Simply typed convertibility is TOWER-complete even for safe lambda-terms
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2023
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| _version_ | 1866913494590291968 |
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| author | Nguyên, Lê Thành Dũng |
| author_facet | Nguyên, Lê Thành Dũng |
| contents | We consider the following decision problem: given two simply typed $λ$-terms, are they $β$-convertible? Equivalently, do they have the same normal form? It is famously non-elementary, but the precise complexity - namely TOWER-complete - is lesser known. One goal of this short paper is to popularize this fact.
Our original contribution is to show that the problem stays TOWER-complete when the two input terms belong to Blum and Ong's safe $λ$-calculus, a fragment of the simply typed $λ$-calculus arising from the study of higher-order recursion schemes. Previously, the best known lower bound for this safe $β$-convertibility problem was PSPACE-hardness. Our proof proceeds by reduction from the star-free expression equivalence problem, taking inspiration from the author's work with Pradic on "implicit automata in typed $λ$-calculi".
These results also hold for $βη$-convertibility. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2305_12601 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Simply typed convertibility is TOWER-complete even for safe lambda-terms Nguyên, Lê Thành Dũng Logic in Computer Science Programming Languages We consider the following decision problem: given two simply typed $λ$-terms, are they $β$-convertible? Equivalently, do they have the same normal form? It is famously non-elementary, but the precise complexity - namely TOWER-complete - is lesser known. One goal of this short paper is to popularize this fact. Our original contribution is to show that the problem stays TOWER-complete when the two input terms belong to Blum and Ong's safe $λ$-calculus, a fragment of the simply typed $λ$-calculus arising from the study of higher-order recursion schemes. Previously, the best known lower bound for this safe $β$-convertibility problem was PSPACE-hardness. Our proof proceeds by reduction from the star-free expression equivalence problem, taking inspiration from the author's work with Pradic on "implicit automata in typed $λ$-calculi". These results also hold for $βη$-convertibility. |
| title | Simply typed convertibility is TOWER-complete even for safe lambda-terms |
| topic | Logic in Computer Science Programming Languages |
| url | https://arxiv.org/abs/2305.12601 |