Star Height Research Articles

Digraph complexity measures and applications in formal language theory

Automata, Logic and Semantics We investigate structural complexity measures on digraphs, in particular the cycle rank. This concept is intimately related to a classical topic in formal language theory, namely the star height of regular languages. We explore this connection, and obtain several new algorithmic insights regarding both cycle rank and star height. Among other results, we show that computing the cycle rank is NP-complete, even for sparse digraphs of maximum outdegree 2. Notwithstanding, we provide both a polynomial-time approximation algorithm and an exponential-time exact algorithm for this problem. The former algorithm yields an O((log n)^(3/2))- approximation in polynomial time, whereas the latter yields the optimum solution, and runs in time and space O*(1.9129^n) on digraphs of maximum outdegree at most two. Regarding the star height problem, we identify a subclass of the regular languages for which we can precisely determine the computational complexity of the star height problem. Namely, the star height problem for bideterministic languages is NP-complete, and this holds already for binary alphabets. Then we translate the algorithmic results concerning cycle rank to the bideterministic star height problem, thus giving a polynomial-time approximation as well as a reasonably fast exact exponential algorithm for bideterministic star height.

COMPLEX VARIABILITY OF THE Hα EMISSION LINE PROFILE OF THE T TAURI BINARY SYSTEM KH 15D: THE INFLUENCE OF ORBITAL PHASE, OCCULTATION BY THE CIRCUMBINARY DISK, AND ACCRETION PHENOMENA

We have obtained 48 high resolution echelle spectra of the pre-main sequence eclipsing binary system KH~15D (V582 Mon, P = 48.37 d, $e$ $\sim$ 0.6, M$_{A}$ = 0.6 M$_{\odot}$, M$_{B}$ = 0.7 M$_{\odot}$). The eclipses are caused by a circumbinary disk seen nearly edge on, which at the epoch of these observations completely obscured the orbit of star B and a large portion of the orbit of star A. The spectra were obtained over five contiguous observing seasons from 2001/2002 to 2005/2006 while star A was fully visible, fully occulted, and during several ingress and egress events. The H$\alpha$ line profile shows dramatic changes in these time series data over timescales ranging from days to years. A fraction of the variations are due to "edge effects" and depend only on the height of star A above or below the razor sharp edge of the occulting disk. Other observed variations depend on the orbital phase: the H$\alpha$ emission line profile changes from an inverse P Cygni type profile during ingress to an enhanced double-peaked profile, with both a blue and red emission component, during egress. Each of these interpreted variations are complicated by the fact that there is also a chaotic, irregular component present in these profiles. We find that the complex data set can be largely understood in the context of accretion onto the stars from a circumbinary disk with gas flows as predicted by the models of eccentric T Tauri binaries put forward by Artymowicz & Lubow, G\"{u}nther & Kley, and de Val-Borro et al. In particular, our data provide strong support for the pulsed accretion phenomenon, in which enhanced accretion occurs during and after perihelion passage.

A characterization of regular expressions under bisimulation

We solve an open question of Milner [1984]. We define a set of so-called well-behaved finite automata that, modulo bisimulation equivalence, corresponds exactly to the set of regular expressions, and we show how to determine whether a given finite automaton is in this set. As an application, we consider the star height problem.

Classifying regular languages by a split game

In this paper, we introduce the split game, a variant of the Ehrenfeucht–Fraïssé game from logic, which is useful for analyzing the expressive power of classes of generalized regular expressions. An extension of the split game to generalized ω -regular expressions is also established. To gain insight into how the split game can be applied to attack the long-standing generalized star height 2 problem, we propose and solve the omega power problem, a similar but tractable problem in the context of ω -languages. Namely we show that omega powers, together with boolean combinations and concatenations, are not sufficient to express the class of ω -regular languages.

Regular languages and associative language descriptions

The Associative Language Description model (ALD) is a combination of locally testable and constituent structure ideas. It is consistent with current views on brain organization and can rather conveniently describe typical technical languages such as Pascal or HTML. ALD languages are strictly enclosed in context-free languages but in practice the ALD model equals CF grammars in explanatory adequacy. Various properties of ALD have been investigated, but many theoretical questions are still open. For instance, it is unknown, at the present, whether the ALD family includes the regular languages. Here it is proved that several known classes of regular languages are ALD: threshold locally testable languages, group languages, positive commutative languages and commutative languages on 2-letter alphabets. Moreover, we show that there is an ALD language in each level of (restricted) star height hierarchy. These results seem to show that ALD languages are well-distributed over the class of regular languages.

Distance desert automata and the star height problem

We show that it is decidable in time complexity \(2^{2^{2^{O{(n)}}}}\) whether the language accepted by an n-state non-deterministic automaton is of star height one, which is the first ever complexity result for the star height one problem. To achieve this, we introduce distance desert automata as a joint generalization of distance automata and desert automata, and show the decidability of its limitedness problem by solving the underlying Burnside problem.

On lindenmayerian rational subsets of monoids

We define the family of L rational subsets in an arbitrary monoid. We discuss also L rational relations, L rational transductions and L rational star height.

ON THE STAR HEIGHT OF RATIONAL LANGUAGES

Two problems concerning the star height of a rational language are investigated: the star height one problem and the relationships between the unambiguity of an expression and its star height. For this purpose we consider the class of factorial, transitive and rational (FTR) languages. From the algebraic point of view a FTR language is the set of factors of a rational submonoid M. Two subclasses of FTR languages are introduced: renewal languages, corresponding to the case of M finitely generated, and unambiguous renewal languages, corresponding to the case of M finitely generated and free. We prove that a FTR language has star height one if and only if it is renewal. This gives a simple decision procedure for the star height one problem for this class of languages. As concerns the relationships with the ambiguity, we introduce the notion of unambiguous star height of a rational language and prove that it can be strictly greater than the star height. This is a consequence of the following results: i) a FTR language of unambiguous star height one is unambiguous renewal; ii) there exist renewal languages which are not unambiguous renewal. This last result is obtained by using arguments involving the entropy of a language.

