Kosaburo Hashiguchi

Limitedness theorem on finite automata with distance functions

A finite automaton A with a distance function d is a sextuple, 〈Σ, Q, M, S, F, d〉, such that Σ is the input alphabet, Q is the finite set of states, M: Q × Σ → 2Q is the transition function, S ⊂ Q and F ⊂ Q are the sets of initial and final states, respectively, d: Q × Σ × Q → {0, 1, ∞} is the distance function, where ∞ denotes infinity, and d satisfies the following: for any (q, a, q′) ∈ Q × Σ × Q, d(q, a, q′ = ∞ iff q′ ∉ M(q, a). M is extended toQ × Σ∗ and d is extended to Q × Σ∗ × Q in the usual way. A is said to be limited in distance if there exists a nonnegative integer k such that for any w accepted by A, d(q, w, q′) ⩽ k for some q ∈S and q′ ∈ F. This paper shows that there exists an algorithm for deciding whether or not an arbitrary finite automaton with a distance function is limited in distance.

Algorithms for determining relative star height and star height

Eggan(l963) introduced the notion of the star height to each regular expression which is a nonnegative integer denoting the nestedness of star operators in this expression. The star height of a regular language is the minimum of the star height of regular expressions denoting this language. We remark here that to each regular language, there exist generally infinitely many regular expressions denoting this language. Eggan(l963) showed that for each nonnegative integer k, there exists a regular language of star height k, and posed the problem of determining the star height of any regular language. Dejean and Schutzenberger (1966) showed that for each nonnegative integer k, there exists a regular language of star height k over the two-letter alphabet. McNaughton (1967) presented an algorithm for determining the loop complexity (i.e., the star height) of any regular language whose syntactic monoid is a group. Cohen (1970, 1971) and Cohen and Brzozowski (1970) investigated many properties of star height, some of which provide algorithms for determining the star height of any regular language of certain reset-free type. Hashiguchi and Honda (1979) presented an algorithm for determining the star height of any reset-free language and any reset language. Hashiguchi (1982B) presented an algorithm for deciding whether or not an arbitrary language is of star height one. To obtain this result, it uses the limitedness theorem on finite automata with distance functions (in short, D-automata) in Hashiguci (1982A, 1983).

Representation theorems on regular languages

Let T denote the set of regular operators, concatenation (·), union (∪), and star(∗). For any subset Ω of T, any regular language R, and any finite class C of relgular languages, R is said to have a Ω-representation over C if R can be obtained by a finite number of applications of operators in Ω to C This paper shows that there exist algorithms for deciding whether or not an arbitrary regular language has a Ω-representation over an arbitrary class C of regular languages for any subset Ω of T.

Improved limitedness theorems on finite automata with distance functions

Abstract Let A be a finite automaton with a distance function, and ID( A ) be the set of distances associated with words accepted by A . This paper presents an improved upper bound of ID( A ) when the upper limit of ID( A ) is finite. It also presents one necessary and sufficient condition concerning (word,+)-expressions for the upper limit of ID( A ) to be infinite.

Regular languages of star height one

There exists an algorithm for deciding whether or not an arbitrary regular language is of star height one.

Recognizable closures and submonoids of free partially commutative monoids

Abstract The main result of this paper is the presentation of two sufficient conditions for the closure [ X ] of X and the closure [ X ∗ ] of X ∗ , respectively, in any free partially commutative monoid for any regular language X , to be recognizable. Several previously obtained sufficient conditions for [ X ] to be recognizable can be seen to imply one of those two conditions.

Algorithms for Determining the Smallest Number of Nonterminals (States) Sufficient for Generating (Accepting) a Regular Language

There exist algorithms for determining the number of nonterminals in a nonterminal-minimal (generalized) right-linear grammar generating R, and the number of states in a state-minimal (generalized) nondeterministic finite automaton accepting R for any given regular language R.

Algorithms for determining relative inclusion star height and inclusion

Abstract Let C =(R 1 ,⋯,R m ) be a finite class of regular languages over a finite alphabet Σ. Let Δ=(b1,⋯,bm) be an alphabet, and δ be the substitution from Δ∗ into Σ∗ such that δbi)=Ri for all i. Let R 10 ,R 20 ⊂Σ ∗ be two regular languages. The relative inclusion star height h r (R 10 ,R 20 , C ) of (R 10 ,R 20 ) w.r.t. C is the minimum star height of regular languages L⊂Δ ∗ such that R10⊂δ(L)⊂R20. This paper proves the existence of an algorithm for determining relative inclusion star height.

String matching problems over free partially commutative monoids

Abstract This paper studies two string matching problems over free partially commutative monoids. We analyze these two problems in detail, and present two efficient polynomial time algorithms for solving them.

Extended regular expressions of star degree at most two

Abstract This paper introduces the notion of the star operator of degree two, and studies properties of the closure of the family of finite languages under the operations, union, concatenation, the Kleene star, and the star operator of degree two: this closure is a proper subfamily of the family of context-free languages.