Infinite versions of some NP-complete problems

Jeffry L. Hirst; Steffen Lempp

Home/Archive/arXiv

arXiv · math/9503205

Infinite versions of some NP-complete problems

Jeffry L. Hirst·Steffen Lempp

View original source ↗Read source document ↗Export metadata ↓

Abstract

Recently, connections have been explored between the complexity of finite problems in graph theory and the complexity of their infinite counterparts. As is shown in our paper (and in independent work of Tirza Hirst and D. Harel from a different angle) there is no firm connection between these complexities, namely finite problems of equal complexity can have radically different complexity for the infinite versions and vice versa. Furthermore, the complexity of an infinite counterpart can depend heavily on precisely how the finite problem is rephrased in the infinite case. The finite problems we address include colorability of graphs and existence of subgraph isomorphisms. In particular, we give three infinite versions of the 3-colorability problem that vary considerably in their recursion theoretic and proof theoretic complexity. Additionally, we show that three subgraph isomorphism problems of varying finite complexity have infinite versions with identical recursion theoretic and proof theoretic content.

Explore related subjects

math.LO

Keep this discovery

Explore connections, maps & timelines

BibTeX RIS

Jeffry L. Hirst, Steffen Lempp. 1995-03-10. Infinite versions of some NP-complete problems. https://arxiv.org/abs/math/9503205

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

KEEP EXPLORING

Related papers

arXiv2026

Natural Term Logic

In this paper we develop a formal system called Natural Term Logic (NTL). NTL aims to represent key aspects of the logical and grammatical mechanisms of natural language as well as grammatical transformations which preserve core logical meaning. NTL can be seen as a refinement of the ideas of Quine's paper `Variables Explained Away' and the technical concepts introduced by Bealer and Zalta. NTL is more fine-grained than Bealer's first-order intensional logic (BL): there is a many-to-one correspondence $ν$ between NTL terms and closed BL terms as well as a canonical map $β$ which assigns to each closed BL term a corresponding NTL term. The map $ν$ can be seen as assigning a core logical content of the NTL term. We define a series of reductions on NTL terms which intuitivelyy speaking capture meaning-preserving syntactic transformations ( transformations which preserved the basic logical meaning of a term) and our main result is that each NTL term $T$ reduces to a unique normal term $N$. The reductions fall into the structural, predicative and pushing-in categories. Predicative reductions decompose NTL terms so that predication is only applied to a primitive term (such terms are called prenormal). A key ingredient in the proof is the fact that $βνN = N$ when $N$ is normal. This suggests that within NTL the normal form of a term expresses the core logical content of the term.

math.LO↗

arXiv2026

Hyper-hyperfiniteness and complexity

We show that if there exists a countable Borel equivalence relation which is hyper-hyperfinite but not hyperfinite, then the complexity of hyperfinite countable Borel equivalence relationsis as high as possible, namely, $Σ^1_2$-complete. We also establish an implication between the question of the effectivity of hyperfiniteness and its complexity.

math.LO↗

arXiv2026

Coordinate recognition: General theory, Groups, and other surprises

A class of structures \emph{recognizes coordinates} if any reduced product of structures from said class witnesses a certain kind of rigidity phenomenon. We provide several equivalent characterizations of this property. This property has (at least) two remarkable consequences, one set-theoretic and one model-theoretic, for reduced products of structures of the said class. First, under appropriate set-theoretic assumptions every isomorphism between such reduced products associated with the Fréchet ideal lifts (modulo a finite change) to an isomorphism between products of the original structures. Second, with an additional mild assumption, it implies a strong quantifier elimination result. Of note, we show that a class recognizes coordinates if and only if an individual formula witnesses a certain syntactic property. We also consider many concrete classes of structures and determine whether or not they recognize coordinates. We place heavy emphasis on well-known classes of groups, such as permutation groups, acylindircally hyperbolic groups, quasisimple groups, free products, and graph products, but we also discuss other classes of structures.

math.LO↗