More languages of generalised star height 1

Suppose that membership of a word w in a language L is specified in terms of the number (modulo some integer k) of occurrences of a fixed word w′ as a factor of w. Then the language obtained for any m by adding arbitrary numbers of copies of a m anywhere in the words of L has generalized star height at most 1.

Algorithms for determining relative inclusion star height and inclusion star height

Let C=(R 1,⋯,R m) be a finite class of regular languages over a finite alphabet Σ. Let Δ=( b 1,⋯, b m ) be an alphabet, and δ be the substitution from Δ ∗ into Σ ∗ such that δb i )= R i 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 R 10⊂ δ( L)⊂ R 20. This paper proves the existence of an algorithm for determining relative inclusion star height.

Algorithms for determining relative star height and star height

Let C = { R 1 , …, R m } be a finite class of regular languages over a finite alphabet Σ. Let Δ = { b 1 , …, b m } be an alphabet, and δ be the substitution from Δ ∗ into Σ ∗ such that δ ( b i ) = R i for all i (1 ≤ i ≤ m ). Let R be a regular language over Σ which can be defined from C by a finite number of applications of the operators union, concatenation, and star. Then there exist regular languages over Δ which can be transformed onto R by δ. The relative star height of R w.r.t. C is the minimum star height of regular languages over Δ which can be transformed onto R by δ. This paper proves the existence of an algorithm for determining relative star height. This result obviously implies the existence of an algorithm for determining the star height of any regular language.

An application of Cohen's result on star height to the theory of control structures

A result/concept on star height is applied to two problems in the theory of control structures. First, Kosaraju's systems RE n of control structures are generalized and it is shown by using Cohen's result on star height that the generalized systems constitute a pathwise hierarchy with respect to their expressive powers. Second, by using a concept on star height, a result of Peterson et al. ( Comm. ACM 16 (1973), 503–512) that RE ∞ is pathwise complete is sharpened.

Regular languages of star height one

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

The star height of reset-free events and strictly locally testable events

An algorithm is presented for determining the star height of reset-free events and strictly locally testable events.

Characterization of the structure-generating functions of regular sets and the DOL growth functions

The structure-generating functions of regular sets and the DOL growth functions are characterized. Our result is: A rational function f(z) with integral coefficients is a structure-generating function of a regular set if and only if (1) the constant term of its denominator is 1 and that of its numerator is 0, (2) every coefficient an of its Taylor series expansion is nonnegative, and (3) every pole of the minimal absolute value of fi(z) = ∑n=0∞ anM+i zn is of the form rε, where r > 0 and ε is a root of unity for any integer M ⩾ 1 and i = 0, 1,…, M − 1. Also stated are a result on the star height problem and an analogous characterization of the growth functions of DOL systems.

Complexity measures for regular expressions

Several measures of the complexity of a regular expression are defined. (Star height and number of alphabetical symbols are two of them.) Upper and lower estimates for the complexities of expressions for certain sets of paths on graphs are derived.

Homomorphisms that preserve star height

A homomorphism ƒ from the set of words over a finite alphabet Σ 1 to the set of words over another finite alphabet Σ 2 is said to preserve star height if for all regular events R ⊆ Σ 1 * , the star height of R is equal to that of ƒ ( R ). An algorithm is presented for deciding whether an arbitrary homomorphism preserves star height or not: in our result, the injectivity of homomorphisms plays an important role, and it is shown that all homomorphisms which preserve star height are injective.

Rank-non-increasing transformations on transition graphs

This paper is concerned with cycle rank (or, as sometimes called, cycle complexity) of finite transition graphs, a notion which was introduced by Eggan in connection with the notion of star height of regular events. Certain basic types of transformations on transition graphs are introduced and their effect on the cycle rank of the graphs is investigated. It is then shown that any transition graph G can be converted, by a finite series of such transformations, into an equivalent non-deterministic state graph (i.e., transition graph without any empty-word transitions) which has no more nodes than G and no higher cycle rank. A stronger version of Eggan's Star Height Theorem follows.

Substitution expressions

This paper describes a characterization of context-free languages by means of expressions using union, substitution, and an iterative substitution called substitution star. The “substitution expressions” so defined have some properties analogous to those of regular expressions. The well-known cases of linear and regular subfamilies of context-free languages are obtainable from the general model by placing proper restrictions on the basis over which the substitution expressions are defined. The notion of substitution star height of a substitution expression is defined analogously to regular star height for regular expressions. It shown that for each n≥0 there exists a context-free, in fact, linear language whose substitution star height is n.

Techniques for establishing star height of regular sets

This paper deals with restricted star height of regular sets, as defined by L. C. Eggan. Theorems relating the star height of a regular event with the structure of the reduced deterministic automaton recognizing the event are presented. These theorems provide new techniques for establishing star height; in many cases the exact star height of the given regular set can be determined and in others lower bounds to the star height are found. Some results concerning the star height of sets recognized by reset-free state graphs (i.e., partial state graphs in which two transitions labelled by the same input letter never enter the same state) are also obtained, e.g., the star height of any event recognized by a group-free reset-free state graph equals the cycle rank of the state graph.