When do homomorphism counts help in query algorithms?
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866916090627489792 |
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| author | Cate, Balder ten Dalmau, Víctor Kolaitis, Phokion G. Wu, Wei-Lin |
| author_facet | Cate, Balder ten Dalmau, Víctor Kolaitis, Phokion G. Wu, Wei-Lin |
| contents | A query algorithm based on homomorphism counts is a procedure for determining whether a given instance satisfies a property by counting homomorphisms between the given instance and finitely many predetermined instances. In a left query algorithm, we count homomorphisms from the predetermined instances to the given instance, while in a right query algorithm we count homomorphisms from the given instance to the predetermined instances. Homomorphisms are usually counted over the semiring N of non-negative integers; it is also meaningful, however, to count homomorphisms over the Boolean semiring B, in which case the homomorphism count indicates whether or not a homomorphism exists. We first characterize the properties that admit a left query algorithm over B by showing that these are precisely the properties that are both first-order definable and closed under homomorphic equivalence. After this, we turn attention to a comparison between left query algorithms over B and left query algorithms over N. In general, there are properties that admit a left query algorithm over N but not over B. The main result of this paper asserts that if a property is closed under homomorphic equivalence, then that property admits a left query algorithm over B if and only if it admits a left query algorithm over N. In other words and rather surprisingly, homomorphism counts over N do not help as regards properties that are closed under homomorphic equivalence. Finally, we characterize the properties that admit both a left query algorithm over B and a right query algorithm over B. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2304_06294 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | When do homomorphism counts help in query algorithms? Cate, Balder ten Dalmau, Víctor Kolaitis, Phokion G. Wu, Wei-Lin Logic in Computer Science F.4.1; G.2.2; I.1.2 A query algorithm based on homomorphism counts is a procedure for determining whether a given instance satisfies a property by counting homomorphisms between the given instance and finitely many predetermined instances. In a left query algorithm, we count homomorphisms from the predetermined instances to the given instance, while in a right query algorithm we count homomorphisms from the given instance to the predetermined instances. Homomorphisms are usually counted over the semiring N of non-negative integers; it is also meaningful, however, to count homomorphisms over the Boolean semiring B, in which case the homomorphism count indicates whether or not a homomorphism exists. We first characterize the properties that admit a left query algorithm over B by showing that these are precisely the properties that are both first-order definable and closed under homomorphic equivalence. After this, we turn attention to a comparison between left query algorithms over B and left query algorithms over N. In general, there are properties that admit a left query algorithm over N but not over B. The main result of this paper asserts that if a property is closed under homomorphic equivalence, then that property admits a left query algorithm over B if and only if it admits a left query algorithm over N. In other words and rather surprisingly, homomorphism counts over N do not help as regards properties that are closed under homomorphic equivalence. Finally, we characterize the properties that admit both a left query algorithm over B and a right query algorithm over B. |
| title | When do homomorphism counts help in query algorithms? |
| topic | Logic in Computer Science F.4.1; G.2.2; I.1.2 |
| url | https://arxiv.org/abs/2304.06294 |