The Subtrace Order and Counting First-Order Logic

We study the subtrace relation among Mazurkiewicz traces which generalizes the much-studied subword order. Here, we consider the 2-variable fragment of a counting extension of first-order logic with regular predicates. It is shown that all definable trace languages are effectively recognizable implying that validity of a sentence of this logic is decidable (this problem is known to be undecidable for virtually all stronger logics already for the subword relation).

[1]  Graham Higman,et al.  Ordering by Divisibility in Abstract Algebras , 1952 .

[2]  J. McKnight.  Kleene quotient theorems. , 1964 .

[3]  Pierre Cartier,et al.  Problemes combinatoires de commutation et rearrangements , 1969 .

[4]  S. Shelah.  The monadic theory of order , 1975, 2305.00968.

[5]  A. Mazurkiewicz.  Concurrent Program Schemes and their Interpretations , 1977 .

[6]  Jean Berstel,et al.  Transductions and context-free languages , 1979, Teubner Studienbücher : Informatik.

[7]  Volker Diekert,et al.  The Book of Traces , 1995 .

[8]  Paul Gastin,et al.  Infinite Traces , 1995, The Book of Traces.

[9]  Philippe Schnoebelen,et al.  Well-structured transition systems everywhere! , 2001, Theor. Comput. Sci..

[10]  P. Madhusudan,et al.  Model-checking trace event structures , 2003, 18th Annual IEEE Symposium of Logic in Computer Science, 2003. Proceedings..

[11]  Dietrich Kuske,et al.  Theories of orders on the set of words , 2006, RAIRO Theor. Informatics Appl..

[12]  R. McKenzie,et al.  Definability in substructure orderings, I: finite semilattices , 2009 .

[13]  Victor L. Selivanov,et al.  Definability in the Subword Order , 2010, CiE.

[14]  Philippe Schnoebelen,et al.  Decidability in the Logic of Subsequences and Supersequences , 2015, FSTTCS.

[15]  A. Wires.  Definability in the Substructure Ordering of Simple Graphs , 2016 .

[16]  Philippe Schnoebelen,et al.  Decidability, complexity, and expressiveness of first-order logic over the subword ordering , 2017, 2017 32nd Annual ACM/IEEE Symposium on Logic in Computer Science (LICS).

[17]  Ramanathan S. Thinniyam.  Defining Recursive Predicates in Graph Orders , 2018, Log. Methods Comput. Sci..

[18]  Simone Tini,et al.  SOS-based Modal Decomposition on Nondeterministic Probabilistic Processes , 2017, Log. Methods Comput. Sci..

[19]  Philippe Schnoebelen,et al.  The height of piecewise-testable languages and the complexity of the logic of subwords , 2015, Log. Methods Comput. Sci..

[20]  Georg Zetzsche,et al.  Languages ordered by the subword order , 2019, FoSSaCS.

[21]  Dietrich Kuske,et al.  Alternating complexity of counting first-order logic for the subword order , 2022, Acta Informatica.