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Following recent works connecting two-variable logic to circuits and monoids, we establish, for numerical predicate sets P satisfying a certain closure property, a one-to-one correspondence between F O[<, P]-uniform linear circuits, two-variable formulas with P predicates, and weak block products of monoids. In particular, we consider the case of linear TC 0 , majority quantifiers, and finitely typed monoids. This correspondence will hold for any numerical predicate set which is F O[<]-closed and whose predicates do not depend on the input length.

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