STA (Systolic Temporal Arithmetic) was developed earlier by Ling [25] for formal description and reasoning of systolic array designs. In this paper, we present our rule-based expert system, the STA Reasoning System (STARS), for systolic architecture designs. STARS assists users to design systolic array architectures in a rule-based specification language. A refinement of STA which incorporates object orientation is presented. We also describe how a systolic array for LU decomposition can be specified in our system. The techniques used are mathematical induction and backward chaining. The induction technique is adopted to exploit the regularity and locality nature of systolic array architectures. A number of verification tactics are developed in the system. The underlying implementation language is Prolog. Using the powerful symbolic computation ability of Prolog, particularly pattern matching, automatic backtracking, and the depth-first searching rules, the system performs efficiently.