Parallel Computation for Well-Endowed Rings and Space-Bounded Probabilistic Machines

It is shown that a probabilistic Turing acceptor or transducer running within space bound S can be simulated by a time S2 parallel machine and therefore by a space S2 deterministic machine. (Previous simulations ran in space S6.) In order to achieve these simulations, known algorithms are extended for the computation of determinants in small arithmetic parallel time to computations having small Boolean parallel time, and this development is applied to computing the completion of stochastic matrices. The method introduces a generalization of the ring of integers, called well-endowed rings. Such rings possess a very efficient parallel implementation of the basic (+,−,×) ring operations.

[1]  John T. Gill,et al.  Computational complexity of probabilistic Turing machines , 1974, STOC '74.

[2]  John Gill,et al.  Deterministic Simulation of Tape-Bounded Probabilistic Turing Machine Transducers , 1980, Theor. Comput. Sci..

[3]  Allan Borodin,et al.  On Relating Time and Space to Size and Depth , 1977, SIAM J. Comput..

[4]  Shmuel Winograd,et al.  On the Time Required to Perform Addition , 1965, JACM.

[5]  Allan Borodin,et al.  The computational complexity of algebraic and numeric problems , 1975, Elsevier computer science library.

[6]  Stuart J. Berkowitz,et al.  On Computing the Determinant in Small Parallel Time Using a Small Number of Processors , 1984, Inf. Process. Lett..

[7]  John E. Savage,et al.  The Complexity of Computing , 1976 .

[8]  H. Jung.  Relationships between Probabilistic and Deterministic Tape Complexity , 1981, MFCS.

[9]  Janos Simon,et al.  On Tape-Bounded Probabilistic Turing Machine Acceptors , 1981, Theor. Comput. Sci..

[10]  Algirdas Avizienis,et al.  Signed-Digit Numbe Representations for Fast Parallel Arithmetic , 1961, IRE Trans. Electron. Comput..

[11]  John Gill,et al.  On Tape-Bounded Probabilistic Turing Machine Transducers (Extended Abstract) , 1978, FOCS 1978.

[12]  Allan Borodin,et al.  Fast parallel matrix and GCD computations , 1982, 23rd Annual Symposium on Foundations of Computer Science (sfcs 1982).

[13]  Nicholas Pippenger,et al.  On simultaneous resource bounds , 1979, 20th Annual Symposium on Foundations of Computer Science (sfcs 1979).

[14]  Janos Simon,et al.  Space-bounded probabilistic turing machine complexity classes are closed under complement (Preliminary Version) , 1981, STOC '81.

[15]  L. Csanky,et al.  Fast Parallel Matrix Inversion Algorithms , 1976, SIAM J. Comput..

[16]  A. Avizeinis,et al.  Signed Digit Number Representations for Fast Parallel Arithmetic , 1961 .

[17]  Shmuel Winograd,et al.  On the Time Required to Perform Multiplication , 1967, JACM.

[18]  Yuri Petrovich Ofman,et al.  On the Algorithmic Complexity of Discrete Functions , 1962 .

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

[20]  John Gill,et al.  Computational Complexity of Probabilistic Turing Machines , 1977, SIAM J. Comput..

[21]  L. Csanky,et al.  Fast parallel matrix inversion algorithms , 1975, 16th Annual Symposium on Foundations of Computer Science (sfcs 1975).

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