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Register a new userA General Verification for Functional Completeness by Abstract Operators
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Yang Tian
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An operator set is functionally incomplete if it can not represent the full set $lbrace eg,vee,wedge,rightarrow,leftrightarrowrbrace$. 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 $widehat{R}$ and $breve{R}$, both of which have no fixed form and are only defined by several weak constraints. Specially, $widehat{R}_{geq}$ and $breve{R}_{geq}$ are the abstract operators defined with the total order relation $geq$. Then, we prove that any operator set $mathfrak{R}$ is functionally complete if and only if it can represent the composite operator $widehat{R}_{geq}circbreve{R}_{geq}$ or $breve{R}_{geq}circwidehat{R}_{geq}$. Otherwise $mathfrak{R}$ is determined to be functionally incomplete. This theory can be generally applied to any untested operator set to determine whether it is functionally complete.
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