Thats a unique idea.You should consider the following idea to enhance your idea so that it would impact uniquely:  study Gaifman and Hanf locality of an extension of first-order logic with modulo p counting quantifiers (FO+MODp, for short) with arbitrary numerical predicates.  require that the validity of formulas is independent of the particular interpretation of the numerical predicates and refer to such formulas as arb-invariant formulas.  gives a detailed picture of locality and non-locality properties of arb-invariant FO+MODp. For example, on the class of all finite structures, for any p >= 2, arb-invariant FO+MODp is neither Hanf nor Gaifman local with respect to a sublinear locality radius. Hover, in case that p is an odd prime por, it is akly Gaifman local with a polylogarithmic locality radius. And when restricting attention to the class of string structures, for odd prime pors p, arb-invariant FO+MODp is both Hanf and Gaifman local with a polylogarithmic locality radius. Our negative results build on examples of order-invariant FO+MODp formulas presented in Niemisto's PhD thesis. Our positive results make use of the close connection beten FO+MODp and Boolean circuits built from NOT-gates and AND-, OR-, and MODp-gates of arbitrary fan-in.
['Frederik Harwath', 'Nicole Schweikardt']
I am writing a research paper on a idea On the locality of arb-invariant first-order logic with modulo counting quantifiers
[{"from": "human", "value": "I am writing a research paper on a idea On the locality of arb-invariant first-order logic with modulo counting quantifiers"}, {"from": "gpt", "value": "Thats a unique idea.You should consider the following idea to enhance your idea so that it would impact uniquely:  study Gaifman and Hanf locality of an extension of first-order logic with modulo p counting quantifiers (FO+MODp, for short) with arbitrary numerical predicates.  require that the validity of formulas is independent of the particular interpretation of the numerical predicates and refer to such formulas as arb-invariant formulas.  gives a detailed picture of locality and non-locality properties of arb-invariant FO+MODp. For example, on the class of all finite structures, for any p >= 2, arb-invariant FO+MODp is neither Hanf nor Gaifman local with respect to a sublinear locality radius. Hover, in case that p is an odd prime por, it is akly Gaifman local with a polylogarithmic locality radius. And when restricting attention to the class of string structures, for odd prime pors p, arb-invariant FO+MODp is both Hanf and Gaifman local with a polylogarithmic locality radius. Our negative results build on examples of order-invariant FO+MODp formulas presented in Niemisto's PhD thesis. Our positive results make use of the close connection beten FO+MODp and Boolean circuits built from NOT-gates and AND-, OR-, and MODp-gates of arbitrary fan-in."}]