By David Hopkin, Barbara Moss (auth.)

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**Example text**

If the input word is not in L, then whatever derivation has been followed, at some stage there is a terminal at the top of the string which does not agree with the next character of input, so the string is rejected. :t, defined by Q, ~, n, qo and g, then there is a context-free grammar which generates L. Proof Since the configuration of Mis determined by the entire contents of the pushdown stack and by the state of M, the auxiliary letters of a grammar for L must correspond to combinations of pushdown symbols and states.

So any non- trivial generator must be non-deterministic. The theorems of this section are most easily demonstrated if we also allow acceptors to be nondeterministic. 4 A non -deterministic automaton consists of (a) a finite alphabet (b) a set of states Q, with specified initial state qo, and a subset Q designated as final states (c) a defining mapping g that associates with each pair (qj, Sj) a set of possible moves to new states, with associated output. This set may be empty. It is convenient to regard g as a function from Q x ~ to IP (~ x Q), or to IP (Q) in the case of an acceptor.

A grammar G is defined on JA ={S, A, B}, VT ={a, b} by S -+- aBA, S -+- bB, A-+- aSB, B -+- bA, A ~ a, S -+- b. Construct a pushdown acceptor for L(G) and check that its action on the two input strings bba and ababbaa simulates derivations of these words in G. 4 Context-sensitive Languages Context-sensitive languages form the next level in the hierarchy and are the closest to natural languages. 4). 7). We then give a grammar which does generate this language, and this grammar will be a member of the more general class of context-sensitive grammars.