## First-order definable languages (2008)

Venue: | Logic and Automata: History and Perspectives, Texts in Logic and Games |

Citations: | 10 - 3 self |

### BibTeX

@INPROCEEDINGS{Diekert08first-orderdefinable,

author = {Volker Diekert and Paul Gastin},

title = {First-order definable languages},

booktitle = {Logic and Automata: History and Perspectives, Texts in Logic and Games},

year = {2008},

pages = {261--306},

publisher = {Amsterdam University Press}

}

### OpenURL

### Abstract

We give an essentially self-contained presentation of some principal results for first-order definable languages over finite and infinite words. We introduce the notion of a counter-free Büchi automaton; and we relate counter-freeness to aperiodicity and to the notion of very weak alternation. We also show that aperiodicity of a regular ∞-language can be decided in polynomial space, if the language is specified by some Büchi automaton. 1

### Citations

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361 |
On a decision method in restricted second order arithmetic
- Büchi
- 1960
(Show Context)
Citation Context ...ich characterizes unary temporal logic [7] over infinite words. The equivalence between definability in monadic second order logic, regular languages, and acceptance by Büchi automata is due to Büchi =-=[3]-=-. However, Büchi automata are inherently non-deterministic. In order to have deterministic automata one has to move to other acceptance conditions such as Muller or Rabin-Streett conditions. This impo... |

296 |
Decidability of Second-Order Theories and Automata on Infinite Trees
- Rabin
- 1969
(Show Context)
Citation Context ...ry. It relates logic, combinatorics, and algebra to automata theory; and it is widely applied in all branches of computer sciences. Moreover it is the core for generalizations, e.g., to tree automata =-=[26]-=- or to partially ordered structures such as Mazurkiewicz traces [6]. In the present contribution we treat first-order languages over finite and infinite words. First-order definability leads to a subc... |

256 |
Tense Logic and the Theory of Linear Order
- Kamp
- 1968
(Show Context)
Citation Context ...variables. Thus, the implications from 4 to 5 and from 5 to 1 are trivial. The highly non-trivial step is to conclude from 1 (or 2 or 3) to 4. This is usually called Kamp’s Theorem and is due to Kamp =-=[13]-=- and Gabbay, Pnueli, Shelah, and Stavi [9]. In this survey we follow the algebraic proof of Wilke which is in his habilitation thesis [38] and which is also published in [39]. Wilke gave the proof for... |

240 |
On the temporal analysis of fairness
- Gabbay, Pnueli, et al.
- 1980
(Show Context)
Citation Context ... 5 and from 5 to 1 are trivial. The highly non-trivial step is to conclude from 1 (or 2 or 3) to 4. This is usually called Kamp’s Theorem and is due to Kamp [13] and Gabbay, Pnueli, Shelah, and Stavi =-=[9]-=-. In this survey we follow the algebraic proof of Wilke which is in his habilitation thesis [38] and which is also published in [39]. Wilke gave the proof for finite words, only. In order to generaliz... |

230 |
Counter-free automata
- McNaughton, Papert
- 1971
(Show Context)
Citation Context ...antor topology. The truth is that combinatorics on infinite objects is more complicated. The equivalence of first-order definability and star-freeness for finite words is due to McNaughton and Papert =-=[20]-=-. The generalization to infinite words is due to Ladner [15] and Thomas [31, 32]. These results have been refined, e.g. by Perrin and Pin in [24]. Based on the logical framework of Ehrenfeucht-Fraïssé... |

191 |
The book of traces
- Diekert
- 1995
(Show Context)
Citation Context ... and it is widely applied in all branches of computer sciences. Moreover it is the core for generalizations, e.g., to tree automata [26] or to partially ordered structures such as Mazurkiewicz traces =-=[6]-=-. In the present contribution we treat first-order languages over finite and infinite words. First-order definability leads to a subclass of regular languages and again: it relates logic, combinatoric... |

147 |
On a decision method in restricted second-order arithmetics
- Büchi
- 1960
(Show Context)
Citation Context ...temporal logic [7] over infinite words.4 V. Diekert, P. Gastin The equivalence between definability in monadic second order logic, regular languages, and acceptance by Büchi automata is due to Büchi =-=[3]-=-. However, Büchi automata are inherently non-deterministic. In order to have deterministic automata one has to move to other acceptance conditions such as Muller or Rabin-Streett conditions. This impo... |

