Article

# A Note on Two-Dimensional Probabilistic Finite Automata.

Information Sciences (Impact Factor: 3.64). 01/1998; 110:303-314. DOI: 10.1016/S0020-0255(98)00008-5

Source: DBLP

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**ABSTRACT:**This paper investigates closure properties of the classes of sets recognized by spacebounded two-dimensional probabilistic Turing machines with error probability less than 12. Let 2-PTM(L(m,n)) be the class of sets recognized by L(m,n) space-bounded two-dimensional probabilistic Turing machines with error probability less than 12, where L(m, n): N2 → N (N denotes the set of all the positive integers) be a function with two variables m (= the number of rows of input tapes) and n (= the number of columns of input tapes). We first show that (i) for any function ƒ(m) = o(logm) (resp., ƒ(m) = o(log m/log log m)) and any monotonic nondecreasing function g(n) which can be constructed by some one-dimensional deterministic Turing machine, 2-PTM(L(m, n)) is not closed under row catenation, row closure, and projection, where L(m, n) = ƒ(m)+ g(n) (resp., L(m,n) = ƒ(m) × g(n)), and (ii) for any function g(n) = o(log n) (resp., g(n) = o(logn/loglogn)) and any monotonic nondecreasing function ƒ(m) which can be constructed by some one-dimensional deterministic Turing machine, 2-PTM(L(m, n)) is not closed under column catenation, column closure, and projection, where L(m,n) = ƒ(m)+g(n)(resp., L(m, n) = ƒ(m) × g(n)). We then show that 2-PTMT(L(m, n)) is closed under union, intersection, and complementation for any L(m, n), where 2-PTMT(L(m, n)) denotes the class of sets recognized by L(m, n) space-bounded two-dimensional probabilistic Turing machines with error probability less than 12 which always halt in the accepting or rejecting state for all the input tapes.Information Sciences 01/1999; 115:61-81. · 3.64 Impact Factor -
##### Conference Paper: A Space Lower Bound of Two-Dimensional Probabilistic Turing Machines.

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**ABSTRACT:**This paper shows a sublogarithmic space lower bound for two-dimensional probabilistic Turing machines (2-ptm’s) over square tapes with bounded error, and shows, using this space lower bound theorem, that a specific set is not recognized by any o(log n) space-bounded 2- ptm. Furthermore, the paper investigates a relationship between 2-ptm's and two-dimensional Turing machines with both nondeterministic and probabilistic states, which we call “two-dimensional stochastic Turing machines (2-stm’s)”, and shows that for any loglog n ≤ L(n) = o(log n), L(n) space-bounded 2-ptm’s with bounded error are less powerful than L(n) space-bounded 2-stm’s with bounded error which start in nondeterministic mode, and make only one alternation between nondeterministic and probabilistic modes.Developments in Language Theory, 6th International Conference, DLT 2002, Kyoto, Japan, September 18-21, 2002, Revised Papers; 01/2002 -
##### Conference Paper: A Survey on Picture-Walking Automata.

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**ABSTRACT:**Picture walking automata were introduced by M. Blum and C. Hewitt in 1967 as a generalization of one-dimensional two-way finite automata to recognize pictures, or two-dimensional words. Several variants have been investigated since then, including deterministic, non-deterministic and alternating transition rules; four-, three- and two-way movements; single- and multi-headed variants; automata that must stay inside the input picture, or that may move outside. We survey results that compare the recognition power of different variants, consider their basic closure properties and study decidability questions.Algebraic Foundations in Computer Science - Essays Dedicated to Symeon Bozapalidis on the Occasion of His Retirement; 01/2011

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