Non-closure Properties of 1-Inkdot Nondeterministic Turing Machines and Alternating Turing Machines with Only Universal States Using Small Space

This paper deals with two topics concerning two-dimensional automata operating in parallel. We first investigate a relationship between the accepting powers of two-dimensional alternating finite automata (2-AFAs) and nondeterministic bottom-up pyramid cellular acceptors (NUPCAs), and show that Ω ( diameter × log diameter ) time is necessary for NUPCAs to simulate 2-AFAs. We then investigate space complexity of two-dimensional alternating Turing machines (2-ATMs) operating in small space, and show that if L (n) is a two-dimensionally space-constructible function such that lim n → ∞ L (n)/ loglog n > 1 and L (n) ≤ log n, and L′ (n) is a function satisfying L′ (n) =o (L(n)), then there exists a set accepted by some strongly L (n) space-bounded two-dimensional deterministic Turing machine, but not accepted by any weakly L′ (n) space-bounded 2-ATM, and thus there exists a rich space hierarchy for weakly S (n) space-bounded 2-ATMs with loglog n ≤ S (n) ≤ log n.

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Preface

International Journal of Pattern Recognition and Artificial Intelligence ◽

10.1142/s0218001497000676 ◽

1997 ◽

Vol 11 (07) ◽

pp. 1023-1024

Author(s):

Serge Miguet ◽

Annick Montanvert ◽

P. S. P. Wang

Keyword(s):

Finite State Machine ◽

Finite Automata ◽

Upper Bounds ◽

Two Dimensional ◽

Turing Machines ◽

Closure Properties ◽

Working Space ◽

Finite State ◽

The Individual ◽

Alternating Turing Machines

Several nonclosure properties of each class of sets accepted by two-dimensional alternating one-marker automata, alternating one-marker automata with only universal states, nondeterministic one-marker automata, deterministic one-marker automata, alternating finite automata, and alternating finite automata with only universal states are shown. To do this, we first establish the upper bounds of the working space used by "three-way" alternating Turing machines with only universal states to simulate those "four-way" non-storage machines. These bounds provide us a simplified and unified proof method for the whole variants of one-marker and/or alternating finite state machine, without directly analyzing the complex behavior of the individual four-way machine on two-dimensional rectangular input tapes. We also summarize the known closure properties including Boolean closures for all the variants of two-dimensional alternating one-marker automata.

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Some observations concerning alternating turing machines using small space

Information Processing Letters ◽

10.1016/0020-0190(87)90085-8 ◽

1987 ◽

Vol 25 (1) ◽

pp. 1-9 ◽

Cited By ~ 27

Author(s):

Jik H. Chang ◽

Oscar H. Ibarra ◽

Bala Ravikumar ◽

Leonard Berman

Keyword(s):

Turing Machines ◽

Small Space ◽

Alternating Turing Machines

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

International Journal of Pattern Recognition and Artificial Intelligence ◽

10.1142/s0218001499000306 ◽

1999 ◽

Vol 13 (04) ◽

pp. 503-521 ◽

Cited By ~ 1

Author(s):

TOKIO OKAZAKI ◽

KATSUSHI INOUE ◽

AKIRA ITO ◽

YUE WANG

Keyword(s):

Closure Property ◽

Two Dimensional ◽

Turing Machines ◽

Small Space ◽

Language Classes ◽

Alternating Turing Machines ◽

Pushdown 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.

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Nonclosure Properties of Two-Dimensional One-Marker Automata

International Journal of Pattern Recognition and Artificial Intelligence ◽

10.1142/s0218001497000469 ◽

1997 ◽

Vol 11 (07) ◽

pp. 1025-1050 ◽

Cited By ~ 3

Author(s):

Akira Ito ◽

Katsushi Inoue ◽

Yue Wang

Keyword(s):

Finite State Machine ◽

Finite Automata ◽

Upper Bounds ◽

Two Dimensional ◽

Turing Machines ◽

Closure Properties ◽

Working Space ◽

Finite State ◽

The Individual ◽

Alternating Turing Machines

Several nonclosure properties of each class of sets accepted by two-dimensional alternating one-marker automata, alternating one-marker automata with only universal states, nondeterministic one-marker automata, deterministic one-marker automata, alternating finite automata, and alternating finite automata with only universal states are shown. To do this, we first establish the upper bounds of the working space used by "three-way" alternating Turing machines with only universal states to simulate those "four-way" non-storage machines. These bounds provide us a simplified and unified proof method for the whole variants of one-marker and/or alternating finite state machine, without directly analyzing the complex behavior of the individual four-way machine on two-dimensional rectangular input tapes. We also summarize the known closure properties including Boolean closures for all the variants of two-dimensional alternating one-marker automata.

