Quotient Tree Arithmetic: Deferred-Division Computation with Bounded Symbolic Depth and Cross-Subtree Cancellation
A new computational framework called Quotient Tree Arithmetic (QTA) is introduced, where values are represented as deferred quotient pairs (N, D) evaluated lazily at a materialization boundary. The framework supports exact rational arithmetic within IEEE 754 doubles' 2^53 exactness window and extends to transcendental values like exp(x) and sqrt(x). Three structural theorems underpin QTA: Bounded Depth Growth limits tree depth increase to at most 1 per operation, ensuring O(m) size after m operations; Cross-Subtree Cancellation allows subtrees in both numerator and denominator to cancel via reference identity without arithmetic; Deferred Stability ensures a single IEEE division at the materialization boundary yields stable results. This approach avoids combinatorial explosion and enables efficient computation with shared transcendental values.
Key facts
- Quotient Tree Arithmetic (QTA) represents values as deferred quotient pairs (N, D).
- Evaluation is lazy at a designated materialization boundary.
- Works with IEEE 754 doubles as exact integer containers within 2^53 window.
- Extends to transcendental values including exp(x) and sqrt(x).
- Bounded Depth Growth theorem: each operation increases tree depth by at most 1.
- Cross-Subtree Cancellation: subtrees in numerator and denominator cancel via reference identity.
- Deferred Stability: single IEEE division at materialization boundary.
- No combinatorial explosion; O(m) tree size after m operations.
Entities
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