134 |
On finite monoids having only trivial subgroups
- Schützenberger
- 1965
(Show Context)
Citation Context ..., but also the length of quantifier blocks one gets even finer results as studied by Blanchet-Sadri in [2]. The equivalence of star-freeness and aperiodicity for finite words is due to Schützenberger =-=[28]-=-. The generalization to infinite words is due to Perrin [23] using the syntactic congruence of Arnold [1]. These results are the basis allowing to decide whether a regular language is first-order defi... |

126 | Fast LTL to büchi automata translation
- Gastin, Oddoux
(Show Context)
Citation Context ...abeled with this state then ψ is ultimately true for this word. q.e.d. As we see below, it is easy to transform a very weak alternating automaton into a Büchi automaton. We follow the construction of =-=[11]-=-. However, for this purpose it is convenient to generalize the acceptance conditions. A generalized Büchi automaton is a tuple A = (Q, Σ, δ, I, F, T1, . . .,Tr) where Q is a finite set of states, Σ is... |

97 |
Testing and generating infinite sequences by a finite automaton
- McNaughton
- 1966
(Show Context)
Citation Context ...y non-deterministic. In order to have deterministic automata one has to move to other acceptance conditions such as Muller or Rabin-Streett conditions. This important result is due to McNaughton, see =-=[19]-=-. Based on this, Thomas [32] extended the notion of deterministic counter-free automaton to deterministic counter-free automaton with Rabin-Streett condition and obtained thereby another characterizat... |

87 |
Classifying regular events in symbolic logic
- Thomas
- 1982
(Show Context)
Citation Context ...Erich Grädel, Thomas Wilke (eds.). Logic and Automata: History and Perspectives. Texts in Logic and Games 2, Amsterdam University Press 2007, pp. 261–306.262 V. Diekert, P. Gastin of Wolfgang Thomas =-=[31, 32, 33, 34, 35]-=-. We also refer to his influential contributions in the handbooks of Theoretical Computer Science [36] and of Formal Languages [37]. We do not compete with these surveys. Our plan is more modest. We t... |

86 |
Lower bounds for natural proof systems
- KOZEN
- 1977
(Show Context)
Citation Context ... reader, the test x ∈ h(Σ ∗ ) is PSPACE-hard, in general [10, Problem MS5]. This problem is closely related to the intersection problem of regular languages, where the PSPACE–hardness is due to Kozen =-=[14]-=-.First-order languages 297 • Q is a finite set of states, • Σ is a finite alphabet, • I ∈ B + (Q) is the (alternating) initial condition, • δ : Q × Σ → B + (Q) is the (alternating) transition functio... |

68 |
Alternating automata, the weak monadic theory of trees and its complexity, Theoret
- Muller, Saoudi, et al.
- 1992
(Show Context)
Citation Context ...tion 11 and seems to be an original part in the paper. We have also added a section about very weak alternating automata. The notion of weak alternating automaton is due to Muller, Saoudi, and Schupp =-=[21]-=-. A very weak alternating automaton is a special kind of weak alternating automaton and this notion has been introduced in the PhD thesis of Rhode [27] in a more general context of ordinals. (In the p... |

61 |
Infinite words
- Perrin, Pin
- 2004
(Show Context)
Citation Context ...|M|+|M| 2 classes. This fact can be derived by some standard Ramsey argument about infinite monochromatic subgraphs. We repeat the argument below in order to keep the article self-contained, see also =-=[3, 12, 25]-=-. It shows the existence of a so-called Ramsey factorization. Lemma 5.2. Let h : Σ ∗ → M be a morphism to a finite monoid M and w = u0u1u2 · · · be an infinite word with ui ∈ Σ + for i ≥ 0. Then there... |

30 |
Complexity of some problems from the theory of automata
- Stern
- 1985
(Show Context)
Citation Context ...ing automata characterize firstorder definability as well. More precisely, we have a cycle from 3 to 6 to 7 and back to 3, and we establish a bridge from 4 to 8 and from 8 to 7. It was shown by Stern =-=[29]-=- that deciding whether a deterministic finite automaton accepts an aperiodic language over finite words can be done in polynomial space, i.e., in PSPACE. Later Cho and Huynh showed in [4] that this pr... |

27 |
First-order logic with two variables and unary temporal logic
- Wilke
- 1997
(Show Context)
Citation Context ...sentence can be translated to a formula in FO 3 . This is sharp, because it is known that there are first-order properties which are not expressible in FO 2 , which characterizes unary temporal logic =-=[7]-=- over infinite words.4 V. Diekert, P. Gastin The equivalence between definability in monadic second order logic, regular languages, and acceptance by Büchi automata is due to Büchi [3]. However, Büch... |

