Formalizing Hyperspaces and Operations on Subsets of Polish spaces over Abstract Exact Real Numbers (2410.13508v1)

Abstract: Building on our prior work on axiomatization of exact real computation by formalizing nondeterministic first-order partial computations over real and complex numbers in a constructive dependent type theory, we present a framework for certified computation on hyperspaces of subsets by formalizing various higher-order data types and operations. We first define open, closed, compact and overt subsets for generic spaces in an abstract topological way that allows short and elegant proofs with computational content coinciding with standard definitions in computable analysis and constructive mathematics. From these proofs we can extract programs for testing inclusion, overlapping of sets, et cetera. To enhance the efficiency of the extracted programs, we then focus on Polish spaces, where we give more efficient encodings based on metric properties of the space. As various computational properties depend on the continuity of the encoding functions, we introduce a nondeterministic version of a continuity principle which is natural in our formalization and valid under the standard type-2 realizability interpretation. Using this principle we further derive the computational equivalence between the generic and the metric encodings. Our theory is fully implemented in the Coq proof assistant. From proofs in this Coq formalization, we can extract certified programs for error-free operations on subsets. As an application, we provide a function that constructs fractals in Euclidean space, such as the Sierpinski triangle, from iterated function systems using the limit operation. The resulting programs can be used to draw such fractals up to any desired resolution.

• The paper introduces a formal framework for exact real computation on hyperspaces that enables error-free, certified programs.
• It precisely defines key topological concepts for open, closed, compact, and overt subsets in Polish spaces to optimize metric computations.
• The implementation in Coq demonstrates practical applications, including the generation of fractals like the Sierpinski triangle with arbitrary precision.

Formalizing Hyperspaces and Operations on Subsets of Polish Spaces

This paper presents a comprehensive framework for certified computations on hyperspaces, specifically focusing on subsets of Polish spaces using exact real numbers. It builds on previous work by the authors on the axiomatization of exact real computation within constructive dependent type theory and is fully implemented in the Coq proof assistant. This framework enables the extraction of error-free, certified programs for real computation, particularly when dealing with fractals and similar complex structures.

Key Contributions

1. Formal Definitions of Topological Notions: The paper introduces formal definitions for open, closed, compact, and overt subsets of generic spaces. These definitions are designed to align with standard definitions in computable analysis and constructive mathematics, allowing for the extraction of computational content, such as programs for testing set inclusion and overlapping.
2. Metric Spaces and Polish Spaces: By focusing on Polish spaces, the paper offers more efficient encodings and computational equivalences based on their metric properties. Polish spaces are complete, separable metric spaces, making them a suitable setting for many applications within computable analysis.
3. Continuity Principle: The authors introduce a nondeterministic version of a continuity principle, which is crucial for ensuring computational equivalence between generic and metric encodings. The principle is shown to be valid under standard type-2 realizability interpretations.
4. Compact and Overt Sets: The paper explores the relationships between compactness and overtness. Compact-overt subsets are proven to coincide with Bishop-compactness in Polish spaces, linking computational and classical topological properties.
5. Concrete Applications: Practical implementations include the generation of fractals, such as the Sierpinski triangle, up to any desired resolution. This demonstrates the framework's potential for concrete applications where accuracy and certified computation are paramount.

Numerical Results and Implications

The framework yields certified programs that allow drawing of complex structures with arbitrary precision. Utilizing the Poligh space properties and nondeterministic computations enhances the accuracy and efficiency of operations on subsets.

Future Developments

This research framework opens up multiple avenues for future exploration. Firstly, expanding the scope to include more complex computational operations on function spaces and other Polish spaces is a potential direction. This could encompass aspects like integration and solutions to differential equations. Moreover, optimizing the processes of computing images under transformations—such as polynomial functions—could result in more practical applications.

Conclusion

This paper successfully bridges theoretical developments in exact real computation and constructive mathematics with practical implementations using proof assistants. By formalizing operations on hyperspaces, particularly Polish spaces, the authors have developed a robust framework that combines efficiency with certified correctness. As computational demands grow, particularly in areas requiring high precision, such as computational geometry and fractal analysis, frameworks like this will be indispensable.