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Some observations concerning alternating turing machines using small space

Information Processing Letters ◽

10.1016/0020-0190(88)90081-6 ◽

1988 ◽

Vol 27 (1) ◽

pp. 53 ◽

Cited By ~ 8

Author(s):

Jik H. Chang ◽

Oscar H. Ibarra ◽

Bala Ravikumar ◽

Leonard Berman

Keyword(s):

Turing Machines ◽

Small Space ◽

Alternating Turing Machines

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ON THE POWER OF ONE-WAY SYNCHRONIZED ALTERNATING MACHINES WITH SMALL SPACE

International Journal of Foundations of Computer Science ◽

10.1142/s0129054192000073 ◽

1992 ◽

Vol 03 (01) ◽

pp. 65-79 ◽

Cited By ~ 8

Author(s):

JURAJ HROMKOVIČ ◽

KATSUSHI INOUE ◽

BRANISLAV ROVAN ◽

ANNA SLOBODOVÁ ◽

ITSUO TAKANAMI ◽

...

Keyword(s):

Finite Automata ◽

Turing Machines ◽

Small Space ◽

Closure Properties ◽

Open Problems ◽

Infinite Hierarchy ◽

Context Sensitive ◽

The Family

This paper continues the investigation of the concept of synchronized alternation. The open problems from Ref. 4 are solved by showing that one-way synchronized alternating (multihead) automata are as powerful as two-way ones. More precisely it is shown that: (i) one-way synchronized alternating finite automata recognize exactly context-sensitive languages, and (ii) NSPACE(nk) is exactly the family of languages recognized by one-way (two-way) synchronized alternating k-head finite automata, for k≥1. Finaly, the synchronization complexity of one-way synchronized Turing machines (1satm's) is investigated and an infinite hierarchy among classes of sets accepted by 1satm's with space and synchronization bounds between log log n and log n is established. Some closure properties of the classes in this hierarchy are also proved.

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ALTERNATING TURING MACHINES WITH MODIFIED ACCEPTING STRUCTURE

International Journal of Foundations of Computer Science ◽

10.1142/s0129054191000194 ◽

1991 ◽

Vol 02 (04) ◽

pp. 401-417

Author(s):

KATSUSHI INOUE ◽

AKIRA ITO ◽

ITSUO TAKANAMI

Keyword(s):

Turing Machine ◽

Input Word ◽

Turing Machines ◽

Closure Properties ◽

Infinite Hierarchy ◽

Computation Tree ◽

On Line ◽

Alternating Turing Machines

We introduce an alternating Turing machine with modified accepting structure (denoted by MATM), which is an alternating Turing machine whose accepting condition differs from that of an ordinary alternating Turing machine (denoted by ATM). An MATM has a set of accepting state sets rather than a set of accepting states. An input word x is accepted by an MATM M if there is a computation tree of M on x such that the set of states associated with the leaves of the tree is equal to an accepting state set. Let UTM (MUTM) denote an ATM (MATM) with no existential state. We first investigate a relationship between ATM’s and MATM’s, and show that (i) for any function L(n), L(n) space bounded on-line (off-line) ATM’s are equivalent to L(n) space bounded on-line (off-line) MATM’s, and (ii) for any L(n) such that L(n)≥ log log n and limn→∞L(n)/n=0, L(n) space bounded on-line MUTM’s are more powerful than L(n) space bounded on-line UTM’s. We then investigate a relationship between online and off-line, and show for example that for any L(n) such that L(n)≥ log n and limn→∞L(n)/n=0, L(n) space bounded off-line MUTM’s are more powerful than L(n) space bounded on-line MUTM’s. We next show that there exists an infinite hierarchy among accepting powers of L(n) space bounded on-line (off-line) MATM’s and MUTM’s with L(n)≥ log log n and limn→∞L(n)/n=0. Finally, we investigate closure properties of space bounded on-line (off-line) MATM’s and MUTM’s.

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