Thats a unique idea.You should consider the following idea to enhance your idea so that it would impact uniquely: Motivated by Fagin's characterization of NP, Saluja et al. have introduced a logic based frame- work for expressing counting problems. In this setting, a counting problem (seen as a mapping C from structures to non-negative integers) is `defined’ by a first-order sentence phi if for every instance A of the problem, the number of possible satisfying assignments of the variables of phi in A is equal to C(A). The logic RHPI_1 has been introduced by Dyer et al. in their study of the counting complexity class #BIS. The interest in the class #BIS stems from the fact that, it is quite plausible that the problems in #BIS are not #P-hard, nor they admit a fully polynomial randomized approximation scheme. In the present paper  investigate which counting constraint satisfaction problems #CSP(H) are definable in the monotone fragment of RHPI_1.  prove that #CSP(H) is definable in monotone RHPI_1 whenever H is invariant under meet and join operations of a distributive lattice.  prove that the converse also holds if H contains the equality relation.  also prove similar results for counting CSPs expressible by linear Datalog. The results in this case are very similar to those for monotone RHPI1, with the addition that H has, additionally, \top (the greatest element of the lattice) as a polymorphism.
['Andrei A. Bulatov', 'Víctor Dalmau', 'Marc Thurley']
I am writing a research paper on a idea Descriptive Complexity of approximate counting CSPs
[{"from": "human", "value": "I am writing a research paper on a idea Descriptive Complexity of approximate counting CSPs"}, {"from": "gpt", "value": "Thats a unique idea.You should consider the following idea to enhance your idea so that it would impact uniquely: Motivated by Fagin's characterization of NP, Saluja et al. have introduced a logic based frame- work for expressing counting problems. In this setting, a counting problem (seen as a mapping C from structures to non-negative integers) is `defined\u2019 by a first-order sentence phi if for every instance A of the problem, the number of possible satisfying assignments of the variables of phi in A is equal to C(A). The logic RHPI_1 has been introduced by Dyer et al. in their study of the counting complexity class #BIS. The interest in the class #BIS stems from the fact that, it is quite plausible that the problems in #BIS are not #P-hard, nor they admit a fully polynomial randomized approximation scheme. In the present paper  investigate which counting constraint satisfaction problems #CSP(H) are definable in the monotone fragment of RHPI_1.  prove that #CSP(H) is definable in monotone RHPI_1 whenever H is invariant under meet and join operations of a distributive lattice.  prove that the converse also holds if H contains the equality relation.  also prove similar results for counting CSPs expressible by linear Datalog. The results in this case are very similar to those for monotone RHPI1, with the addition that H has, additionally, \\top (the greatest element of the lattice) as a polymorphism."}]