Formal grammars Lectures ??–??. Linear grammars and trellis automata

The linear case of context-free, conjunctive and Boolean grammars. Trellis automata. Examples. Equivalence of grammars and automata. 1 Linear Boolean grammars Definition 1. A Boolean grammar G = (Σ, N, P, S) is said to be linear if every rule A→ α1& . . .&αm&¬β1& . . .&¬βn, has αi, βj ∈ Σ∗ ∪ (Σ ∪N)∗. If a linear Boolean grammar is conjunctive (context-free), it is called a linear conjunctive (linear context-free) grammar. The following normal form similar to the binary normal form can be defined for linear Boolean grammars. Definition 2. A linear Boolean grammar G = (Σ, N, P, S) is said to be in the linear normal form if every rule in P is of the form A→ bB1& . . . bBm&C1c& . . .&Cnc&¬bD1& . . .&¬bDk& &¬E1c& . . .&¬E`c (m,n, k, ` > 0; m+ n > 1), A→ a S → ε (only if S does not appear in right hand sides of rules) Theorem 1. For every linear Boolean grammar there exists a linear Boolean grammar in the linear normal form generating the same language. If the grammar is linear conjunctive (linear context-free), then so is the resulting grammar in the linear normal form. The construction is basically the same as used to transform a Boolean grammar to the binary normal form. Note that the given methods for eliminating ε conjuncts and unit conjuncts preserve linearity of a grammar. Motivation for TA: consider the basic cubic-time recognition algorithm for linear Boolean grammars, etc., etc.