27 |
bounds for natural proof systems
- Lower
- 1977
(Show Context)
Citation Context ...ed reader, the test x∈h(Σ ∗ ) is PSPACE-hard, in general [10, Problem MS5]. This problem is closely related to the intersection problem of regular languages, where the PSPACE–hardness is due to Kozen =-=[14]-=-.First-order languages 37 • Q is a finite set of states, • Σ is a finite alphabet, • I ∈ B + (Q) is the (alternating) initial condition, • δ : Q × Σ → B + (Q) is the (alternating) transition function... |

27 |
Application of model theoretic games to discrete linear orders and automata
- Ladner
- 1977
(Show Context)
Citation Context ...objects is more complicated. The equivalence of first-order definability and star-freeness for finite words is due to McNaughton and Papert [20]. The generalization to infinite words is due to Ladner =-=[15]-=- and Thomas [31, 32]. These results have been refined, e.g. by Perrin and Pin in [24]. Based on the logical framework of Ehrenfeucht-Fraïssé-games, Thomas also related the quantifier depth to the so-c... |

27 |
First-order logic and star-free sets
- Perrin, Pin
- 1986
(Show Context)
Citation Context ...eness for finite words is due to McNaughton and Papert [19]. The generalization to infinite words is due to Ladner [15] and Thomas [31, 32]. These results have been refined, e.g. by Perrin and Pin in =-=[24]-=-. Based on the logical framework of Ehrenfeucht-Fraïssé-games, Thomas also related the quantifier depth to the so-called dot-depth hierarchy, [33, 35]. Taking not only the quantifier alternation into ... |

24 | Diamonds are Forever: The Variety DA
- Tesson, Thérien
- 2002
(Show Context)
Citation Context ...ver principle results about the fragmentFirst-order languages 263 of first-order logic which corresponds to unary temporal logic. These diamonds will continue to shine, but not here, and we refer to =-=[30]-=- for more background. As mentioned above, we use Büchi automata, but we do not discuss deterministic models such as deterministic Muller automata. The history of Theorem 1.1 is related to some of the ... |

21 | Alternating automata and logics over infinite words
- Löding, Thomas
- 2000
(Show Context)
Citation Context ...automaton is a special kind of weak alternating automaton and this notion has been introduced in the PhD thesis of Rhode [27] in a more general context of ordinals. (In the paper by Löding and Thomas =-=[17]-=- these automata are called linear alternating.) Section 13 shows that very weak alternating automata characterize firstorder definability as well. More precisely, we have a cycle from 3 to 6 to 7 and ... |

21 |
Finite-automaton aperiodicity is PSPACE-complete
- Cho, Huynh
- 1991
(Show Context)
Citation Context ...wn by Stern [29] that deciding whether a deterministic finite automaton accepts an aperiodic language over finite words can be done in polynomial space, i.e., in PSPACE. Later Cho and Huynh showed in =-=[4]-=- that this problem is actually PSPACE-complete. So, the PSPACE-hardness transfers to (non-deterministic) Büchi automata. It might belong to folklore that the PSPACE-upper bound holds for Büchi automat... |

17 |
On the temporal basis of fairness
- Gabbay, Pnueli, et al.
- 1980
(Show Context)
Citation Context ... 5 and from 5 to 1 are trivial. The highly non-trivial step is to conclude from 1 (or 2 or 3) to 4. This is usually called Kamp’s Theorem and is due to Kamp [13] and Gabbay, Pnueli, Shelah, and Stavi =-=[9]-=-. In this survey we follow the algebraic proof of Wilke which is in his habilitation thesis [38] and which is also published in [39]. Wilke gave the proof for finite words, only. In order to generaliz... |

16 | Pure future local temporal logics are expressively complete for Mazurkiewicz traces
- Diekert, Gastin
- 2004
(Show Context)
Citation Context ...; and it is also widely applied in all branches of computer sciences. Let us mention that first-order definability for Mazurkiewicz traces leads essentially to the same picture as for words (see e.g. =-=[5]-=-), but nice charactizations for first-order definable sets of trees are still missing. The investigation on first-order languages has been of continuous interest over the past decades and many importa... |

13 |
A syntactical congruence for rational ω-languages
- Arnold
- 1985
(Show Context)
Citation Context .... The equivalence of star-freeness and aperiodicity for finite words is due to Schützenberger [28]. The generalization to infinite words is due to Perrin [23] using the syntactic congruence of Arnold =-=[1]-=-. These results are the basis allowing to decide whether a regular language is first-order definable. Putting these results together one sees that statements 1, 2, and 3 in Theorem 1.1 are equivalent.... |

