According to Dung, the sets of arguments which can be considered as admissible from an argumentation framework can be regarded as logic models of a given logic program. Clark's completions defines a basic logic programming semantics which has influenced modern non-monotonic semantics such as Answer Set Semantics. The Complete Semantics is a fundamental argumentation semantics which identifies a set of admissible sets which contains the grounded, stable, preferred and ideal semantics. In this paper we introduce a characterization of the complete semantics in terms of logic models using Clark's completions. Given that we use a unique mapping which characterizes the grounded, stable, preferred and ideal semantics, our characterization argues for a strong bridge between argumentation semantics and logic programming semantics with negation as failure. This paper also seeks to draw attention to the correspondence we found between the complete semantics of argumentation frameworks and models of Clark's completion, since this correspondence also allowed us to identify the possibility of computing argumentation frameworks based on integer programming.