test

Reference Implementation

A C++ reference implementation accompanies the theoretical framework, demonstrating computational viability. It is provided as supporting evidence; the mathematical claims of the paper are stated and justified in the chapters and do not depend on this implementation.

Overview

This directory provides:

• Core data structure. A class IntervalNumber (src/IntervalNumber.hpp) using double for endpoint storage, with normalization to ensure $x_0 \le x_1$ and propagation of NaN semantics.
• Arithmetic operations. All operations defined in Section 4 with explicit handling of extended-real limits, including the indeterminate forms $0 \cdot \infty$, $\infty - \infty$, and related cases.
• Unit tests. A test suite (src/main.cpp) employing the Google Test framework, validating mathematical properties, edge cases, and the behavior of native IEEE-754 floating-point limits used as a baseline.

Build Instructions

The project uses CMake (version 3.14 or higher).

cd test
cmake -S . -B build
cmake --build build --config Release

Google Test is fetched automatically via CMake's FetchContent module; no manual installation is required.

Running the Tests

The location of the test executable depends on the CMake generator:

# Single-config generators (Ninja, Unix Makefiles)
./build/interval_test          # add .exe on Windows

# Multi-config generators (Visual Studio, Xcode)
./build/Release/interval_test.exe
./build/Debug/interval_test.exe

Alternatively, run the suite via CTest from the build directory:

ctest --test-dir build --output-on-failure -C Release

Verification: Mapping Mathematical Claims to Tests

The 93 unit tests in src/main.cpp verify that the reference implementation conforms to the formal definitions of Chapter 4 and to the stated examples and edge cases. Selected correspondences:

Claim	Reference	Test name(s)
Rule I: $0 \cdot \infty = [0, \infty]_{in}$	§3.4	RuleI_ZeroTimesInfinity
Rule II: $0 \cdot (-\infty) = [-\infty, 0]_{in}$	§3.4	RuleII_ZeroTimesNegativeInfinity
Central thesis identity $0 \cdot \infty = -1 \cdot (0 \cdot (-\infty))$	§6.1	Thesis01, Thesis02
Non-associativity of multiplication	Proposition 5.2	MultiplicationNotAssociative
Combined-operation examples	§6.2	ReadmeExample01, ReadmeExample02
$\tfrac{0}{0} = [-\infty, \infty]_{in}$	§4.5	DivisionByZeroIndeterminateForm
$\tfrac{\infty}{\infty} = [0, \infty]_{in}$	§4.5	DivisionInfinityByInfinity
$0^0 = 1^\infty = \infty^0 = [0, \infty]_{in}$	§4.7	Power_IndeterminateForm_0_pow_0, Power_IndeterminateForm_1_pow_Infinity, Power_IndeterminateForm_Infinity_pow_0

All 93 tests pass under the build instructions above. Test names not appearing in this table cover normalization invariants, arithmetic identities on point intervals, IEEE-754 baseline comparisons, and regression tests for reciprocal/division/abs/pow on edge cases.

Scope and Future Refinements

The reference implementation is intended as a proof of computational viability rather than a production library. Possible extensions include:

• Arbitrary precision arithmetic (e.g., via mpfr or boost::multiprecision) to eliminate floating-point artifacts.
• Symbolic representation to preserve exact algebraic identities through interval operations.
• Language bindings (Python, Julia) for use in scientific computing workflows.
• Formal verification of operation correctness using a proof assistant such as Coq or Lean.