12 | The descriptive complexity approach to LOGCFL
- LAUTEMANN, MCKENZIE, et al.
- 2001
(Show Context)
Citation Context ...ar. On the other hand, there is a quite natural notion of counter-freeness as well as of aperiodicity for non-deterministic Büchi automata. (Aperiodic non-deterministic finite automata are defined in =-=[16]-=-, too.) For non-deterministic automata, aperiodicity describes a larger class of automata, but both counter-freeness and aperiodicity can be used to characterize first-order definable ω-languages. Thi... |

11 |
Infinite traces
- Gastin, Petit
- 1995
(Show Context)
Citation Context ...|M|+|M| 2 classes. This fact can be derived by some standard Ramsey argument about infinite monochromatic subgraphs. We repeat the argument below in order to keep the article self-contained, see also =-=[3, 12, 25]-=-. It shows the existence of a so-called Ramsey factorization. Lemma 5.2. Let h : Σ ∗ → M be a morphism to a finite monoid M and w = u0u1u2 · · · be an infinite word with ui ∈ Σ + for i ≥ 0. Then there... |

11 |
Counter-free automata, The M.I.T
- McNaughton, Papert
- 1971
(Show Context)
Citation Context ...antor topology. The truth is that combinatorics on infinite objects is more complicated. The equivalence of first-order definability and star-freeness for finite words is due to McNaughton and Papert =-=[19]-=-. The generalization to infinite words is due to Ladner [15] and Thomas [31, 32]. These results have been refined, e.g. by Perrin and Pin in [24]. Based on the logical framework of Ehrenfeucht-Fraïssé... |

10 |
Finite automaton aperiodicity is PSPACEcomplete, Theoret
- Cho, Huynh
- 1991
(Show Context)
Citation Context ...wn by Stern [29] that deciding whether a deterministic finite automaton accepts an aperiodic language over finite words can be done in polynomial space, i.e., in PSPACE. Later Cho and Huynh showed in =-=[4]-=- that this problem is actually PSPACE-complete. So, the PSPACE-hardness transfers to (non-deterministic) Büchi automata. It might belong to folklore that the PSPACE-upper bound holds for Büchi automat... |

9 |
An application of the Ehrenfeucht-Fraïssé game in formal language theory. Mémoires de la S.M.F
- Thomas
- 1984
(Show Context)
Citation Context ...Erich Grädel, Thomas Wilke (eds.). Logic and Automata: History and Perspectives. Texts in Logic and Games 2, Amsterdam University Press 2007, pp. 261–306.262 V. Diekert, P. Gastin of Wolfgang Thomas =-=[31, 32, 33, 34, 35]-=-. We also refer to his influential contributions in the handbooks of Theoretical Computer Science [36] and of Formal Languages [37]. We do not compete with these surveys. Our plan is more modest. We t... |

8 |
Recent results on automata and infinite words
- Perrin
- 1984
(Show Context)
Citation Context ...er results as studied by Blanchet-Sadri in [2]. The equivalence of star-freeness and aperiodicity for finite words is due to Schützenberger [28]. The generalization to infinite words is due to Perrin =-=[23]-=- using the syntactic congruence of Arnold [1]. These results are the basis allowing to decide whether a regular language is first-order definable. Putting these results together one sees that statemen... |

6 |
A syntactic congruence for rational omega-language
- Arnold
- 1985
(Show Context)
Citation Context .... The equivalence of star-freeness and aperiodicity for finite words is due to Schützenberger [28]. The generalization to infinite words is due to Perrin [23] using the syntactic congruence of Arnold =-=[1]-=-. These results are the basis allowing to decide whether a regular language is first-order definable. Putting these results together one sees that statements 1, 2, and 3 in Theorem 1.1 are equivalent.... |

5 |
A combinatorial approach to the theory of omegaautomata
- Thomas
- 1981
(Show Context)
Citation Context ...Erich Grädel, Thomas Wilke (eds.). Logic and Automata: History and Perspectives. Texts in Logic and Games 2, Amsterdam University Press 2007, pp. 261–306.262 V. Diekert, P. Gastin of Wolfgang Thomas =-=[31, 32, 33, 34, 35]-=-. We also refer to his influential contributions in the handbooks of Theoretical Computer Science [36] and of Formal Languages [37]. We do not compete with these surveys. Our plan is more modest. We t... |

