Separating Regular Languages with Two Quantifiers Alternations

We investigate the quantifier alternation hierarchy of first-order logic over finite words. To do so, we rely on the separation problem. For each level in the hierarchy, this problem takes two regular languages as input and asks whether there exists a formula of the level that accepts all words in the first language and no word in the second one. Usually, obtaining an algorithm that solves this problem requires a deep understanding of the level under investigation. We present such an algorithm for the level a#x03A3; 3 (formulas having at most 2 alternations beginning with an existential block). We also obtain as a corollary that one can decide whether a regular language is definable by a a#x03A3; 4 formula (formulas having at most 3 alternations beginning with an existential block).

[1]  Thomas Place.  Separating Regular Languages with Two Quantifiers Alternations , 2015, 2015 30th Annual ACM/IEEE Symposium on Logic in Computer Science.

[2]  Paul Gastin,et al.  First-order definable languages , 2008, Logic and Automata.

[3]  Howard Straubing,et al.  FINITE SEMIGROUP VARIETIES OF THE FORM V,D , 1985 .

[4]  Jean-Éric Pin,et al.  Theme and Variations on the Concatenation Product , 2011, CAI.

[5]  Thomas Place,et al.  Separation for dot-depth two , 2017, 2017 32nd Annual ACM/IEEE Symposium on Logic in Computer Science (LICS).

[6]  Mikolaj Bojanczyk.  Factorization Forests , 2009, Developments in Language Theory.

[7]  Imre Simon,et al.  Piecewise testable events , 1975, Automata Theory and Formal Languages.

[8]  Thomas Place,et al.  Concatenation Hierarchies: New Bottle, Old Wine , 2017, CSR.

[9]  Robert Knast,et al.  A Semigroup Characterization of Dot-Depth one Languages , 1983, RAIRO Theor. Informatics Appl..

[10]  R. McNaughton,et al.  Counter-Free Automata , 1971 .

[11]  Thomas Place,et al.  Adding successor: A transfer theorem for separation and covering , 2017 .

[12]  Thomas Place,et al.  The tale of the quantifier alternation hierarchy of first-order logic over words , 2015, SIGL.

[13]  Jean-Éric Pin.  Bridges for Concatenation Hierarchies , 1998, ICALP.

[14]  Thomas Colcombet.  Factorization forests for infinite words and applications to countable scattered linear orderings , 2010, Theor. Comput. Sci..

[15]  Howard Straubing,et al.  Monoids of upper triangular boolean matrices , 1981 .

[16]  Thomas Place,et al.  Separation and the Successor Relation , 2015, STACS.

[17]  Thomas Place,et al.  Going Higher in the First-Order Quantifier Alternation Hierarchy on Words , 2014, ICALP.

[18]  Imre Simon.  Factorization Forests of Finite Height , 1990, Theor. Comput. Sci..

[19]  Marcel Paul Schützenberger,et al.  On Finite Monoids Having Only Trivial Subgroups , 1965, Inf. Control..

[20]  Pascal Weil,et al.  Polynomial closure and unambiguous product , 1995, Theory of Computing Systems.

[21]  Dominique Perrin,et al.  First-Order Logic and Star-Free Sets , 1986, J. Comput. Syst. Sci..

[22]  Paul Gastin,et al.  A Survey on Small Fragments of First-Order Logic over Finite Words , 2008, Int. J. Found. Comput. Sci..

[23]  Howard Straubing,et al.  A Generalization of the Schützenberger Product of Finite Monoids , 1981, Theor. Comput. Sci..

[24]  Thomas Place,et al.  The Covering Problem: A Unified Approach for Investigating the Expressive Power of Logics , 2016, MFCS.

[25]  Janusz A. Brzozowski,et al.  Dot-Depth of Star-Free Events , 1971, Journal of computer and system sciences (Print).

[26]  Wolfgang Thomas,et al.  Classifying Regular Events in Symbolic Logic , 1982, J. Comput. Syst. Sci..

[27]  Thomas Wilke,et al.  Over words, two variables are as powerful as one quantifier alternation , 1998, STOC '98.

[28]  Thomas Place,et al.  Separating Regular Languages by Piecewise Testable and Unambiguous Languages , 2013, MFCS.

[29]  Wim Martens,et al.  Efficient Separability of Regular Languages by Subsequences and Suffixes , 2013, ICALP.

[30]  Manfred Kufleitner.  The Height of Factorization Forests , 2008, MFCS.

[31]  Howard Straubing,et al.  Semigroups and Languages of Dot-Depth Two , 1988, Theor. Comput. Sci..

[32]  Mustapha Arfi.  Polynomial Operations on Rational Languages , 1987, STACS.

[33]  Denis Thérien,et al.  Classification of Finite Monoids: The Language Approach , 1981, Theor. Comput. Sci..

[34]  Jean-Éric Pin,et al.  The Dot-Depth Hierarchy, 45 Years Later , 2017, The Role of Theory in Computer Science.

[35]  Christian Glaßer,et al.  Languages of Dot-Depth 3/2 , 2000, STACS.

[36]  Janusz A. Brzozowski,et al.  The Dot-Depth Hierarchy of Star-Free Languages is Infinite , 1978, J. Comput. Syst. Sci..