E. GR)DEL AND M. OTTo, Inductive definability with counting on finite structures, in Com- puter Science Logic, 6th Workshop, CSL '92, San Miniato  Home/Search   Document Details and Download   Summary   Related Articles

....x (over the first sort) among its free variables. Then x[9] is a 55 term in the second sort, with the set of free variables free(#x[qO] free(p) z . The value of #x[qo] is the number of elements a that satisfy qo(a) Counting logics of this form have been introduced by Gr idel and Otto [49] and been studied in detail in [82] We start with first order logic with counting, denoted (FO C) which is the closure of two sorted first order logic under counting terms. Here are two simple examples that illustrate the use of counting terms. Example 4.2. On a undirected graph G = V,E) ....

.... quantifiers rather than counting terms) The equivalence of LFP and IFP readily translates to (LFP C) IFP C) Further, there are a number of other logical formalizations of the concept inductive definability with counting that turn out to have the same expressive power as (IFP C) see [49] and the paragraph on Datalog with counting below for details) 56 Example 4.5. An interesting example for a (IFP C) definable query is the method of stable coloutings for graph canonization. Given a graph G with a colouring f : V 0, r of its vertices, we define a refinement f of f, ....

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E. GR)DEL AND M. OTTo, Inductive definability with counting on finite structures, in Com- puter Science Logic, 6th Workshop, CSL '92, San Miniato

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