|
|
| Line 1: |
Line 1: |
| {{multiple issues|
| | Legal Executive Renaldo from Milton, likes to spend time bonsai trees, property developers [http://simply-affordable.com/activity/p/584/ housing in singapore] singapore and yoyo. Finds the entire world an fascinating place we have spent 2 months at Ningaloo Coast. |
| {{Unreferenced|date=December 2009}}
| |
| {{Context|date=October 2009}}
| |
| {{Technical|date=July 2012}}
| |
| }}
| |
| | |
| The [[Niklaus Wirth|Wirth]]-[[Helmut Weber|Weber]] relationship between a pair of symbols <math>(V_t \cup V_n)</math> is necessary to determine if a [[formal grammar]] is a [[Simple precedence grammar]], and in such case the [[Simple precedence parser]] can be used.
| |
| | |
| The goal is to identify the when the [[viable prefix]]es have the '''pivot''' and must be reduced. A <math>\gtrdot</math> means that the '''pivot''' is found, a <math>\lessdot</math> means that a potential '''pivot''' is starting, and a <math>\dot =</math> means that we are still in the same '''pivot'''.
| |
| | |
| __TOC__
| |
| | |
| ==Formal definition==
| |
| <math> G = <V_n, V_t, S, P> </math>
| |
| | |
| * <math> X \dot = Y \iff \begin{cases} A \to \alpha X Y \beta \in P \\ A \in V_n \\ \alpha , \beta \in (V_n \cup V_t)^* \\ X, Y \in (V_n \cup V_t) \end{cases} </math>
| |
| | |
| * <math> X \lessdot Y \iff \begin{cases} A \to \alpha X B \beta \in P \\ B \Rightarrow^+ Y \gamma \\ A, B \in V_n \\ \alpha , \beta, \gamma \in (V_n \cup V_t)^* \\ X, Y \in (V_n \cup V_t) \end{cases} </math>
| |
| | |
| * <math> X \gtrdot a \iff \begin{cases} A \to \alpha B Y \beta \in P \\ B \Rightarrow^+ \gamma X \\ Y \Rightarrow^* a \delta \\ A, B \in V_n \\ \alpha , \beta, \gamma, \delta \in (V_n \cup V_t)^* \\ X, Y \in (V_n \cup V_t) \\ a \in V_t \end{cases} </math>
| |
| | |
| ==Precedence Relations Computing Algorithm==
| |
| We will define three Sets for a symbol:
| |
| | |
| * <math>Head^+(X) = \{Y|X \Rightarrow^+ Y \alpha \}</math>
| |
| * <math>Tail^+(X) = \{Y|X \Rightarrow^+ \alpha Y \}</math>
| |
| * <math>Head^*(X) = (Head^+(X) \cup \{ X \}) \cap V_t</math>
| |
| <br>
| |
| ''Note that Head<sup>*</sup>(X) is X if X is a terminal, and if X is a non-terminal, Head<sup>*</sup>(X) is the set with only the terminals belonging to Head<sup>+</sup>(X). This set is equivalent to '''First-set''' or '''Fi(X)''' described in [[LL parser]]''
| |
| | |
| ''Note that Head<sup>+</sup>(X) and Tail<sup>+</sup>(X) are <math>\empty</math> if X is a terminal.''
| |
| | |
| <br>
| |
| The pseudocode for computing relations is:
| |
| | |
| * RelationTable := <math>\empty</math>
| |
| * For each production <math> A \to \alpha \in P </math>
| |
| ** For each two adjacent symbols X Y in α
| |
| *** add(RelationTable,<math>X \dot = Y</math>)
| |
| *** add(RelationTable,<math>X \lessdot Head^+(Y)</math>)
| |
| *** add(RelationTable,<math>Tail^+(X) \gtrdot Head^*(Y)</math>)
| |
| * add(RelationTable,<math>$ \lessdot Head^+(S)</math>) where S is the initial non terminal of the grammar, and $ is a limit marker
| |
| * add(RelationTable,<math>Tail^+(S) \gtrdot $ </math>) where S is the initial non terminal of the grammar, and $ is a limit marker
| |
| | |
| ''Note that <math>\lessdot</math> and <math>\gtrdot</math> are used with sets instead of elements as they were defined, in this case you must add all the [[cartesian product]] between the sets/elements''
| |
| | |
| ==Examples==
| |
| <math>S \to aSSb | c</math>
| |
| | |
| * Head+(a) = <math>\empty</math>
| |
| * Head+(S) = { a, c}
| |
| * Head+(b) = <math>\empty</math>
| |
| * Head+(c) = <math>\empty</math>
| |
| * Tail+(a) = <math>\empty</math>
| |
| * Tail+(S) = { b, c}
| |
| * Tail+(b) = <math>\empty</math>
| |
| * Tail+(c) = <math>\empty</math>
| |
| * Head*(a) = a
| |
| * Head*(S) = { a, c}
| |
| * Head*(b) = b
| |
| * Head*(c) = c
| |
| | |
| * <math>S \to aSSb</math>
| |
| ** '''a''' Next to '''S'''
| |
| *** a <math>\dot =</math> S
| |
| *** a <math>\lessdot</math> Head+(S)
| |
| **** a <math>\lessdot</math> a
| |
| **** a <math>\lessdot</math> c
| |
| ** '''S''' Next to '''S'''
| |
| *** S <math>\dot =</math> S
| |
| *** S <math>\lessdot</math> Head+(S)
| |
| **** S <math>\lessdot</math> a
| |
| **** S <math>\lessdot</math> c
| |
| *** Tail+(S) <math>\gtrdot</math> Head*(S)
| |
| **** b <math>\gtrdot</math> a
| |
| **** b <math>\gtrdot</math> c
| |
| **** c <math>\gtrdot</math> a
| |
| **** c <math>\gtrdot</math> c
| |
| ** '''S''' Next to '''b'''
| |
| *** S <math>\dot =</math> b
| |
| *** Tail+(S) <math>\gtrdot</math> Head*(b)
| |
| **** b <math>\gtrdot</math> b
| |
| **** c <math>\gtrdot</math> b
| |
| * <math>S \to c</math>
| |
| ** there is only one symbol, so no relation is added.
| |
| | |
| precedence table:
| |
| {|border="1" cellpadding="2"
| |
| | ||S||a||b||c||$
| |
| |-
| |
| |S||<math>\dot =</math>|| <math>\lessdot</math> || <math>\dot =</math> || <math>\lessdot</math>||
| |
| |-
| |
| |a||<math>\dot =</math>||<math>\lessdot</math>|| ||<math>\lessdot</math>||
| |
| |-
| |
| |b|| ||<math>\gtrdot</math>|| <math>\gtrdot</math> ||<math>\gtrdot</math>||<math>\gtrdot</math>
| |
| |-
| |
| |c|| ||<math>\gtrdot</math>||<math>\gtrdot</math>||<math>\gtrdot</math>||<math>\gtrdot</math>
| |
| |-
| |
| |$|| ||<math>\lessdot</math>|| ||<math>\lessdot</math>||
| |
| |}
| |
| | |
| {{DEFAULTSORT:Wirth-Weber Precedence Relationship}}
| |
| [[Category:Formal languages]]
| |
Legal Executive Renaldo from Milton, likes to spend time bonsai trees, property developers housing in singapore singapore and yoyo. Finds the entire world an fascinating place we have spent 2 months at Ningaloo Coast.