Vants and Turmites.

On space functions constructed by two-dimensional turing machines

A computation universal Turing machine, U, with 2 states, 4 letters, 1 head and 1 two-dimensional tape is constructed by a translation of a universal register-machine language into networks over some simple abstract automata and, finally, of such networks into U. As there exists no universal Turing machine with 2 states, 2 letters, 1 head and 1 two-dimensional tape only the 2-state, 3-letter case for such machines remains an open problem. An immediate consequence of the construction of U is the existence of a universal 2-state, 2-letter, 2-head, 1 two-dimensional tape Turing machine, giving a first sharp boundary of the necessary complexity of universal Turing machines.

Two kinds of three-way isometric array grammars arc proposed as subclasses of an isometric monotonic array grammar. They are a three-way horizontally context-sensitive array grammar (3HCSAG) and a three-way immediately terminating array grammar (3ITAG). In these three-way grammars, patterns of symbols can grow only in the leftward, rightward and downward directions. We show that their generating abilities of rectangular languages are precisely characterized by some kinds of three-way two-dimensional Turing machines or related acceptors. In this paper. the following results are proved. First, 3HCSAG is characterized by a nondeterministic two-dimensional three-way Turing machine with space-bound n (n is the width of a rectangular input) and a nondeterministic one-way parallel/sequential array acceptor. Second, 3ITAG is characterized by a nondeterministic two-dimensional three-way real-time (or linear-time) restricted Turing machine, a nondeterministic one-dimensional bounded cellular acceptor and a nondeterministic two-dimensional one-line tessellation acceptor.

This paper introduces a two-dimensional alternating Turing machine (2-ATM) which is an extension of an alternating Turing machine to two-dimensions. This paper also introduces a three-way two-dimensional alternating Turing machine (TR2-ATM) which is an alternating version of a three-way two-dimensional Turing machine. We first investigate a relationship between the accepting powers of space-bounded 2-ATM's (or TR2-ATM's) and ordinary space-bounded two-dimensional Turing machines (or three-way two-dimensional Turing machines). We then introduce a simple, natural complexity measure for 2-ATM's (or TR2-ATM's), called -&-ldquo;leaf-size-&-rdquo;, and provides a spectrum of complexity classes based on leaf-size bounded computations. We finally investigate the recognizability of connected patterns by 2-ATM's (or TR2-ATM's).

Two-dimensional alternating turing machines

This paper introduces a three-way two-dimensional probabilistic Turing machine (tr2-ptm), and investigates several properties of the machine. The tr2-ptm is a two-dimensional probabilistic Turing machine (2-ptm) whose input head can only move left, right, or down, but not up. Let 2-ptms (resp. tr2-ptms) denote a 2-ptm (resp. tr2-ptm) whose input tape is restricted to square ones, and let 2-PTMs(S(n)) (resp. TR2-PTMs(S(n))) denote the class of sets recognized by S(n) space-bounded 2-ptms's (resp. tr2-ptms's) with error probability less than ½, where S(n): N→N is a function of one variable n (= the side-length of input tapes). Let TR2-PTM(L(m,n)) denote the class of sets recognized by L(m,n) space-bounded tr2-ptm's with error probability less than ½, where L(m,n): N2→N is a function of two variables m (= the number of rows of input tapes) and n (= the number of columns of input tapes). The main results of this paper are: (1) 2-NFAs - TR2-PTMs(S(n))≠ϕ for any S(n)=o(log n), where 2-NFAs denotes the class of sets of square tapes accepted by two-dimensional nondeterministic finite automata, (2) TR2-PTMsS(n)[Formula: see text]2-PTMs(S(n)) for any S(n)=o(log n), and (3) for any function g(n)=o(log n) (resp. g(n)=o(log n/log log n)) and any monotonic nondecreasing function f(m) which can be constructed by some one-dimensional deterministic Turing machine, TR2-PTM(f(m)+g(n)) (resp. TR2-PTM(f(m)×g(n))) is not closed under column catenation, column closure, and projection. Additionally, we show that two-dimensional nondeterministic finite automata are equivalent to two-dimensional probabilistic finite automata with one-sided error in accepting power.

A note on two-dimensional probabilistic Turing machines

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.

This paper investigates closure property of the classes of sets accepted by space-bounded two-dimensional alternating Turing machines (2-atm's) and space-bounded two-dimensional alternating pushdown automata (2-apda's), and space-bounded two-dimensional alternating counter automata (2-aca's). Let L(m, n): N2 → N (N denotes the set of all 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 show that (i) for any function f(m) = o( log m) (resp. f(m) = o( log m/ log log m)) and any monotonic nondecreasing function g(n) space-constructible by a two-dimensional Turing machine (2-Tm) (resp. two-dimensional pushdown automaton (2-pda)), the class of sets accepted by L(m,n) space-bounded 2-atm's (2-apda's) is not closed under row catenation, row + or projection, and (ii) for any function f(m) = o(m/ log ) (resp. for any function f(m) such that log f(m) = o( log m)) and any monotonic nondecreasing function g(n) space-constructible by a two-dimensional counter automaton (2-ca), the class of sets accepted by L(m, n) space-bounded 2-aca's is not closed under row catenation, row + or projection, where L(m, n) = f(m) + g(n) (resp. L(m, n) = f(m) × g(n)).

In this paper, we deal with three-dimensional computational model, k-neighborhood template A-type two-dimensional bounded cellular acceptor on three-dimensional tapes, and discuss some basic properties. This model consists of a pair of a converter and a configuration-reader. The former converts the given three-dimensional tape to two-dimensional configuration. The latter determines whether or not the derived two-dimensional configuration is accepted, and concludes the acceptance or non-acceptance of given three-dimensional tape. We mainly investigate some open problems about k-neighborhood template A-type two-dimensional bounded cellular acceptor on three-dimensional tapes whose configuration-readers are L(m) space-bounded deterministic (nondeterministic) two-dimensional Turing machines.

Two-dimensional alternating turing machines with only universal states

We show that emptiness is decidable for three-way two-dimensional nondeterministic finite automata as well as the universe problem for the corresponding class of deterministic automata. Emptiness is undecidable for three-way (and even two-way) two-dimensional alternating finite automata over a single-letter alphabet. Consequently inclusion, equivalence, and disjointness for these automata are undecidable properties.

A hierarchy result for 2-dimensional TM's operating in small space

On space functions fully constructed by two-dimensional turing machines

A note on deterministic three-way tape-bounded two-dimensional Turing machines