5 |
A concatenation game and the dot-depth hierarchy
- Thomas
- 1987
(Show Context)
Citation Context |

4 |
Some logical characterizations of the dot-depth hierarchy and applications
- Blanchet-Sadri
- 1995
(Show Context)
Citation Context ... so-called dot-depth hierarchy, [33, 35]. Taking not only the quantifier alternation into account, but also the length of quantifier blocks one gets even finer results as studied by Blanchet-Sadri in =-=[2]-=-. The equivalence of star-freeness and aperiodicity for finite words is due to Schützenberger [28]. The generalization to infinite words is due to Perrin [23] using the syntactic congruence of Arnold ... |

4 |
Star-free regular sets of omega-sequences
- Thomas
- 1979
(Show Context)
Citation Context |

3 |
The local algebras of an associative algebra and their applications
- Lopez, Barroso
- 2002
(Show Context)
Citation Context ...s. We shall use an induction on the size of the monoid M, for this we recall first a construction due to [5]. The construction is very similar to a construction of what is known as local algebra, see =-=[8, 18]-=-. For a moment let M be any monoid and m ∈ M an element. Then mM ∩ Mm is obviously a subsemigroup, but it may not have a neutral element. Hence it is not a monoid, in general. Note that, if m ̸= 1M an... |

2 |
Lectures on algebras and triple systems
- Mayberg
- 1972
(Show Context)
Citation Context ...s. We shall use an induction on the size of the monoid M, for this we recall first a construction due to [5]. The construction is very similar to a construction of what is known as local algebra, see =-=[8, 18]-=-. For a moment let M be any monoid and m ∈ M an element. Then mM ∩ Mm is obviously a subsemigroup, but it may not have a neutral element. Hence it is not a monoid, in general. Note that, if m ̸= 1M an... |

2 |
Lectures on algebras and triple systems
- Meyberg
- 1972
(Show Context)
Citation Context ...phic image of the submonoid N of M. In particular, if M is aperiodic, then (mM ∩ Mm, ◦, m) is aperiodic, too. The construction is very similar to a construction of what is known as local algebra, see =-=[8, 20]-=-. Therefore we call (mM ∩ Mm, ◦, m) the local divisor of M at the element m. 8.2 Closing the cycle Proposition 8.1. We have AP ⊆ LTL. More precisely, let L ⊆ Σ ∞ be a language recognized by an aperiod... |

2 |
Utilisation des automates alternants pour un model-checking efficace des logiques temporelles linéaires
- Oddoux
- 2003
(Show Context)
Citation Context ...3.2. The automata A and A ′ accept the same language. The proof that A and A ′ accept the same language is a little bit technical, but not very hard. Details are left to the reader or can be found in =-=[22]-=-. We now state and prove the converse of Proposition 13.1. Proposition 13.3. Let L ⊆ Σ ∞ be accepted by some very weak alternating automaton. Then L is aperiodic. Proof. Let A = (Q, Σ, δ, I, F, R) be ... |

2 |
Alternating Automata and the temporal logic of ordinals
- Rhode
- 1997
(Show Context)
Citation Context ...g automaton is due to Muller, Saoudi, and Schupp [21]. A very weak alternating automaton is a special kind of weak alternating automaton and this notion has been introduced in the PhD thesis of Rhode =-=[27]-=- in a more general context of ordinals. (In the paper by Löding and Thomas [17] these automata are called linear alternating.) Section 13 shows that very weak alternating automata characterize firstor... |

1 |
The descriptive complexity approach to LOGCFL
- Schwentick, Vollmer
(Show Context)
Citation Context ...ar. On the other hand, there is a quite natural notion of counter-freeness as well as of aperiodicity for non-deterministic Büchi automata. (Aperiodic non-deterministic finite automata are defined in =-=[16]-=-, too.) For non-deterministic automata, aperiodicity describes a larger class of automata, but both counter-freeness and aperiodicity can be used to characterize first-order definable ω-languages. Thi... |

1 |
Infinite traces. In The book of traces
- Gastin, Petit
- 1995
(Show Context)
Citation Context ...|M|+|M| 2 classes. This fact can be derived by some standard Ramsey argument about infinite monochromatic subgraphs. We repeat the argument below in order to keep the article self-contained, see also =-=[3, 12, 25]-=-. It shows the existence of a so-called Ramsey factorization. Lemma 5.2. Let h : Σ ∗ → M be a morphism to a finite monoid M and w = u0u1u2 · · · be an infinite word with ui ∈ Σ + for i ≥ 0. Then there... |