Year: 2024
Type: article
Abstract: Exact real computation is an alternative to floating-point arithmetic where operations on real numbers are performed exactly, without the introduction of rounding errors. When proving the correctness of an implementation, one can focus solely on the mathematical properties of the problem without thinking about the subtleties of representing real numbers. We propose a new axiomatization of the real numbers in a dependent type theory with the goal of extracting certified exact real computation programs from constructive proofs. Our formalization differs from similar approaches, in that we formalize the reals in a conceptually similar way as some mature implementations of exact real computation. Primitive operations on reals can be extracted directly to the corresponding operations in such an implementation, producing more efficient programs. We particularly focus on the formalization of partial and nondeterministic computation, which is essential in exact real computation. We prove the soundness of our formalization with regards to the standard realizability interpretation from computable analysis and show how to relate our theory to a classical formalization of the reals. We demonstrate the feasibility of our theory by implementing it in the Coq proof assistant and present several natural examples. From the examples we have automatically extracted Haskell programs that use the exact real computation framework AERN for efficiently performing exact operations on real numbers. In experiments, the extracted programs behave similarly to native implementations in AERN in terms of running time and memory usage.
Source: Journal of Logic and Computation
Authors: Michal Konečný, Sewon Park, Holger Thies
Institutions: Aston University, Kyoto University
Language: English
Subfield: Artificial Intelligence
Field: Computer Science
Domain: Physical Sciences
Open Access status: green
Awards: Verified exact computation over continuous higher types, Computational complexity of continuous systems, Research on Computable Analysis and Verification of Efficient Exact Real Computation, Computational complexity and practice of verified and efficient algorithms for dynamical systems, G4803253602, 連続な空間上の計算とその複雑さの研究, G8282741467
OpenAlex ID: W4403514846