A NEW PROOF OF SCHUTZENBERGER'S THEOREM

2000 ◽  
Vol 10 (02) ◽  
pp. 217-220 ◽  
Author(s):  
PETER M. HIGGINS
Keyword(s):  
Finite Automata ◽  
Forward Direction ◽  
Regular Languages ◽  
Finite Monoid

We prove the theorem of Schutzenberger that a language is star-free if and only if it is recognized by a finite monoid with trivial subgroups. The forward direction follows the lines of the proof of Kleene's Theorem which characterizes regular languages as those recognized by finite automata. The converse is a relatively short induction.

Space-bounded OTMs and REG ∞

Computability ◽  
2021 ◽  
pp. 1-16
Author(s):  
Merlin Carl
Keyword(s):  
Complexity Theory ◽  
Turing Machine ◽  
Finite Automata ◽  
Regular Languages ◽  
The Other ◽  
Deterministic Finite Automata ◽  
Logarithmic Space ◽  
Working Space ◽  
Important Theorem

An important theorem in classical complexity theory is that REG = LOGLOGSPACE, i.e., that languages decidable with double-logarithmic space bound are regular. We consider a transfinite analogue of this theorem. To this end, we introduce deterministic ordinal automata (DOAs) and show that they satisfy many of the basic statements of the theory of deterministic finite automata and regular languages. We then consider languages decidable by an ordinal Turing machine (OTM), introduced by P. Koepke in 2005 and show that if the working space of an OTM is of strictly smaller cardinality than the input length for all sufficiently long inputs, the language so decided is also decidable by a DOA, which is a transfinite analogue of LOGLOGSPACE ⊆ REG; the other direction, however, is easily seen to fail.

Concise representations of regular languages by degree and probabilistic finite automata

1993 ◽  
Vol 26 (4) ◽  
pp. 379-395 ◽  
Author(s):  
Chandra M. R. Kintala ◽  
Kong -Yee Pun ◽  
Detlef Wotschke
Keyword(s):  
Finite Automata ◽  
Regular Languages ◽  
Probabilistic Finite Automata

scholarly journals Nondeterministic Syntactic Complexity

2021 ◽  
pp. 448-468
Author(s):  
Robert S. R. Myers ◽  
Stefan Milius ◽  
Henning Urbat
Keyword(s):  
Finite Automata ◽  
Regular Languages ◽  
Syntactic Complexity ◽  
Deterministic Finite Automata ◽  
Syntactic Monoid ◽  
Nondeterministic Finite Automata ◽  
Structural Work ◽  
Algebraic Interpretation ◽  
Nondeterministic State

AbstractWe introduce a new measure on regular languages: their nondeterministic syntactic complexity. It is the least degree of any extension of the ‘canonical boolean representation’ of the syntactic monoid. Equivalently, it is the least number of states of any subatomic nondeterministic acceptor. It turns out that essentially all previous structural work on nondeterministic state-minimality computes this measure. Our approach rests on an algebraic interpretation of nondeterministic finite automata as deterministic finite automata endowed with semilattice structure. Crucially, the latter form a self-dual category.

UNWEIGHTED AND WEIGHTED HYPER-MINIMIZATION

2012 ◽  
Vol 23 (06) ◽  
pp. 1207-1225 ◽  
Author(s):  
ANDREAS MALETTI ◽  
DANIEL QUERNHEIM
Keyword(s):  
Finite Automata ◽  
Reduction Technique ◽  
Regular Languages ◽  
Minimization Algorithm ◽  
Compression Method ◽  
Closure Properties ◽  
State Reduction ◽  
Deterministic Finite Automata ◽  
Input Strings ◽  
Weighted Finite Automata

Hyper-minimization of deterministic finite automata (DFA) is a recently introduced state reduction technique that allows a finite change in the recognized language. A generalization of this lossy compression method to the weighted setting over semifields is presented, which allows the recognized weighted language to differ for finitely many input strings. First, the structure of hyper-minimal deterministic weighted finite automata is characterized in a similar way as in classical weighted minimization and unweighted hyper-minimization. Second, an efficient hyper-minimization algorithm, which runs in time [Formula: see text], is derived from this characterization. Third, the closure properties of canonical regular languages, which are languages recognized by hyper-minimal DFA, are investigated. Finally, some recent results in the area of hyper-minimization are recalled.

scholarly journals Reversible parallel communicating finite automata systems

2021 ◽  
Vol 58 (4) ◽  
pp. 263-279
Author(s):  
Henning Bordihn ◽  
György Vaszil
Keyword(s):  
Finite Automata ◽  
Regular Languages ◽  
The Other ◽  
Deterministic Case ◽  
Computational Power ◽  
Deterministic Finite Automata ◽  
Relationship Of ◽  
The Relationship ◽  
Context Free

AbstractWe study the concept of reversibility in connection with parallel communicating systems of finite automata (PCFA in short). We define the notion of reversibility in the case of PCFA (also covering the non-deterministic case) and discuss the relationship of the reversibility of the systems and the reversibility of its components. We show that a system can be reversible with non-reversible components, and the other way around, the reversibility of the components does not necessarily imply the reversibility of the system as a whole. We also investigate the computational power of deterministic centralized reversible PCFA. We show that these very simple types of PCFA (returning or non-returning) can recognize regular languages which cannot be accepted by reversible (deterministic) finite automata, and that they can even accept languages that are not context-free. We also separate the deterministic and non-deterministic variants in the case of systems with non-returning communication. We show that there are languages accepted by non-deterministic centralized PCFA, which cannot be recognized by any deterministic variant of the same type.

Regular Languages Definable by Lindström Quantifiers (Preliminary Version)

2002 ◽  
Vol 9 (20) ◽  
Author(s):  
Zoltán Ésik ◽  
Kim G. Larsen
Keyword(s):  
Semidirect Product ◽  
Expressive Power ◽  
Regular Languages ◽  
Finite Monoid ◽  
Preliminary Version

In our main result, we establish a formal connection between Lindström quantifiers with respect to regular languages and the double semidirect product of finite monoid-generator pairs. We use this correspondence to characterize the expressive power of Lindström quantifiers associated with a class of regular languages.<br /><br />Superseded by BRICS-RS-03-28.

Roots and Powers in Regular Languages: Recognizing Nonregular Properties by Finite Automata

2020 ◽  
Vol 175 (1-4) ◽  
pp. 173-185
Author(s):  
Fabian Frei ◽  
Juraj Hromkovič ◽  
Juhani Karhumäki
Keyword(s):  
Lower Bound ◽  
Regular Language ◽  
Finite Automata ◽  
Regular Languages ◽  
Fractional Powers ◽  
Concrete Construction ◽  
Square Roots ◽  
Language L ◽  
Interesting Behavior ◽  
Fractional Length

It is well known that the set of powers of any given order, for example squares, in a regular language need not be regular. Nevertheless, finite automata can identify them via their roots. More precisely, we recall that, given a regular language L, the set of square roots of L is regular. The same holds true for the nth roots for any n and for the set of all nontrivial roots; we give a concrete construction for all of them. Using the above result, we obtain decision algorithms for many natural problems on powers. For example, it is decidable, given two regular languages, whether they contain the same number of squares at each length. Finally, we give an exponential lower bound on the size of automata identifying powers in regular languages. Moreover, we highlight interesting behavior differences between taking fractional powers of regular languages and taking prefixes of a fractional length. Indeed, fractional roots in a regular language can typically not be identified by finite automata.

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