representational reasoning

an encoding of algorithmic identity provides a language-independent description of a method. sufficient to distinguish algorithmic families such as quicksort versus mergesort. it abstracts from machine details and retains only defining structure, recurrence, or partitioning.

good encodings are judged by:

• clarity of essential steps
• omission of incidental implementation detail

• represent domains abstractly when only their properties matter.
• a concrete representation is needed only if the algorithm depends on those specifics.
• example: quicksort requires a total order but not the bit width of elements.
• if overflow or rounding affects correctness, drop the abstraction and specify concretely.
• the principle is to keep abstraction at the level of properties essential to identity, refining only when representation influences correctness.

• operational semantics: state transitions, imperative pseudocode, abstract state machines
• denotational semantics: input - output mappings, recurrences, set-theoretic relations
• algebraic semantics: equations and laws, algebraic specifications, horn clauses

• begin from a high-level specification that defines identity, e.g. "permutation and sortedness."
• refine stepwise into a partition-based recurrence, then pseudocode, then machine code.
• each refinement preserves correctness; the refined form implies the original.

• natural language description
• functions (functional form as canonical)
• imperative pseudocode
• mathematical recurrence equations
• horn clauses / logic programming
• algebraic specifications (equational logic)
• set-theoretic relations (input/output and invariants)
• category-theoretic combinators (folds, catamorphisms)
• abstract state machines
• type-theoretic or refinement-type specifications
• flowcharts or control-flow graphs
• state-machine morphisms (mealy or moore; specific reactive subclass)

• identify orthogonal primary variables {v₁, …, vₙ} spanning the conceptual space.
• for each variable, define orthogonal subvariables to refine coverage.
• incomplete vectors reveal missing dimensions; formulate questions that resolve them.

find the smallest high-leverage fact set that yields broad understanding. use a "20-questions" framing: each question carves the search space, reduces ambiguity, and links categories.

recognize that there can exist triplets where any two elements determine the third.

this structure:

• guarantees completeness as any two imply a unique third
• enables inversion-based reasoning: reverse execution, synthesis, recovery
• supports bidirectionality by swapping knowns and unknowns symmetrically
• detects underdetermination early, single known yields multiple completions
• supports automation as one relation serves multiple tasks by permutation

• examples:

  • (query, dataset) -> result

  • (initial, rules) -> final

  • (program, input) -> output

• evaluate options against objectives, constraints, and trade-offs.
• choose optimal paths

• create tuples of variables to map conceptual space.
• build word clusters, idea lists, vocabulary and keyword sets, and use cleverly chosen group headings to form categories.