A Monoidal View on Fixpoint Checks
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arXiv
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| Hauptverfasser: | , , , , |
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| Format: | Preprint |
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2023
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| _version_ | 1866916856540954624 |
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| author | Baldan, Paolo Eggert, Richard König, Barbara Matt, Timo Padoan, Tommaso |
| author_facet | Baldan, Paolo Eggert, Richard König, Barbara Matt, Timo Padoan, Tommaso |
| contents | Fixpoints are ubiquitous in computer science as they play a central role in providing a meaning to recursive and cyclic definitions. Bisimilarity, behavioural metrics, termination probabilities for Markov chains and stochastic games are defined in terms of least or greatest fixpoints. Here we show that our recent work which proposes a technique for checking whether the fixpoint of a function is the least (or the largest) admits a natural categorical interpretation in terms of gs-monoidal categories. The technique is based on a construction that maps a function to a suitable approximation. We study the compositionality properties of this mapping and show that under some restrictions it can naturally be interpreted as a (lax) gs-monoidal functor. This guides the development of a tool, called UDEfix that allows us to build functions (and their approximations) like a circuit out of basic building blocks and subsequently perform the fixpoints checks. We also show that a slight generalisation of the theory allows one to treat a new relevant case study: coalgebraic behavioural metrics based on Wasserstein liftings. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2305_02957 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | A Monoidal View on Fixpoint Checks Baldan, Paolo Eggert, Richard König, Barbara Matt, Timo Padoan, Tommaso Logic in Computer Science Fixpoints are ubiquitous in computer science as they play a central role in providing a meaning to recursive and cyclic definitions. Bisimilarity, behavioural metrics, termination probabilities for Markov chains and stochastic games are defined in terms of least or greatest fixpoints. Here we show that our recent work which proposes a technique for checking whether the fixpoint of a function is the least (or the largest) admits a natural categorical interpretation in terms of gs-monoidal categories. The technique is based on a construction that maps a function to a suitable approximation. We study the compositionality properties of this mapping and show that under some restrictions it can naturally be interpreted as a (lax) gs-monoidal functor. This guides the development of a tool, called UDEfix that allows us to build functions (and their approximations) like a circuit out of basic building blocks and subsequently perform the fixpoints checks. We also show that a slight generalisation of the theory allows one to treat a new relevant case study: coalgebraic behavioural metrics based on Wasserstein liftings. |
| title | A Monoidal View on Fixpoint Checks |
| topic | Logic in Computer Science |
| url | https://arxiv.org/abs/2305.02957 |