In this work, we consider the satisfiability problem in a logic that combines word equations over string variables denoting words of unbounded lengths, regular languages to which words belong and Pres-burger constraints on the length of words. We present a novel decision procedure over two decidable fragments that include quadratic word equations (i.e., each string variable occurs at most twice). The proposed procedure reduces the problem to solving the satisfiability in the Pres-burger arithmetic. The procedure combines two main components: (ⅰ) an algorithm to derive a complete set of all solutions of conjunctions of word equations and regular expressions; and (ⅱ) two methods to precisely compute relational constraints over string lengths implied by the set of all solutions. We have implemented a prototype tool and evaluated it over a set of satisfiability problems in the logic. The experimental results show that the tool is effective and efficient.
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