogous notion for punctual structures. We show that such notions coincide for non-$Δ_{1}^{0}$-categorical injection structures, and construct an example of a $Δ_{1}^{0}$-categorical injection structure for which these notions differ. Additionally, we also show that in every non-zero c.e.~Turing degree, there exists a PR-degree that is low for punctual isomorphism (to be defined), and also a PR-degree that is a degree of punctual categoricity.

math.LO↗

arXiv2022

Computable topological groups

We investigate what it means for a (Hausdorff, second-countable) topological group to be computable. We compare several potential definitions in the literature. We relate these notions with the well-established definitions of effective presentability for discrete and profinite groups, and compare these results with similar results in computable topology. Most of these definitions can be separated by counter-examples. Remarkably, we prove that two such definitions are equivalent for locally compact Polish and abelian Polish groups. More specifically, we prove that in these broad classes of groups, every computable topological group admits a right-c.e.~(upper semi-computable) presentation with a left-invariant metric, and a computable dense sequence of points. In the locally compact case, we also show that if the group is additionally effectively locally compact, then we can produce an effectively proper left-invariant metric.

math.LO↗