Franz-Josef Brandenburg

Uniformly growing k-th power-free homomorphisms

A string is called kth power-free, if it does not have xk as a nonempty substring. For all nonnegative rational numbers k, kth power-free strings and kth power-free homomorphisms are investigated and the shortest uniformly growing square-free (k=2) and cube-free (k=3) homomorphisms mapping into least alphabets with three and two-letters are introduced. It is shown that there exist exponentially many square-free and cube-free strings of each length over these alphabets. Sharpening the kth power-freeness to the repetitive threshold RT(n) of n letter alphabets, we provide arguments for the nonexistence of various RT(n)th power-free homomorphisms.

Multiple equality sets and post machines

Abstract Equality sets of finite sets of homomorphisms are studied as part of formal language theory. Some particular equality sets, called Merge k ( k -COPY), are investigated. These languages are combinatorially difficult, and are full semiAFL generators of the recursively enumerable sets, and are semiAFL generators of the class MULTI-RESET, provided k ⩾ 3. To accomplish this characterization, equality sets are related to multihead and multitape Post machines operating in real time. A Post machine has a one-way input tape and Post tapes as storage tapes, which in the multihead version are scanned from left to right by a write head and several read heads. By simulating Post machines by multiple reset machines, and vice versa, several new characterisations of the class MULTI-RESET are obtained, and it is shown that for multihead and multitape Post machines linear time is no more powerful than real time, and two Post tapes or, alternatively, three heads on one Post tape are as powerful as any finite number of heads or tapes. Finally, some complexity bounds for equality sets and Post machines are discussed.

Intersections of Some Families of Languages

What do a pushdown and a queue have in common? What is their intersection? Is it a counter? If we add the one-reversal restriction, is a one-reversal counter exactly the intersection of a one-reversal pushdown and a queue or, symmetrically, the intersection of a one-reset tape and a pushdown? These and similar assumptions can be heard here and there, and there are some conjectures by Autebert et al. [1], Book et al. [3] and Rodriguez

Three write heads are as good ask

Nondeterministic one-wayk-head writing finite automata are investigated. It is shown that 3-head automata with one read-write head followed by two read-only heads are as powerful as any multihead automaton. Furthermore, a new way to detect a coincidence of heads is introduced, which has as a consequence that it makes no difference, whether the heads make stationary moves ore-moves.

Analogies of PAL and COPY

The LIFO or pushdown principle and the FIFO or queue principle are compared in the framework of language theory. To this effect, languages which characteristically describe these principles are studied comparing the least cones or semiAFLs containing them. Although the classes of languages so obtained are different and often are incomparable, very interesting analogies are established between LIFO type languages and FIFO type languages. Thus our investigations show both, the common and the contrasting properties of LIFO and FIFO structures.

On the Height of Syntactical Graphs

Syntactical graphs are representations of derivations of arbitrary grammars. The height of syntactical graphs is introduced here as a complexity measure. It is shown that polynomial height-bounded e-free grammars are equivalent to polynomial time-bounded nondeterministic Turing machines, and that polynomial height-bounded arbitrary grammars are equivalent to polynomial space-bounded Turing machines. Furthermore, context-free languages are linear height-bounded and regular languages are logarithmic height-bounded, even for context-free grammars.

A Truely Morphic Characterization of Recursively Enumerable Sets

Continuing recent research of [3–9] et al. we study the composition of homomorphisms, inverse homomorphisms and twin-morphisms 〈g,h〉 and 〈g,h〉−1, where 〈g,h〉 (w) = g(w) ∩ h(w) and 〈g,h〉−1 (w) = g−1(w) ∩ h−1 (w). We investigate some properties of these morphic mappings and concentrate on a characterization of the recursively enumerable sets, which says: For every recursively enumerable set L there exist four homomorphisms such that L=f1 ° ° f 4 -1 ({

The Contextsensitivity Bounds of Contextsensitive Grammars and Languages

}) and L=h1 ° -1 ° h 4 -1 ({

Uniform simulations of nondeterministic real time multitape turing machines

}), and four homomorphisms are minimal for such representations.

Extended Chomsky-Schützenberger Theorems

In this paper we study the derivational complexity of contextsensitive grammars and languages by placing bounds on their contextsensitivity. The contextsensitivity of a grammar is defined on its derivations, and it is determined by the maximal length of the strings of ancestors of any symbol occurring at any position of the derived strings. A total recursive function f bounds the (right-) contextsensitivity function of grammar G, if for every terminal string x of length n generated by G there is a (right-canonical) derivation from S to x in G whose contextsensitivity is less than or equal to f(n).