What are Precedence Functions in compiler design?

Precedence relations between any two operators or symbols in the precedence table can be converted to two precedence functions f & g that map terminals symbols to integers.

• If a <. b, then f (a) <. g (b)
• If a = b, then f (a) =. g (b)
• If a .> b, then f (a) .> g (b)

Here a, b represents terminal symbols. f (a) and g (b) represents the precedence functions that have an integer value.

Computations of Precedence Functions

• For each terminal a, create the symbol f_a&g_a.
• Make a node for each symbol.

               If a =. b, then fa & gb are in same group or node.

               If a =. b & c =. b, then fa & fc must be in same group or node.

• (a) If a <. b, Mark an edge from g_b to f_a.

             (b) If a .>b, Mark an edge from f_a to g_b.

• If the graph constructed has a cycle, then no precedence functions exist.
• If there are no cycles.

        (a) f_a = Length of longest path beginning at the group of f_a.

        (b) g_a = Length of the longest path from the group of g_a.

Example1 − Construct precedence graph & precedence function for the following table.

Solution

Step1 − Create Symbols

Step2 − No symbol has equal precedence, as can be seen in the given table; therefore, each symbol will remain in a different node.

Step3 − If a <. b, create an edge from f_a → g_a

              If a .>b, create an edge from g_b → f_a

Since, $ <. +,*, id. therefore, make an edge from g₊, g_*, g_id to f_s

Similarity + <. ,∗, id. ∴ make an edge from g_*, g_id to f₊

Similarity * <. id. Therefore, Mark an edge from g_id to f_*.

Since, +,*, id . > $ therefore, Mark an edge from f₊, f_*, f_id to g_s.

Similarity +,*, id . > +. Mark an edge from f₊, f_*, f_id to g₊.

Similarity *, id . > *. Mark an edge from f_*, f_id to g.

Combining all the edges we get

Step4 − Computing the maximum length of the path from each node, we get the following precedence functions

Example2 − Construct precedence graph & precedence function for the following table.

Solution

As we have (=.). Therefore f & g will be in the same group.

Computation of precedence graph

Computation of Precedence Function Table

Since f_$ and g_$ have no outgoing edges, f($ ) = g($ ) = 0.

Since f and g have no outgoing edges, f(( ) = g( )) = 0.

For all others, compute the path of the longest length starting from it.

For Example −

Take f₊,

It has three outgoing edges and traces out its paths.

f₊ → g_$

f₊ → f₍

f₊ → g₊ → f₍ and f₊ → g₊ → f_$

Select the path of maximum length, and the length is 2.

Hence, f₊ = 2. computing all paths of f and g, we get Precedence table

Ginni

Updated on: 30-Oct-2021

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