Preface

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.

Download Full-text

TWO TOPICS CONCERNING TWO-DIMENSIONAL AUTOMATA OPERATING IN PARALLEL

International Journal of Pattern Recognition and Artificial Intelligence ◽

10.1142/s0218001492000126 ◽

1992 ◽

Vol 06 (02n03) ◽

pp. 211-225 ◽

Cited By ~ 1

Author(s):

KATSUSHI INOUE ◽

ITSUO SAKURAMOTO ◽

MAKOTO SAKAMOTO ◽

ITSUO TAKANAMI

Keyword(s):

Turing Machine ◽

Finite Automata ◽

Space Complexity ◽

Two Dimensional ◽

Turing Machines ◽

Small Space ◽

Bottom Up ◽

Constructible Function ◽

Deterministic Turing Machine ◽

Alternating Turing Machines

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.

Download Full-text

CONSTANT LEAF-SIZE HIERARCHY OF TWO-DIMENSIONAL ALTERNATING TURING MACHINES

International Journal of Pattern Recognition and Artificial Intelligence ◽

10.1142/s0218001494000267 ◽

1994 ◽

Vol 08 (02) ◽

pp. 509-524

Author(s):

AKIRA ITO ◽

KATSUSHI INOUE ◽

ITSUO TAKANAMI ◽

YASUYOSHI INAGAKI

Keyword(s):

Finite Automata ◽

Leaf Size ◽

Two Dimensional ◽

Turing Machines ◽

Size Hierarchy ◽

Minimum Number ◽

Space Function ◽

Number Of Leaves ◽

Alternating Turing Machines ◽

Necessary And Sufficient

“Leaf-size” (or “branching”) is the minimum number of leaves of some accepting computation trees of alternating devices. For example, one leaf corresponds to nondeterministic computation. In this paper, we investigate the effect of constant leaves of two-dimensional alternating Turing machines, and show the following facts: (1) For any function L(m, n), k leaf- and L(m, n) space-bounded two-dimensional alternating Turing machines which have only universal states are equivalent to the same space bounded deterministic Turing machines for any integer k≥1, where m (n) is the number of rows (columns) of the rectangular input tapes. (2) For square input tapes, k+1 leaf- and o(log m) space-bounded two-dimensional alternating Turing machines are more powerful than k leaf-bounded ones for each k≥1. (3) The necessary and sufficient space for three-way deterministic Turing machines to simulate k leaf-bounded two-dimensional alternating finite automata is nk+1, where we restrict the space function of three-way deterministic Turing machines to depend only on the number of columns of the given input tapes.

Download Full-text

CONSTANT LEAF-SIZE HIERARCHY OF TWO-DIMENSIONAL ALTERNATING TURING MACHINES

Series in Machine Perception and Artificial Intelligence - Parallel Image Analysis and Processing ◽

10.1142/9789814447096_0008 ◽

1994 ◽

pp. 95-110

Author(s):

AKIRA ITO ◽

KATSUSHI INOUE ◽

ITSUO TAKANAMI ◽

YASUYOSHI INAGAKI

Keyword(s):

Leaf Size ◽

Two Dimensional ◽

Turing Machines ◽

Size Hierarchy ◽

Alternating Turing Machines

Download Full-text

LIMITED AUTOMATA AND REGULAR LANGUAGES

International Journal of Foundations of Computer Science ◽

10.1142/s0129054114400140 ◽

2014 ◽

Vol 25 (07) ◽

pp. 897-916 ◽

Cited By ~ 15

Author(s):

GIOVANNI PIGHIZZINI ◽

ANDREA PISONI

Keyword(s):

Finite Automata ◽

Upper And Lower Bounds ◽

Finite State Automata ◽

Turing Machines ◽

Double Exponential ◽

Finite State ◽

A Cell ◽

Fixed Constant ◽

The One ◽

Context Free

Limited automata are one-tape Turing machines that are allowed to rewrite the content of any tape cell only in the first d visits, for a fixed constant d. In the case d = 1, namely, when a rewriting is possible only during the first visit to a cell, these models have the same power of finite state automata. We prove state upper and lower bounds for the conversion of 1-limited automata into finite state automata. In particular, we prove a double exponential state gap between nondeterministic 1-limited automata and one-way deterministic finite automata. The gap reduces to a single exponential in the case of deterministic 1-limited automata. This also implies an exponential state gap between nondeterministic and deterministic 1-limited automata. Another consequence is that 1-limited automata can have less states than equivalent two-way nondeterministic finite automata. We show that this is true even if we restrict to the case of the one-letter input alphabet. For each d ≥ 2, d-limited automata are known to characterize the class of context-free languages. Using the Chomsky-Schützenberger representation for contextfree languages, we present a new conversion from context-free languages into 2-limited automata.

Download Full-text

Closure properties of the classes of sets recognized by space-bounded two-dimensional probabilistic turing machines

Information Sciences ◽

10.1016/s0020-0255(98)10084-1 ◽

1999 ◽

Vol 115 (1-4) ◽

pp. 61-81 ◽

Cited By ~ 1

Author(s):

Tokio Okazaki ◽

Katsushi Inoue ◽

Akira Ito ◽

Yue Wang

Keyword(s):

Two Dimensional ◽

Turing Machines ◽

Closure Properties

Download Full-text

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

International Journal of Pattern Recognition and Artificial Intelligence ◽

10.1142/s0218001401001398 ◽

2001 ◽

Vol 15 (07) ◽

pp. 1143-1165 ◽

Cited By ~ 1

Author(s):

TOKIO OKAZAKI ◽

KATSUSHI INOUE ◽

AKIRA ITO ◽

YUE WANG

Keyword(s):

Turing Machine ◽

Nondecreasing Function ◽

Closure Property ◽

Two Dimensional ◽

Turing Machines ◽

Pushdown Automaton ◽

Alternating Turing Machines ◽

Positive Integers ◽

Pushdown 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)).

Download Full-text