Range concatenation grammar

Formal definition A positive range concatenation grammar (PRCG) is a tuple G = (N,~T,~V,~S,~P), where: • N, T and V are disjoint finite sets of (respectively) predicate names, terminal symbols and variable names. Each predicate name has an associated arity given by the function \dim: N \rightarrow \mathbb{N}\setminus\{0\}. • S \in N is the start predicate name and verify \dim(S)=1. • P is a finite set of clauses of the form \psi_0 \rightarrow \psi_1 \ldots \psi_m, where the \psi_i are predicates of the form A_i(\alpha_1, \ldots, \alpha_{\dim(A_i)}) with A_i \in N and \alpha_i \in (T \cup V)^\star. A negative range concatenation grammar (NRCG) is defined like a PRCG, but with the addition that some predicates occurring in the right-hand side of a clause can have the form \overline{A_i(\alpha_1, \ldots, \alpha_{\dim(A_i)})}. Such predicates are called negative predicates. A range concatenation grammar is a positive or a negative one. Although PRCGs are technically NRCGs, the terms are used to highlight the absence (PRCG) or presence (NRCG) of negative predicates. A range in a word w \in T^\star is a couple \langle l, r \rangle_w, with 0 \leq l \leq r \leq n, where n is the length of w. Variables bind to ranges, not to arbitrary strings of nonterminals. Two ranges \langle l_1, r_1 \rangle_w and \langle l_2, r_2 \rangle_w can be concatenated iff r_1 = l_2, and we then have: \langle l_1, r_1 \rangle_w \cdot \langle l_2, r_2 \rangle_w = \langle l_1, r_2 \rangle_w. When instantiating a clause, where an argument consists of multiple elements from T \cup V, their ranges must concatenate. For a word w = w_1w_2\ldots w_n, with w_i \in T, the dotted notation for ranges is: \langle l, r \rangle_w = w_1\ldots w_{l-1} \bullet w_l\ldots w_{r-1} \bullet w_r\ldots w_n. Recognition of strings The strings of predicates being rewritten represent constraints that the string being tested has to satisfy (if positive), or in the case of negative predicates not satisfy. The order of predicates is irrelevant. Rewrite steps amount to replacing one constraint by zero or more simpler constraints. Like LMGs, RCG clauses have the general schema A(x_1, ..., x_n) \to \alpha, where in an RCG, \alpha is either the empty string or a string of predicates. The arguments x_i consist of strings of terminal symbols and/or variable symbols, which pattern match against actual argument values like in LMG. Adjacent variables constitute a family of matches against partitions, so that the argument xy, with two variables, matches the literal string ab in three different ways: x = \epsilon,\ y = ab;\ x = a,\ y = b;\ x = ab,\ y = \epsilon. These would give rise to three different instantiations of the clause containing that argument xy. Predicate terms come in two forms, positive (which produce the empty string on success), and negative (which produce the empty string on failure/if the positive term does not produce the empty string). Negative terms are denoted the same as positive terms, with an overbar, as in \overline{A(x_1, ..., x_n)}. The rewrite semantics for RCGs is rather simple, identical to the corresponding semantics of LMGs. Given a predicate string A(\alpha_1, ..., \alpha_n), where the symbols \alpha_i are terminal strings, if there is a rule A(x_1, ..., x_n) \to \beta in the grammar that the predicate string matches, the predicate string is replaced by \beta, substituting for the matched variables in each x_i. For example, given the rule A(x, ayb) \to B(axb, y), where x and y are variable symbols and a and b are terminal symbols, the predicate string A(a, abb) can be rewritten as B(aab, b), because A(a, abb) matches A(x, ayb) when x = a,\ y = b. Similarly, if there were a rule A(x, ayb) \to A(x, x)\ A(y, y), A(a, abb) could be rewritten as A(a, a)\ A(b, b). A proof/recognition of a string \alpha is done by showing that S(\alpha) produces the empty string. For the individual rewrite steps, when multiple alternative variable matches are possible, any rewrite which could lead the whole proof to succeed is considered. Thus, if there is at least one way to produce the empty string from the initial string S(\alpha), the proof is considered a success, regardless of how many other ways to fail exist. ==Example==