Membership Algorithm for Single-Occurrence Regular Expressions with Shuffle and Counting

Since shuffle introduced into regular expressions makes the membership problem NP-hard, and an efficient membership algorithm facilitates processing many membership-based applications, it is an essential work to devise an efficient membership algorithm for regular expressions which can support shuffle. In this paper, we focus on the membership problem for single-occurrence regular expressions with shuffle and counting (SOREFCs). First, we define single-occurrence finite automata with shuffles and counters (SFA(&,#)s), which can recognize the languages defined by SOREFCs. We prove that the membership problem for SFA(&,#)s is decidable in polynomial time. Then, we devise a membership algorithm for SOREFCs by constructing equivalent SFA(&,#)s. Experimental results demonstrate that our algorithm is efficient in membership checking.