Two-dimensional Turing Machines Research Articles

A physically universal Turing machine

We construct a two-dimensional Turing machine that is physically universal in both the moving tape and moving head model. In particular, it is mixing of all finite orders in both models. We also provide a variant that is physically universal in the moving tape model, but not in the moving head model.

K-Neighborhood Template A-Type Two-Dimensional Bounded Cellular Acceptors

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.

Universality in two dimensions

Turing, in his immortal 1936 paper, observed that '[human] computing is normally done by writing... symbols on [two- dimensional] paper', but noted that use of a second dimension 'is always avoidable' and that 'the two-dimensional character of paper is no essential of computation'. We propose to promote two-dimensional models of computation and exploit the naturalness of two-dimensional representations of data. In particular, programs for a two-dimensional Turing machine can be recorded most naturally on its own two-dimensional input-output grid in such a transparent fashion that schoolchildren would have no difficulty comprehending their behaviour. This two-dimensional rendering allows, furthermore, for a most perspicacious rendering of Turing's universal machine.

A note on one-pebble two-dimensional Turing machines

This paper investigates a relationship among the accepting powers of deterministic, nondeterministic, and alternating one-pebble two-dimensional Turing machines, and shows that nondeterminism are more powerful than determinism for o(logn) space-bounded one-pebble two-dimensional Turing machines whose input tapes are restricted to square ones, and alternation are more powerful than nondeterminism for f(m)+g(n) (resp., f(m)×g(n)) space-bounded one-pebble two-dimensional Turing machines, where f:N→N is an arbitrary monotonic nondecreasing function space-constructible by a deterministic one-pebble two-dimensional Turing machine, and g:N→N is an arbitrary function such that g(n)=o(logn) (resp., g(n)=o(logn/loglogn)).

A note on one-pebble two-dimensional Turing machines

This paper investigates a relationship among the accepting powers of deterministic, nondeterministic, and alternating one-pebble two-dimensional Turing machines, and shows that 1. nondeterminism are more powerful than determinism for o(log n) space-bounded one-pebble two-dimensional Turing machines whose input tapes are restricted to square ones, and 2. alternation are more powerful than nondeterminism for f( m)+ g( n) (resp., f( m)× g( n)) space-bounded one-pebble two-dimensional Turing machines, where f: N→ N is an arbitrary monotonic nondecreasing function space-constructible by a deterministic one-pebble two-dimensional Turing machine, and g: N→ N is an arbitrary function such that g( n)=o(log n) (resp., g(n)= o( logn/ log log n) ).

A note on one-pebble two-dimensional Turing machines

Abstract This paper investigates a relationship among the accepting powers of deterministic, nondeterministic, and alternating one-pebble two-dimensional Turing machines, and shows that (i) nondeterminism are more powerful than determinism for o(log n) space-bounded one-pebble two-dimensional Turing machines whose input tapes are restricted to square ones, and (ii) alternation are more powerful than nondeterminism for f(m)+g(n) (resp., f(m) × g(n)) space-bounded one-pebble two-dimensional Turing machines, where f : N → N is an arbitrary monotonic nondecreasing funcion space-constractible by a deterministic one-pebble two-dimensional Turing machine, and g : N → N is an arbitrary function such that g(n) = o(log n) (resp., g(n) = o(log n/loglog n)).

CLOSURE PROPERTY OF SPACE-BOUNDED TWO-DIMENSIONAL ALTERNATING TURING MACHINES, PUSHDOWN AUTOMATA, AND COUNTER AUTOMATA

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)).

A NOTE ON THREE-WAY TWO-DIMENSIONAL PROBABILISTIC 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.

CHARACTERIZING THE ABILITY OF PARALLEL ARRAY GENERATORS ON REVERSIBLE PARTITIONED CELLULAR AUTOMATA

A PCAAG introduced by Morita and Ueno is a parallel array generator on a partitioned cellular automaton (PCA) that generates an array language (i.e. a set of symbol arrays). A "reversible" PCAAG (RPCAAG) is a backward deterministic PCAAG, and thus parsing of two-dimensional patterns can be performed without backtracking by an "inverse" system of the RPCAAG. Hence, a parallel pattern recognition mechanism on a deterministic cellular automaton can be directly obtained from a RPCAAG that generates the pattern set. In this paper, we investigate the generating ability of RPCAAGs and their subclass. It is shown that the ability of RPCAAGs is characterized by two-dimensional deterministic Turing machines, i.e. they are universal in their generating ability. We then investigate a monotonic RPCAAG (MRPCAAG), which is a special type of an RPCAAG that satisfies monotonic constraint. We show that the generating ability of MRPCAAGs is exactly characterized by two-dimensional deterministic linear-bounded automata.

