of Third Order

In this work we establish that the language MIX = {w ∈ {a; b; c}||w|a = |w|b = |w|c} and the language O2 = {w ∈ {a; a; b; b}||w|a = |w|a ∧ |w|b = |w|b} are 2-MCFLs. As 2-MCFLs form a class of languages that is included in both the IO and OI hierarchies, and as O2 is the group language of a simple presentation of Z we exhibit here the first, to our knowledge, non-virtually-free group language (i.e. non-context-free group language) that is captured by the IO and OI hierarchies. Moreover, it was a long-standing open problem whether MIX was a mildly context sensitive language or not, and it was conjectured that it was not, so we close this conjecture by giving it a negative answer.

[1]  Arnold L. Rosenberg,et al.  Real-time solutions of the origin-crossing problem , 2005, Mathematical systems theory.

[2]  Seymour Ginsburg,et al.  AFL with the Semilinear Property , 1971, J. Comput. Syst. Sci..

[3]  Geoffrey K. Pullum,et al.  Context-Freeness and the Computer Processing of Human Languages , 1983, ACL.

[4]  R. Ho.  Algebraic Topology , 2022 .

[5]  David E. Muller,et al.  Groups, the Theory of Ends, and Context-Free Languages , 1983, J. Comput. Syst. Sci..

[6]  Aravind K. Joshi,et al.  Natural language parsing: Tree adjoining grammars: How much context-sensitivity is required to provide reasonable structural descriptions? , 1985 .

[7]  Werner Damm,et al.  An Automata-Theoretical Characterization of the OI-Hierarchy , 1986, Inf. Control..

[8]  Makoto Kanazawa,et al.  Generating control languages with Abstract Categorial Grammars , 2007 .

[9]  David J. Weir.  Linear Context-Free Rewriting Systems and Deterministic Tree-Walking Transducers , 1992, ACL.

[10]  Michael J. Fischer,et al.  Grammars with Macro-Like Productions , 1968, SWAT.

[11]  S S.  11 Encoding second order string ACG with deterministic tree walking transducers , 2007 .

[12]  Klaus Aehlig,et al.  Safety Is not a Restriction at Level 2 for String Languages , 2005, FoSSaCS.

[13]  Y. Lafont.  Word Problem , 2019, 99 Variations on a Proof.

[14]  David J. Weir,et al.  The convergence of mildly context-sensitive grammar formalisms , 1990 .

[15]  A. V. Anisimov.  Group languages , 1971 .

[16]  Sylvain Pogodalla,et al.  On the Expressive Power of Abstract Categorial Grammars: Representing Context-Free Formalisms , 2004, J. Log. Lang. Inf..

[17]  Jens Michaelis.  An Additional Observation on Strict Derivational Minimalism , 2009 .

[18]  David J. Weir,et al.  Characterizing mildly context-sensitive grammar formalisms , 1988 .

[19]  Seymour Ginsburg,et al.  Studies In Abstract Families Of Languages , 1969 .

[20]  Pawel Urzyczyn,et al.  Higher-Order Pushdown Trees Are Easy , 2002, FoSSaCS.

[21]  Eric Bach,et al.  Discontinuous constituents in generalized categorial grammars , 1981 .

[22]  Daniel Götzmann.  Multiple Context-Free Grammars , 2007 .

[23]  Andrzej S. Murawski,et al.  Collapsible Pushdown Automata and Recursion Schemes , 2008, 2008 23rd Annual IEEE Symposium on Logic in Computer Science.

[24]  Werner Damm,et al.  The IO- and OI-Hierarchies , 1982, Theor. Comput. Sci..

[25]  Makoto Kanazawa.  The Pumping Lemma for Well-Nested Multiple Context-Free Languages , 2009, Developments in Language Theory.

[26]  Emmon Bach,et al.  Categorial Grammars as Theories of Language , 1988 .

[27]  Michel Latteux,et al.  Cônes rationnels commutatifs , 1979, J. Comput. Syst. Sci..

[28]  Maurice Nivat,et al.  Transduction des langages de Chomsky , 1968 .