Pseudo-polynomial transformation¶
Map instances between numerical decision problems while preserving yes/no membership and polynomially controlling computation, encoding growth, and the target's largest numerical parameter.
Core Idea¶
A pseudo-polynomial transformation is a many-one instance map between numerical decision problems that preserves yes/no membership and controls both ordinary encoding size and the magnitudes of the numerical parameters. For encodings \(L_1,L_2\), it supplies a function \(f\) such that \(w\in L_1\) exactly when \(f(w)\in L_2\). Unlike an ordinary polynomial-time reduction, its computation may be polynomial in the source encoding length together with the source problem's maximum numerical parameter. Additional size inequalities prevent the map from hiding an exponential blow-up or mapping polynomially bounded source instances to numerically unbounded target instances.
Scope of Application¶
The abstraction is literal wherever practitioners can identify the same constitutive roles, apply the same boundary tests, and obtain the same kind of output. The following habitats are uses of Pseudo-polynomial transformation itself, not metaphors based only on resemblance.
- Strong NP-completeness proofs. Transporting hardness from a bounded-numeric restriction of a known source problem.
- Scheduling. Mapping jobs, processing times, or deadlines while controlling their generated magnitudes.
- Packing and partitioning. Relating numerical capacity problems without losing value-sensitive complexity.
- Combinatorial optimization. Auditing whether a construction proves ordinary or strong hardness.
- Encoding analysis. Separating bit-length growth from numeric-value growth.
- Reduction verification. Checking all four defining obligations instead of membership preservation alone.
Clarity¶
A clear account of Pseudo-polynomial transformation must preserve the recognition invariant stated in the Core Idea rather than rely on the title alone. Declare both encodings and the exact numerical parameters used by each magnitude function. State all polynomial bounds separately rather than calling the construction efficient without a cost model. Prove both directions of decision preservation for every legal source instance. Name the strongly NP-complete source restriction and separately verify the target's class membership.
Manages Complexity¶
Pseudo-polynomial transformation manages complexity by replacing a diffuse field of observations or possible operations with a bounded role structure: source problem and encoding supplies a numerical decision problem supplies legal strings, membership, and a declared bit representation.; target problem and encoding supplies a second decision problem receives the constructed instance under its own representation conventions.; magnitude function supplies the largest designated numerical parameter separates value-sensitive cost from ordinary bit length.; instance map supplies a total computable function constructs one target encoding from each source encoding.; membership preservation supplies yes-instances map to yes-instances and no-instances to no-instances..
Abstract Reasoning¶
- Fix source and target languages and reject malformed encodings explicitly. 2. Define the largest-numerical-parameter functions before analyzing runtime. 3. Construct the target instance and prove yes/no equivalence in both directions. 4. Bound construction time in source length and numerical magnitude. 5. Prove the polynomial relation between input and output encoding lengths. 6. Bound every target numerical parameter by a polynomial in source length and magnitude.
Knowledge Transfer¶
The strict upward abstraction is Function Mapping. Pseudo-polynomial Transformation instantiates Function Mapping because it is literally a function from encoded source instances to encoded target instances, specialized by decision preservation and quantitative bounds. Within strong np completeness reductions, the full mechanism transfers literally when the same roles and boundary tests recur. Beyond that domain, only the parent-level skeleton should travel. Reusing the label Pseudo-polynomial transformation after removing its constitutive vocabulary would hide a change of mechanism behind an analogy. The honest transfer rule is therefore two-stage: recognize the domain-specific pattern first, then lift only the parent relation that remains invariant under a substrate change.
Relationships to Other Abstractions¶
Current abstraction Pseudo-polynomial transformation Domain-specific
Parents (1) — more general patterns this builds on
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Pseudo-polynomial transformation is a kind of Function (Mapping) Prime
Pseudo-polynomial Transformation instantiates Function Mapping because it is literally a function from encoded source instances to encoded target instances, specialized by decision preservation and quantitative bounds.
Hierarchy path (1) — routes to 1 parentless root
- Pseudo-polynomial transformation → Function (Mapping)
Neighborhood in Abstraction Space¶
Pseudo-polynomial transformation sits in a sparse region of the domain-specific corpus (82nd percentile for distinctiveness): few abstractions share its structure, so a faithful description tends to retrieve it precisely.
Family — Complexity Reductions & Decidability (10 abstractions)
Nearest neighbors
- PTAS Reduction — 0.82
- Variable-Length Encoding — 0.81
- L-Reduction — 0.81
- Reduction (complexity) — 0.81
- FNP (complexity) — 0.81
Computed from structural-signature embeddings · 2026-09-08