SPACE HIERARCHIES OF TWO-DIMENSIONAL ALTERNATING TURING MACHINES, PUSHDOWN AUTOMATA AND COUNTER AUTOMATA

This paper investigates the space hierarchies of the language classes for two-dimensional Turing machines (2-TM's), two-dimensional pushdown automata (2-PDA's) and two-dimensional counter automata (2-CA's) with small space. We show that (1) if L(n) is space constructible by a 2-TM, L(n) ≤ log n and L′(n) = o(L(n)), then strong 2-DSPACE(L(n)) – weak 2-ASPACE(L′(n)) ≠ ∅, (2) if L(n) is space constructible by a 2-PDA, L(n) ≤ log n and L′(n) = o(L(n)), then strong 2-DPDA(L(n)) – weak 2-ASPACE(L′(n)) ≠ ∅, and (3) if L(n) is space-constructible by a 2-CA, L(n) ≤ n and L′(n) = o(L(n)), then strong 2-DCA(L(n)) – weak 2-ACA(L′(n)) ≠ ∅, (4) where strong 2-DSPACE(L(n)) (strong 2-DPDA(L(n)), strong 2-DCA(L(n))) denotes the class of sets accepted by strongly L(n) space-bounded deterministic 2-TM's (2-PDA's, 2-CA's), and weak 2-ASPACE(L′(n)) (weak 2-ACA(L′(n))) denotes the class of sets accepted by weakly L′(n) space-bounded alternating 2-TM's (2-CA's). We also investigate the closure property of space-bounded alternating 2-PDA's and 2-CA's under complementation, and show that (1) if L(n) = o( log log n), then the class of sets accepted by L(n) space-bounded alternating 2-PDA's is not closed under complementation, and (2) if L(n) is space-constructible by a 2-CA, L(n) ≤ n and [Formula: see text], then the class of sets accepted by L′(n) space-bounded alternating 2-CA's is not closed under complementation.

A note on two-dimensional probabilistic Turing machines

This paper introduces two-dimensional probabilistic Turing machines (2-ptm's), and investigates several properties of them. We first investigate a relationship between two-dimensional alternating finite automata (2-afa's) and 2-ptm's with error probability less than 1 2 and with sublogarithmic space, and show that there is a set of square tapes accepted by 2-afa, but not recognized by any o(log- n) space-bounded 2-ptm with error probability less than 1 2 . This partially solves an open problem in Okazaki et al. (Inform. Sci., to appear). We next investigate a space hierarchy of 2-ptm's with error probability less than 1 2 and with sublogarithmic space, and show that if L(n) is space-constructible by a two-dimensional Turing machine, there is a set of square tapes accepted by a strongly L(n) space-bounded two-dimensional deterministic Turing machine, but not recognized by any L(n) space-bounded 2-ptm with error probability less than 1 2 .

On space functions fully constructed by two-dimensional turing machines

On space functions fully constructed by two-dimensional turing machines

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

A two-dimensional Turing machine (2-TM) is a TM with a two-dimensional input tape whose input head can move in all four directions. We show that if S( n) is a two-dimensionally space-constructible function such that S( n) ⩽ log n and T( n) is a function satisfying T( n) = o( S( n)), then there is a language (i.e., set of square patterns over some finite alphabet) that is accepted by a strongly S( n) space-bounded 2-DTM, but not by any strongly T( n) space-bounded 2-ATM. This partially answers a question posed by Inoue and Takanami. Since functions such as log n, loglog n, logloglog n, …, are all two-dimensionally space-constructible, this result implies a rich space hierarchy for strongly O( log n) space-bounded 2-ATM's.

On space functions constructed by two-dimensional turing machines

It is the well-known fact that for one-dimensional Turing machines there is a gap in the space complexity between the constant functions and log log n. In this paper we show that this is not the case for two-dimensional Turing machines with square inputs and that the functions log ∗n and log k n, for every k, can be constructed by Turing machines with square inputs. This means that for two-dimensional Turing machines the space hierarchy starts much lower than for one-dimensional Turing machines. Moreover, we show that log log n is fully space constructible by a two-dimensional Turing machine with square inputs.

A survey of two-dimensional automata theory

The main purpose of this paper is to survey several properties of alternating, nondeterministic, and deterministic two-dimensional Turing machines (including two-dimensional finite automata and marker automata), and to briefly survey cellular types of two-dimensional automata.

TWO-DIMENSIONAL THREE-WAY ARRAY GRAMMARS AND THEIR ACCEPTORS

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.

On three-way two-dimensional turing machines

This paper solves several open problems concerning closure properties of three-way tape-bounded Turing machines. It is shown that: (1) the class of sets of square tapes accepted by nondeterministic three-way L( m) tape-bounded Turing machines is closed under complementation if L( m)⩾ m 2 is constructible, (2) the class of sets of square tapes accepted by nondeterministic three-way L( m) tape-bounded Turing machines is closed neither under row nor column cyclic closure if L( m) < log m, (3) if L( m, n) = mg( n) or L( m, n) = g( m) n, then L( m, n)⩾ mn space is necessary for the class of sets of (general) tapes accepted by deterministic three-way L( m, n) tape-bounded Turing machines to be closed under row or column catenation, row or column closure or row or column cyclic closure.

Three-way two-dimensional finite automata with rotated inputs

This paper introduces a new type of two-dimensional automaton called a three-way two-dimensional finite automaton with rotated inputs, and investigates a relationship among the accepting powers of these new automata, two-dimensional finite automata, and tape-bounded three-way two-dimensional Turing machines. We show, for example, that m log m space ( m 2 space) is necessary and sufficient for deterministic three-way two-dimensional Turing machines to simulate deterministic (nondeterministic) three-way two-dimensional finite automata with rotated inputs.

A space-hierarchy result on two-dimensional alternating Turing machines with only universal states

This paper investigates a space hierarchy of the classes of sets accepted by alternating space-bounded two-dimensional Turing machines which have only universal states and whose input tapes are restricted to square ones and shows that there exists a dense hierarchy for the classes of sets accepted by these Turing machines with spaces of size less than or equal to log m.

A relationship between two-dimensional finite automata and three-way tape-bounded two-dimensional Turing machines

This note supplements a result of Inoue and Takanami (1980) who showed that for any function L( m) such that (i) L( m)⩾log m, and (ii) lim m→∞ [ L(m) m 2 ]= 0 ( resp lim m→∞ [ L(m) m log m ] = 0) , the class of sets of square tapes accepted by deterministic three-way L( m) tape-bounded two-dimensional Turing machines is incomparable with the class of sets of square tapes accepted by nondeterministic (resp. deterministic) two-dimensional finite automata.