A generic characterization of generalized unary temporal logic and two-variable first-order logic
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We investigate an operator on classes of languages. For each class C, it outputs a new class FO^2(I_C) associated with a variant of two-variable first-order logic equipped with a signatureI_C built from C. For C = {∅, A^*}, we get the variant FO^2(<) equipped with the linear order. For C = {∅, {ε},A^+, A^*}, we get the variant FO^2(<,+1), which also includes the successor. If C consists of all Boolean combinations of languages A^*aA^* where a is a letter, we get the variant FO^2(<,Bet), which also includes “between relations”. We prove a generic algebraic characterization of the classes FO^2(I_C). It smoothly and elegantly generalizes the known ones for all aforementioned cases. Moreover, it implies that if C has decidable separation (plus mild properties), then FO^2(I_C) has a decidable membership problem. We actually work with an equivalent definition of in terms of unary temporal logic. For each class C, we consider a variant TL(C) of unary temporal logic whose future/past modalities depend on C and such that TL(C) = FO^2(I_C). Finally, we also characterize FL(C) and PL(C), the pure-future and pure-past restrictions of TL(C). These characterizations as well imply that if is a class with decidable separation, then FL(C) and PL(C) have decidable membership.
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