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This paper deals with the calculations performed in the reasoning process of rule-based expert systems, where inference chains are applied. It presents a logic model for representing the rules and the rule base of a given system. Also, the fact base of the same expert system is involved in the logic model. The proposed equivalent representation manifests itself in a logic network. After that, a four-valued logic algebra is introduced. This algebra is used for the calculations where forward chaining is carried out. Next, the notion of line-value justification is described. This operation is applied in the backward chaining process, also on the base of the previously introduced four-valued logic. The paper describes two exact algorithms which serve for the forward and backward chaining processes. These algorithms make it possible to be implemented by a computer program, resulting in an efficient inference engine of an expert system. The achieved result enhances the reliability and usability of the intelligent software systems which is extremely important in embedded environments.