Deterministic finite automaton: Difference between revisions

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Gold's algorithm assumes that <math>S^+</math> and <math>S^-</math> contain a ''characteristic set'' of the regular language; otherwise, the constructed DFA will be inconsistent either with <math>S^+</math> or <math>S^-</math>.

Other notable DFA identification algorithms include the RPNI algorithm,<ref>{{Cite book | doi=10.1142/9789812797902_0004| chapter=Inferring Regular Languages in Polynomial Updated Time| title=Pattern Recognition and Image Analysis| volume=1| pages=49–61| series=Series in Machine Perception and Artificial Intelligence| year=1992| last1=Oncina| first1=J.| last2=García| first2=P.| isbn=978-981-02-0881-3}}</ref> the Blue-Fringe evidence-driven state-merging algorithm,<ref>{{Cite book |doi = 10.1007/BFb0054059|chapter = Results of the Abbadingo one DFA learning competition and a new evidence-driven state merging algorithm|title = Grammatical Inference|volume = 1433|pages = 1–12|series = Lecture Notes in Computer Science|year = 1998|last1 = Lang|first1 = Kevin J.|last2 = Pearlmutter|first2 = Barak A.|last3 = Price|first3 = Rodney A.|isbn = 978-3-540-64776-8|url = http://eprints.maynoothuniversity.ie/10250/1/BP-Results-1998.pdf}}</ref>

Windowed-EDSM.<ref>{{Cite webbook | url=https://dl.acm.org/doi/abs/10.5555/645519.655966 | title=Beyond EDSM &#124; Proceedings of the 6th International Colloquium on Grammatical Inference: Algorithms and Applications| date=23 September 2002| pages=37–48| isbn=9783540442394}}</ref>

Another research direction is the application of [[evolutionary algorithms]]: the smart state labeling evolutionary algorithm<ref>{{Cite journal |doi = 10.1109/TPAMI.2005.143|pmid = 16013754|title = Learning deterministic finite automata with a smart state labeling evolutionary algorithm|journal = IEEE Transactions on Pattern Analysis and Machine Intelligence|volume = 27|issue = 7|pages = 1063–1074|year = 2005|last1 = Lucas|first1 = S.M.|last2 = Reynolds|first2 = T.J.|s2cid = 14062047}}</ref> allowed to solve a modified DFA identification problem in which the training data (sets <math>S^+</math> and <math>S^-</math>) is ''noisy'' in the sense that some words are attributed to wrong classes.

Yet another step forward is due to application of [[Boolean satisfiability problem|SAT]] solvers by [[Marijn Heule|Marjin J. H. Heule]] and S. Verwer: the minimal DFA identification problem is reduced to deciding the satisfiability of a Boolean formula.<ref name=HW1>{{cite conference|last1=Heule|first1=M. J. H.|title=Grammatical Inference: Theoretical Results and Applications |author-link=Marijn Heule|titlechapter=Exact DFA Identification Using SAT Solvers|series=Lecture Notes in Computer Science |conference=Grammatical Inference: Theoretical Results and Applications. ICGI 2010. Lecture Notes in Computer Science|date=2010|volume=6339|pages=66–79|doi=10.1007/978-3-642-15488-1_7|isbn=978-3-642-15487-4 |ref=Heule2010}}</ref> The main idea is to build a augmented prefix-tree acceptor (a [[trie]] containing all input words with corresponding labels) based on the input sets and reduce the problem of finding a DFA with <math>C</math> states to ''coloring'' the tree vertices with <math>C</math> states in such a way that when vertices with one color are merged to one state, the generated automaton is deterministic and complies with <math>S^+</math> and <math>S^-</math>.

Though this approach allows finding the minimal DFA, it suffers from exponential blow-up of execution time when the size of input data increases.

Therefore, Heule and Verwer's initial algorithm has later been augmented with making several steps of the EDSM algorithm prior to SAT solver execution: the DFASAT algorithm.<ref>{{Cite journal |doi = 10.1007/s10664-012-9222-z|title = Software model synthesis using satisfiability solvers|journal = Empirical Software Engineering|volume = 18|issue = 4|pages = 825–856|year = 2013|last1 = Heule|first1 = Marijn J. H.|author-link=Marijn Heule|last2 = Verwer|first2 = Sicco|hdl = 2066/103766|s2cid = 17865020|url = https://lirias.kuleuven.be/handle/123456789/370182|hdl-access = free}}</ref>