Deterministic finite automaton: Difference between revisions

Revision as of 08:01, 12 April 2023

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* the DFA with a minimum number of states for a particular regular language (Minimization Problem)

DFAs are equivalent in computing power to [[nondeterministic finite automata]] (NFAs). This is because, firstly any DFA is also an NFA, so an NFA can do what a DFA can do. Also, given an NFA, using the [[powerset construction]] one can build a DFA that recognizes the same language as the NFA, although the DFA could have exponentially larger number of states than the NFA.<ref name=Sak105>Sakarovitch (2009) p.105</ref><ref name=Law63>Lawson (2004) p.63</ref> However, even though NFAs are computationally equivalent to DFAs, the above -mentioned problems are not necessarily solved efficiently also for NFAs. The non-universality problem for NFAs is [[PSPACE complete]] since there are small NFAs with shortest rejecting word in exponential size. A DFA is universal if and only if all states are final states, but this does not hold for NFAs. The Equality, Inclusion and Minimization Problems are also PSPACE complete since they require forming the complement of an NFA which results in an exponential blow up of size.<ref>https://www7.in.tum.de/um/courses/auto/ws1718/slides1718/04-Implementations_sets.pdf {{Bare URL PDF|date=March 2022}}</ref>

On the other hand, finite-state automata are of strictly limited power in the languages they can recognize; many simple languages, including any problem that requires more than constant space to solve, cannot be recognized by a DFA. The classic example of a simply described language that no DFA can recognize is bracket or [[Dyck language]], i.e., the language that consists of properly paired brackets such as word "(()())". Intuitively, no DFA can recognize the Dyck language because DFAs are not capable of counting: a DFA-like automaton needs to have a state to represent any possible number of "currently open" parentheses, meaning it would need an unbounded number of states. Another simpler example is the language consisting of strings of the form ''aⁿbⁿ'' for some finite but arbitrary number of ''a''&apos;s, followed by an equal number of ''b''&apos;s.<ref name=Law46>Lawson (2004) p.46</ref>

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Another research direction is the application of [[evolutionary algorithm]]s: 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|chapter=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 aan 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>