Conjunctive grammars are a class of formal grammars studied in formal language theory. They extend the basic type of grammars, the context-free grammars, with a conjunction operation. Besides explicit conjunction, conjunctive grammars allow implicit disjunction represented by multiple rules for a single nonterminal symbol, which is the only logical connective expressible in context-free grammars. Conjunction can be used, in particular, to specify intersection of languages. A further extension of conjunctive grammars known as Boolean grammars additionally allows explicit negation. The rules of a conjunctive grammar are of the form where is a nonterminal and , ..., are strings formed of symbols in and (finite sets of terminal and nonterminal symbols respectively). Informally, such a rule asserts that every string over that satisfies each of the syntactical conditions represented by , ..., therefore satisfies the condition defined by . Formal definition A conjunctive grammar is defined by the 4-tuple where is a finite set; each element is called a nonterminal symbol or a variable. Each variable represents a different type of phrase or clause in the sentence. Variables are also sometimes called syntactic categories. is a finite set of terminals, disjoint from , which make up the actual content of the sentence. The set of terminals is the alphabet of the language defined by the grammar . is a finite set of productions, each of the form for some in and . The members of are called the rules or productions of the grammar. is the start variable (or start symbol), used to represent the whole sentence (or program). It must be an element of . It is common to list all right-hand sides for the same left-hand side on the same line, using | (the pipe symbol) to separate them. Rules and can hence be written as . Two equivalent formal definitions of the language specified by a conjunctive grammar exist. One definition is based upon representing the grammar as a system of language equations with union, intersection and concatenation and considering its least solution. The other definition generalizes Chomsky's generative definition of the context-free grammars using rewriting of terms over conjunction and concatenation. Definition by derivation For any strings , we say directly yields , written as , if either there is a rule such that and , or there exists a string such that and . For any string we say generates , written as , if such