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Recently, ranking-based semantics is proposed to rank-order arguments from the most acceptable to the weakest one(s), which provides a graded assessment to arguments. In general, the ranking on arguments is derived from the strength values of the arguments. Categoriser function is a common approach that assigns a strength value to a tree of arguments. When it encounters an argument system with cycles, then the categoriser strength is the solution of the non-linear equations. However, there is no detail about the existence and uniqueness of the solution, and how to find the solution (if exists). In this paper, we will cope with these issues via fixed point technique. In addition, we define the categoriser-based ranking semantics in light of categoriser strength, and investigate some general properties of it. Finally, the semantics is shown to satisfy some of the axioms that a ranking-based semantics should satisfy

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Noncanonical cofactor biomimetics (NCBs) such as nicotinamide mononucleotide (NMN+) provide enhanced scalability for biomanufacturing. However, engineering enzymes to accept NCBs is difficult. Here, we establish a growth selection platform to evolve enzymes to utilize NMN+-based reducing power. This is based on an orthogonal, NMN+-dependent glycolytic pathway in Escherichia coli which can be coupled to any reciprocal enzyme to recycle the ensuing reduced NMN+. With a throughput of >106 variants per iteration, the growth selection discovers a Lactobacillus pentosus NADH oxidase variant with ~10-fold increase in NMNH catalytic efficiency and enhanced activity for other NCBs. Molecular modeling and experimental validation suggest that instead of directly contacting NCBs, the mutations optimize the enzyme’s global conformational dynamics to resemble the WT with the native cofactor bound. Restoring the enzyme’s access to catalytically competent conformation states via deep navigation of protein sequence space with high-throughput evolution provides a universal route to engineer NCB-dependent enzymes.