Approximate Axiomatization for Differentially-Defined Functions
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arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866915335206076416 |
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| author | Platzer, André Qian, Long |
| author_facet | Platzer, André Qian, Long |
| contents | This article establishes a complete approximate axiomatization for the real-closed field $\mathbb{R}$ expanded with all differentially-defined functions, including special functions such as $\sin(x), \cos(x), e^x, \dots$. Every true sentence is provable up to some numerical approximation, and the truth of such approximations converge under mild conditions. Such an axiomatization is a fragment of the axiomatization for differential dynamic logic, and is therefore a finite extension of the axiomatization of real-closed fields. Furthermore, the numerical approximations approximate formulas containing special function symbols by $\text{FOL}_{\mathbb{R}}$ formulas, improving upon earlier decidability results only concerning closed sentences. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2506_08233 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Approximate Axiomatization for Differentially-Defined Functions Platzer, André Qian, Long Logic in Computer Science Logic 03B70, 03F03, 65L70, 65G20 F.4.1; F.3.1; G.1.7; I.2.3 This article establishes a complete approximate axiomatization for the real-closed field $\mathbb{R}$ expanded with all differentially-defined functions, including special functions such as $\sin(x), \cos(x), e^x, \dots$. Every true sentence is provable up to some numerical approximation, and the truth of such approximations converge under mild conditions. Such an axiomatization is a fragment of the axiomatization for differential dynamic logic, and is therefore a finite extension of the axiomatization of real-closed fields. Furthermore, the numerical approximations approximate formulas containing special function symbols by $\text{FOL}_{\mathbb{R}}$ formulas, improving upon earlier decidability results only concerning closed sentences. |
| title | Approximate Axiomatization for Differentially-Defined Functions |
| topic | Logic in Computer Science Logic 03B70, 03F03, 65L70, 65G20 F.4.1; F.3.1; G.1.7; I.2.3 |
| url | https://arxiv.org/abs/2506.08233 |