Complete characterization of context-sensitive languages

Intrinsic complete characterizations of constructive, context-free and regular languages have been formulated by means of configurations of languages. The definition of a semiconfiguration is given here by generalizing the definition of a configuration. By means of semiconfigurations, an intrinsic complete characterization of context-sensitive languages is formulated. 1. Languages and generalized grammars. If V is a set we denote by V* the free monoid over V, i.e. the set of all finite sequences of elements of the set V including the empty sequence A this set being provided by the binary operation of concatenation. We identify one-member-sequences with elements of V; it follows V £ V*. If x = X]X2, ... x " e V* where n is a natural number and xt e V for i = An ordered pair (V L) where V is a set and L £ V* is called a language. The elements of V* are called strings. If (V, L), (U, M) are languages then we define the intersection (V, L) n (U, M) of these languages by the formula (V L) n (U, M) = = (Vn U,LnM). Let Vbe a set, suppose R £ V* x V*. Let us have x, y e V*. We writte x-• y(R) if (x, y) e R. Further, we put x =* • y(R) if there exist such strings u, v, t, z e V* that x = utv, uzv =-y, t-> z(R). Finally, we write x =>* • y(R) if there exist an integer p ^ 0 and some strings x = t0, tu ..., tp = y in V* that tt_x=> r,(R) for i = 1,2,... ..., p. Then the sequence of strings (t,)f=o is called an x-derivation of y of length pinR.