B, C, K, W system¶

A basis for combinatory logic using composition, permutation, constant and duplication combinators as primitives.

Version: v1 · 2026-09-08 · History
Domain-specific #: 3381
Origin domain: combinatory logic
Subdomain: combinatory logic

Core Idea¶

The basis is extensionally equivalent in expressive power to SKI, but translation size and reduction behavior depend on chosen encodings and evaluation strategy. Application trees built from B C K and W reduce by their defining rewrite rules, and combinations reconstruct abstraction and any computable lambda term without bound variables. 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¶

B, C, K, W system belongs to combinatory logic and is useful where the analyst can specify the typed combinatory logic carrier, including objects, relations, parameters, conventions, evidence, boundaries, and comparison targets, then evaluate the application syntax and association, four primitive combinators and rewrite equations, reduction strategy, translation from lambda terms or another basis, proof of combinatory completeness and extensionality convention are explicit. The scope is broad within that domain but bounded by the need for the application syntax and association, four primitive combinators and rewrite equations, reduction strategy, translation from lambda terms or another basis, proof of combinatory completeness and extensionality convention are explicit.

Clarity¶

The abstraction clarifies a crowded vocabulary by making the application syntax and association, four primitive combinators and rewrite equations, reduction strategy, translation from lambda terms or another basis, proof of combinatory completeness and extensionality convention 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 B, C, K, W system. B, C, K, W system 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 combinatory logic carrier, including objects, relations, parameters, conventions, evidence, boundaries, and comparison targets. Reject examples whose alleged carrier belongs to a different problem. 2. Lock the constitutive rule. Express the application syntax and association, four primitive combinators and rewrite equations, reduction strategy, translation from lambda terms or another basis, proof of combinatory completeness and extensionality convention are explicit independently of one notation or implementation.

Knowledge Transfer¶

Knowledge transfers strongly among subfields of combinatory logic because they reuse the typed combinatory logic carrier, including objects, relations, parameters, conventions, evidence, boundaries, and comparison targets, Application trees built from B C K and W reduce by their defining rewrite rules, and combinations reconstruct abstraction and any computable lambda term without bound variables., and type the carrier, state every parameter and convention in the definition, test that the application syntax and association, four primitive combinators and rewrite equations, reduction strategy, translation from lambda terms or another basis, proof of combinatory completeness and extensionality convention are explicit, compare the nearest accepted identity, and report counterexamples, uncertainty, and limiting cases.

Relationships to Other Abstractions¶

Current abstraction B, C, K, W system Domain-specific

Parents (1) — more general patterns this builds on

B, C, K, W system is a kind of Formal System Prime

The proposed strict upward parent is prime:formal_system.

Hierarchy paths (2) — routes to 2 parentless roots

B, C, K, W system → Formal System → Formalization → Representation → Abstraction

Show alternative path (1)

Neighborhood in Abstraction Space¶

B, C, K, W system sits in a crowded region of the domain-specific corpus (18^th percentile for distinctiveness): several abstractions share nearly its structure, so a description that fits it tends to fit its neighbors too.

Family — Formal Logic & Type Theory (34 abstractions)

Nearest neighbors

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