Abstract
In this paper, definitions of \({\mathcal{K}}\) automata, \({\mathcal{K}}\) regular languages, \({\mathcal{K}}\) regular expressions and \({\mathcal{K}}\) regular grammars based on lattice-ordered semirings are given. It is shown that \({\mathcal{K}}\)NFA is equivalent to \({\mathcal{K}}\)DFA under some finite condition, the Pump Lemma holds if \({\mathcal{K}}\) is finite, and \({{\mathcal{K}}}\epsilon\)NFA is equivalent to \({\mathcal{K}}\)NFA. Further, it is verified that the concatenation of \({\mathcal{K}}\) regular languages remains a \({\mathcal{K}}\) regular language. Similar to classical cases and automata theory based on lattice-ordered monoids, it is also found that \({\mathcal{K}}\)NFA, \({\mathcal{K}}\) regular expressions and \({\mathcal{K}}\) regular grammars are equivalent to each other when \({\mathcal{K}}\) is a complete lattice.
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Acknowledgments
This work was supported by NSFC Major Research Program 60496324; NSFC No. 6002530760234010, 60603002; Pre-973 Project 2001CCA03000; 863 High-Tech Project 2001AA113130; 973 Project 2001CB312004; CAS Brain and Mind Science Project; China Postdoctoral Science Foundation.
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Lu, X., Shang, Y. & Lu, R. Automata theory based on lattice-ordered semirings. Soft Comput 15, 269–280 (2011). https://doi.org/10.1007/s00500-010-0565-3
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DOI: https://doi.org/10.1007/s00500-010-0565-3