Kojiro Kobayashi

The firing squad synchronization problem for two-dimensional arrays

We consider the firing squad synchronization problem for two-dimensional arrays. Any finite connected nonempty subset of the rectangular grid is permitted as an array. Any position in an array is permitted as a position of the general. We construct (1) solutions to this problem having minimal firing time for arrays in some classes of arrays and (2) (for each K) a solution to this problem having a “linear” firing time for any array having not more than K edges. We also show an example of string-type arrays (that is, lines of breadth 1 without loops (the position of the general is one of the ends)) of length N whose minimal firing time is less than 2N − 2.

The firing squad synchronization problem for a class of polyautomata networks

Abstract By the work of Rosenstiehl we know that the firing squad synchronization problem for the class of all polyautomata networks N satisfying the following conditions has a solution: (1) N is connected, (2) the connections of N are bilateral (that is, each connection carries information in both directions). We show that the problem has a solution even if we delete condition (2) if we add the following conditions: (3) the “fan-out” of each output terminal of automata is at most one, (4) for each output terminal of an automaton, the automaton knows whether there is a connection from the output terminal or not. The firing time of our solution is at most (2 a 2 + 1) n for sufficiently large n ( n is the number of finite automata in the given network and a is the number of output terminals of the component finite automaton).

On proving time constructibility of functions

We formalize the techniques that have been used to prove time constructibility of functions by means of two theorems. The first theorems gives one sufficient condition for time constructibility of f1(n)+f2(n) and f2(n) to imply that of f1(n). As an application of this theorem, we show that, for a function f(n) such that (∃e>0)(∀∞n)f(n)⩾(1+e)n, f(n) is time constructible if and only if it is computable by a Turing machine within O(f(n)) steps. The second theorem concerns time constructibility of functions f(n) for which there are no ϵ > 0 such that (∀∞n) f(n) ⩾ (1+ϵ)n.

On the structure of one-tape nondeterministic Turing machine time hierarchy

Abstract We show that if the number of available states is fixed and is sufficiently large, then one-tape nondeterministic Turing machines can accept more sets within time bound a 2 ƒ(n) than within a 1 ƒ(n) , for 0 a 1 a 2 . Here, ƒ(n) is any function of the form n b 0 (log n ) b 1 (log 2 n ) b 2 …(log h n ) b h ( b 0 ,…, b h are rational numbers) with order n log n and n 2 . Crossing sequences and Kolmogorov complexity are used to prove it.

On the minimal firing time of the firing squad synchronization problem for polyautomata networks

Abstract For each class Γ of polyautomata networks (networks composed of copies of a same finite automaton and connections from output terminals to input terminals of the copies), the firing squad synchronization problem (fssp) for Γ is naturally defined. Supposed that the fssp for Γ has a solution. For each network N in Γ , the minimal firing time of N is also naturally defined (the minimum value of the firing time of N when all the solutions of the fssp for Γ are considered). We proved that the minimal firing time of N does not decrease even if we permit the use of infinite state automata as component automata of networks. We also give one characterization of the minimal firing time of N . For many classes Γ , this characterization gives algorithms to calculate the minimal firing time of N for each given N .

Characterization of ω-regular languages by first-order formulas

Abstract First-order formulas are used to specify various ways of acceptance of ω-languages by (deterministic) finite automata, and we study the relationship between the ‘arithmetic’ hierarchy of the formulas in prenex normal form and the topological hierarchy of the accepted ω-languages. Among other things it is proved that the ω-languages accepted by finite automata under the accepting conditions specified by Σ 1 -type formulas are precisely open ω-regular languages, those accepted under the conditions specified by Σ 2 -type formulas coincide with the ω-regular languages which are denumerable unions of closed sets, and that as long as the accepting conditions are specified by first-order formulas the accepted ω-languages remain to be ω-regular.

σ 0 n -complete properties of programs and Martin-Lo¨f randomness

Abstract G.J. Chaitin proved that the binary representation of the probability that a randomly selected program halts is Martin-Lof random. In the present paper, we introduce a notion called “Σ0n-completeness with constant-overhead reductions” and show that if a property of programs is Σ0n-complete with constant-overhead reductions then the binary representation of the probability that a randomly selected program has the property is Martin-Lof random. For almost all known Σ0n-complete properties of programs, their proofs of Σ0n-completeness can be easily modified to proofs of Σ0n-completeness in this new sense.

Transformations that preserve malignness of universal distributions

Abstract A function μ(x) that assigns a nonnegative real number μ(x) to each bit string x is said to be malign if, for any algorithm, the worst-case computation time and the average-case computation time of the algorithm are functions of the same order when each bit string x is given to the algorithm as an input with the probability that is proportional to the value μ(x). M. Li and P.M.B. Vitanyi found that functions that are known as “universal distributions” are malign. We say that a function ƒ(x) preserves malignness of universal distributions if ƒ(μ(x)) is malign for any universal distribution μ(x). We show one necessary and sufficient condition for ƒ(x) to preserve malignness of universal distributions under the assumption that ƒ(x) satisfies some additional conditions. As an application of this result, we show that ƒ(x) = x t preserves malignness of universal distributions or does not according as t⩾ 1 or 0

On coding theorems with modified length functions

Let R=(r1, r2,...) be an infinite sequence of real numbers (0 0, v>0 such that u≤ ri≤ 1-v for all i, then the coding theorem for memoryless sources holds even if ¦w¦r is used as the length of a code word w instead of the usual length ¦w¦. We prove that if limi→∞ ri/2−i=0 then the coding thorem with this modified length ¦w¦r does not hold true.

Characterization of Ω-regular languages by modadic second-order formulas

Abstract Many interesting subclasses of ω-regular languages have been characterized in terms of the product topology on Σω. We study the relationship between the topological hierarchy of ω-regular languages and the complexity of the monodic second-order formulas defining them. Among others, it is proved that open ω-regular languages are precisely the ω-languages defined by the formulas of the form ∀v∃x∃P(u, v, x), where u and ν are ω-word variables and x is a number variable. It is also proved that ω-regular languages in the class of denumerable unions of closed sets are the ω-languages defined by the formulas of the form ∃v∃x∀yP(u, v, x, y).