A Note on Mod and Generalised Mod Classes
نویسندگان
چکیده
We characterise Mod classes in terms of #P functions, where the membership is determined by co-primality or gcd testing of the function value (Theorem 3.1), instead of residue (mod k) testing. Imposing a restriction on the range of the functions gives a characterisation of the intersection of Mod classes (Theorem 3.2). These intersection classes, which we denote by Mod ∩k P , are interesting because they share most of the “nice” properties (closure under complementation, normal forms, lowness for itself etc) of ModpP for prime p. We show that the class Mod ∩k P is low for ModkP , and also for Mod ∩k P itself (Theorem 3.3). We also strengthen some of the separation results known for Mod classes. A diagonalisation argument due to Beigel shows that when k is a prime not dividing j, ModjP can be separated from ModkP in some relativised world. We observe that this argument even separates Mod ∩j P from ModkP under the same conditions (Theorem 4.1). Further, if k is not known to be prime, the same argument still diagonalises, but out of a smaller class; it separates Mod ∩j P from Mod ∩k P (Theorem 4.2). The class ModP was defined in [6] as a generalisation of the Mod classes. We define a simple generalisation, ModKP , and show that it coincides with the disjunctive truth table closure of ModP , P dtt (Theorem 5.2). We give neat characterisations of P ModP dtt and P ModP ctt (Theorem 5.3), and also a new characterisation of ModP (Theorem 5.4). The results of section 5 thus give us an overall picture of the relations between the generalised Mod classes as shown in Figure 1. Arrows denote containment, and connections tagged coindicate that the corresponding classes are the Co-classes of each other.
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ورودعنوان ژورنال:
- Inf. Process. Lett.
دوره 55 شماره
صفحات -
تاریخ انتشار 1995