Resolution of The Linear-Bounded Automata Question

This work resolves a longstanding open question in automata theory, i.e., the linear − boundedautomataquestion ( LBA question), which can also be phrased succinctly in the language of computational complexity theory as DSPACE [ n ] ? = NSPACE [ n ]. We show first that DSPACE [ n ] ̸ = NSPACE [ n ]. Our proof technique is primarily based on diagonalization by a universal nondeterministic n space-bounded Turing machine against all deterministic n space-bounded Turing machines. Then, we show by padding argument that DSPACE [log n ] ̸ = NSPACE [log n ], which also resolves a famous and longstanding open question. Our proof also implies the following fundamental consequences:

[1]  T. Preu.  Refuting Tianrong Lin's arXiv: 2110.05942 "Resolution of The Linear-Bounded Automata Question" , 2021, ArXiv.

[2]  Tianrong Lin.  Diagonalization of Polynomial-Time Deterministic Turing Machines Via Nondeterministic Turing Machine , 2021, 2110.06211.

[3]  A. Turing.  On Computable Numbers, with an Application to the Entscheidungsproblem. , 1937 .

[4]  Aaron Potechin,et al.  Bounds on Monotone Switching Networks for Directed Connectivity , 2009, 2010 IEEE 51st Annual Symposium on Foundations of Computer Science.

[5]  Sanjeev Arora,et al.  Computational Complexity: A Modern Approach , 2009 .

[6]  Omer Reingold,et al.  Undirected connectivity in log-space , 2008, JACM.

[7]  Avi Wigderson,et al.  Entropy waves, the zig-zag graph product, and new constant-degree expanders and extractors , 2000, Proceedings 41st Annual Symposium on Foundations of Computer Science.

[8]  Pascal Michel,et al.  A Survey of Space Complexity , 1992, Theor. Comput. Sci..

[9]  Desh Ranjan,et al.  Space Bounded Computations: Review and New Separation Results , 1991, Theor. Comput. Sci..

[10]  Viliam Geffert,et al.  Nondeterministic Computations in Sublogarithmic Space and Space Constructibility , 1990, SIAM J. Comput..

[11]  Róbert Szelepcsényi,et al.  The method of forced enumeration for nondeterministic automata , 1988, Acta Informatica.

[12]  Neil Immerman,et al.  Nondeterministic space is closed under complementation , 1988, [1988] Proceedings. Structure in Complexity Theory Third Annual Conference.

[13]  Burkhard Monien,et al.  On the LBA Problem , 1981, FCT.

[14]  Stephen A. Cook,et al.  Space Lower Bounds for Maze Threadability on Restricted Machines , 1980, SIAM J. Comput..

[15]  Burkhard Monien,et al.  The LBA-problem and the deterministic tape complexity of two-way one-counter languages over a one-letter alphabet , 1977, Acta Informatica.

[16]  A. Meyer.  Review: Walter J. Savitch, Relationships between Nondeterministic and Deterministic Tape Complexities , 1974 .

[17]  Walter J. Savitch,et al.  Relationships Between Nondeterministic and Deterministic Tape Complexities , 1970, J. Comput. Syst. Sci..

[18]  D. C. Cooper,et al.  Theory of Recursive Functions and Effective Computability , 1969, The Mathematical Gazette.

[19]  Richard Stanley.  Review: Peter S. Landweber, Three Theorems on Phrase Structure Grammars of Type 1 , 1967 .

[20]  Richard Edwin Stearns,et al.  Hierarchies of memory limited computations , 1965, SWCT.

[21]  S.-Y. Kuroda,et al.  Classes of Languages and Linear-Bounded Automata , 1964, Inf. Control..

[22]  Peter S. Landweber,et al.  Three Theorems on Phrase Structure Grammars of Type 1 , 1963, Inf. Control..

[23]  A. Church.  Review: A. M. Turing, On Computable Numbers, with an Application to the Entscheidungsproblem , 1937 .

[24]  R. Carmichael.  The Theory of Functions of a Real Variable and the Theory of Fourier's Series , 1928, Nature.

[25]  Tianrong Lin.  Diagonalization of Polynomial-Time Turing Machines Via Nondeterministic Turing Machine , 2021, ArXiv.

[26]  Tianrong Lin,et al.  The Separation of NP and PSPACE , 2021, ArXiv.

[27]  C. Papadimitriou,et al.  Introduction to the Theory of Computation , 2018 .

[28]  Eitan M. Gurari,et al.  Introduction to the theory of computation , 1989 .

[29]  Jeffrey D. Ullman,et al.  Introduction to Automata Theory, Languages and Computation , 1979 .

[30]  Alfred V. Aho,et al.  The Design and Analysis of Computer Algorithms , 1974 .

[31]  Jeffrey D. Ullman,et al.  Formal languages and their relation to automata , 1969, Addison-Wesley series in computer science and information processing.

[32]  Jeffrey D. Ullman,et al.  Some Results on Tape-Bounded Turing Machines , 1969, JACM.

[33]  Seymour Ginsburg,et al.  The mathematical theory of context free languages , 1966 .

[34]  W. Rudin.  Principles of mathematical analysis , 1964 .

[35]  G. H. H.,et al.  The Theory of Functions of a Real Variable and the Theory of Fourier's Series , 1907, Nature.