Wirth-Weber precedence relationship

The Wirth-Weber relationship between a pair of symbols $$(V_t cup V_n) 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 prefixes have the pivot and must be reduced. A $$gtrdot means that the pivot is found, a $$lessdot means that a potential pivot is starting, and a $$dot = means that we are still in the same pivot.

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=Formal definition=$$G =

* $$X dot = Y iff egin{cases} A o alpha X Y eta in P \ A in V_n \ alpha , eta in (V_n cup V_t)^* \ X, Y in (V_n cup V_t) end{cases}

* $$X lessdot Y iff egin{cases} A o alpha X B eta in P \ B Rightarrow^+ Y gamma \ A, B in V_n \ alpha , eta, gamma in (V_n cup V_t)^* \ X, Y in (V_n cup V_t) end{cases}

* $$X gtrdot a iff egin{cases} A o alpha B Y eta in P \ B Rightarrow^+ gamma X \ Y Rightarrow^* a delta \ A, B in V_n \ alpha , eta, gamma, delta in (V_n cup V_t)^* \ X, Y in (V_n cup V_t) \ a in V_t end{cases}

=Precedence Relations Computing Algorithm=

We will define three Sets for a symbol:

* $$Head^+(X) = {Y|X Rightarrow^+ Y alpha }
* $$Tail^+(X) = {Y|X Rightarrow^+ alpha Y }
* $$Head^*(X) = (Head^+(X) cup { X }) cap V_t
"Note that Head^*(X) is X if X is a terminal, and if X is a non-terminal, Head^*(X) is the set with only the terminals belonging to Head⁺(X). This set is equivalent to First-set or Fi(X) described in LL parser"

The pseudocode for computing relations is:

* RelationTable := $$empty
* For each production $$A o alpha in P
** For each two adjacent symbols X Y in α
*** add(RelationTable,$$X dot = Y)
*** add(RelationTable,$$X lessdot Head^+(Y))
*** add(RelationTable,$$Tail^+(X) gtrdot Head^*(Y))
* add(RelationTable,$$$ lessdot Head^+(S)) where S is the initial non terminal of the grammar, and $ is a limit marker
* add(RelationTable,$$Tail^+(S) gtrdot $ ) where S is the initial non terminal of the grammar, and $ is a limit marker

"Note that $$lessdot and $$gtrdot 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=

$$S o aSSb | c

precedence table: