Computational Universality in One-variable Language Equations

It has recently been shown that several computational models, such as trellis automata, recursive functions and Turing machines, admit characterization by resolved systems of language equations with different sets of language-theoretic operations. This paper investigates how simple the systems of equations from the computationally universal types could be while still retaining their universality. It is proved that the universality and the associated hardness of decision problems begins at one-variable equations. Additionally, it is shown that language equations with added quotient with regular languages can specify every set representable in first-order Peano arithmetic.

[1]  Michal Kunc,et al.  Regular solutions of language inequalities and well quasi-orders , 2005, Theor. Comput. Sci..

[2]  Jr. Hartley Rogers.  Theory of Recursive Functions and Effective Computability , 1969 .

[3]  Manfred Kudlek,et al.  A Universal Turing Machine with 3 States and 9 Symbols , 2001, Developments in Language Theory.

[4]  Guo-Qiang Zhang,et al.  Domain µ-calculus , 2003, RAIRO Theor. Informatics Appl..

[5]  Michal Kunc.  The Power of Commuting with Finite Sets of Words , 2005, STACS.

[6]  Brenda S. Baker,et al.  Reversal-Bounded Multipushdown Machines , 1974, J. Comput. Syst. Sci..

[7]  Seymour Ginsburg,et al.  Two Families of Languages Related to ALGOL , 1962, JACM.

[8]  Alexander Okhotin,et al.  On the equivalence of linear conjunctive grammars and trellis automata , 2004, RAIRO Theor. Informatics Appl..

[9]  Alexander Okhotin.  Decision Problems for Language Equations with Boolean Operations , 2003, ICALP.

[10]  Ernst L. Leiss,et al.  Unrestricted Complementation in Language Equations Over a One-Letter Alphabet , 1994, Theor. Comput. Sci..

[11]  Alexander Okhotin.  Conjunctive Grammars and Systems of Language Equations , 2004, Programming and Computer Software.

[12]  Arto Salomaa,et al.  Systolic trellis automatat , 1984 .

[13]  Alexander Okhotin,et al.  On the number of nonterminals in linear conjunctive grammars , 2004, Theor. Comput. Sci..

[14]  Ion Petre,et al.  TWO PROBLEMS ON COMMUTATION OF LANGUAGES , 2004 .

[15]  Lila Kari,et al.  Maximal and Minimal Solutions to Language Equations , 1996, J. Comput. Syst. Sci..

[16]  I. N. Sneddon,et al.  Theory Of Automata , 1969 .

[17]  Alexander Okhotin,et al.  Conjunctive Grammars , 2001, J. Autom. Lang. Comb..

[18]  Jean Berstel,et al.  Context-Free Languages and Pushdown Automata , 1997, Handbook of Formal Languages.

[19]  Henning Fernau.  Nonterminal complexity of programmed grammars , 2003, Theor. Comput. Sci..

[20]  Tiziano Villa,et al.  Solution of parallel language equations for logic synthesis , 2001, IEEE/ACM International Conference on Computer Aided Design. ICCAD 2001. IEEE/ACM Digest of Technical Papers (Cat. No.01CH37281).

[21]  Alexander Okhotin.  On Computational Universality in Language Equations , 2004, MCU.

[22]  Alexander Okhotin.  Boolean grammars , 2004, Inf. Comput..