Matrix mortality problem¶

The decision problem asking whether the zero matrix belongs to the multiplicative semigroup generated by a finite set of integer matrices.

Version: v1 · 2026-09-08 · History
Domain-specific #: 5492
Origin domain: computability and matrix semigroups
Subdomain: computability and matrix semigroups

Core Idea¶

Decidability depends sharply on matrix dimension, generator count and coefficient restrictions: the general problem is undecidable in established dimensions while low-dimensional and restricted cases remain decidable or open. All finite words over the generator set are mapped to matrix products; the question is whether any word evaluates to zero, with reductions from computation proving undecidability in general. The abstraction is therefore identified by a declared carrier, a transformation or constraint over that carrier, and an invariant that tells an analyst whether the named structure is genuinely present.

Scope of Application¶

Matrix mortality problem belongs to computability and matrix semigroups and is useful where the analyst can specify the typed computability and matrix semigroups carrier, defining objects and relations, parameters, conventions, evidence, boundary cases, and comparison targets, then evaluate the coefficient domain, square dimension, finite generator set, product order and empty-word convention, target zero matrix, decision versus witness task, restrictions, known decidability frontier, and reduction assumptions are explicit. The scope is broad within that domain but bounded by the need for the coefficient domain, square dimension, finite generator set, product order and empty-word convention, target zero matrix, decision versus witness task, restrictions, known decidability frontier, and reduction assumptions are explicit.

Clarity¶

The abstraction clarifies a crowded vocabulary by making the coefficient domain, square dimension, finite generator set, product order and empty-word convention, target zero matrix, decision versus witness task, restrictions, known decidability frontier, and reduction assumptions are explicit the center of the account. A claim should name the carrier, the governing operation or relation, the applicable assumptions, and the recognition test.

Manages Complexity¶

Without the abstraction, an analyst must reason directly over many local details: the carrier roles, admissibility assumptions, competing conventions, derived invariants, boundary cases, and proof or validation obligations specific to Matrix mortality problem. Matrix mortality problem compresses them into the roles in the structural signature. That compression permits comparison across instances without erasing the variables that determine validity. It also exposes which details may be varied safely and which are constitutive.

Abstract Reasoning¶

Identify the carrier. State what the elements, states, objects, or observations are: the typed computability and matrix semigroups carrier, defining objects and relations, parameters, conventions, evidence, boundary cases, and comparison targets. Reject examples whose alleged carrier belongs to a different problem. 2. Lock the constitutive rule. Express the coefficient domain, square dimension, finite generator set, product order and empty-word convention, target zero matrix, decision versus witness task, restrictions, known decidability frontier, and reduction assumptions are explicit independently of one notation or implementation.

Knowledge Transfer¶

Knowledge transfers strongly among subfields of computability and matrix semigroups because they reuse the typed computability and matrix semigroups carrier, defining objects and relations, parameters, conventions, evidence, boundary cases, and comparison targets, All finite words over the generator set are mapped to matrix products; the question is whether any word evaluates to zero, with reductions from computation proving undecidability in general., and type the carrier, state every parameter and convention in the definition, test that the coefficient domain, square dimension, finite generator set, product order and empty-word convention, target zero matrix, decision versus witness task, restrictions, known decidability frontier, and reduction assumptions are explicit, compare the nearest accepted identity, and report counterexamples, uncertainty, and limiting cases.

Relationships to Other Abstractions¶

Current abstraction Matrix mortality problem Domain-specific

Parents (1) — more general patterns this builds on

Matrix mortality problem is a kind of Decidability Computability Prime

The proposed strict upward parent is prime:decidability_computability.

Hierarchy paths (2) — routes to 2 parentless roots

Matrix mortality problem → Decidability Computability → Computability → Algorithm → Function (Mapping)

Show alternative path (1)

Neighborhood in Abstraction Space¶

Matrix mortality problem sits in a crowded region of the domain-specific corpus (35^th percentile for distinctiveness): several abstractions share nearly its structure, so a description that fits it tends to fit its neighbors too.

Family — Computability, Enumeration & Reducibility (15 abstractions)

Nearest neighbors

Computed from structural-signature embeddings · 2026-09-08