Hi HN, I want to share an interesting meta-mathematical concept — Embedded Inverser (Ei). It is a universal inversion operator that works at the structural level of an expression and automatically generates its mirrored version.
Features of Ei:
Automatic inversion: Ei detects all operators and the structure of the expression on its own.
Type-independent: works with numbers, strings, arrays, boolean expressions, sets, and even algorithms.
Deterministic: each operation receives a formally defined mirrored counterpart or a MAX result if no direct analog exists.
Structure-preserving: nesting, parentheses, and functions are automatically maintained.
Formal idea:
Let E be any expression composed of operations and operands. The low-level inversion Ei is defined as an operator I acting on the structure of the expression:
I(E) = E[I(O1), I(O2), …, I(On)]
where I(Oi) = mirrored operation Oi or MAX if the operation is unique and has no direct counterpart.
Examples of inversions:
Boolean operators: NOT → TRUE, AND → OR, OR → AND, XOR → XNOR
Sets: UNION → INTERSECT, INTERSECT → UNION, BACKSLASH → symmetric difference
Numbers: + → *, > → <, MAX → MIN
Arrays and strings: append → prepend, sort → reverse sort, concat → reverse concat
Example of Ei in action:
Input: NOT A AND B OR C
Ei result: TRUE A OR B AND C
Structure is preserved, operators are automatically inverted, operand types don’t matter.
Applications of Ei:
Boolean algebra and combinatorics: analyzing symmetries, generating alternative logic circuits
Set theory: building mirrored expressions, finding antimorphisms
Linear algebra: mirrored matrix operations, alternative computation algorithms
Algebraic structures: analyzing symmetries in groups, rings, fields
Information theory and cryptography: alternative encryption schemes, pattern analysis
Automated testing and AI: generating mirrored algorithm variants for code stability checks
Ei is a universal inversion formula that works without user intervention, automatically determines mirrored operations, preserves expression structure, and can be applied in mathematics, programming, cryptography, combinatorial design, AI, and algorithm testing.
Comments URL: https://news.ycombinator.com/item?id=45273848
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Source: news.ycombinator.com
