1 research outputs found
A General Verification for Functional Completeness by Abstract Operators
An operator set is functionally incomplete if it can not represent the full
set . The
verification for the functional incompleteness highly relies on constructive
proofs. The judgement with a large untested operator set can be inefficient.
Given with a mass of potential operators proposed in various logic systems, a
general verification method for their functional completeness is demanded. This
paper offers an universal verification for the functional completeness.
Firstly, we propose two abstract operators and , both
of which have no fixed form and are only defined by several weak constraints.
Specially, and are the abstract
operators defined with the total order relation . Then, we prove that any
operator set is functionally complete if and only if it can
represent the composite operator or
. Otherwise is
determined to be functionally incomplete. This theory can be generally applied
to any untested operator set to determine whether it is functionally complete.Comment: Under the processing of Annals of Pure and Applied Logi