Useful modifications to grammars

1. Useful modifications to grammars • Ambiguity elimination • undecidable in general case whether a grammar is ambiguous • not always possible to remove ambiguity – some inherently ambiguous grammars • no algorithm exists for always removing ambiguity • Left recursion elimination • can result in empty productions – which can be eliminated too • Left factoring • combine productions with same initial RHS subsequence • by introducing new nonterminals • Add sentence-final sentinel ‘$’ to inputs, add production S’ ::= S $ http://csiweb.ucd.ie/staff/acater/comp30330.html Compiler Construction

2. Recursive Descent parsers • A simple top-down parsing technique that requires backtracking • Constructs a leftmost derivation • Decide which production to choose, then match its RHS Proc A () choose the correct production A ::= X1 X2 … Xk for i from 1 to k if Xi is a terminal then if nextToken()=Xi then join it to the parse tree else error() else join subtree for Xi to the parse tree, and use proc Xi http://csiweb.ucd.ie/staff/acater/comp30330.html Compiler Construction

3. Predictive parser • A predictive parser does not need to backtrack • It chooses correct production by lookahead of k symbols • Grammars for which it is possible to construct such parsers are LL(k) grammars. • LL(1) is the important case. • They depend on knowledge of FIRST and FOLLOW sets • which are also useful for bottom-up parsers later • FOLLOW sets are also useful for error recovery • Productions are stored in a predictive parsing table indexed by • the nonterminal to be recognised • the terminal found in next position in input stream http://csiweb.ucd.ie/staff/acater/comp30330.html Compiler Construction

4. FIRST sets • Meaningful for a particular grammar • FIRST( β ) = the set of terminals that may begin strings derived from β, • may include e iff be • Motivation: If there are some productions A ::= γ | δ, and FIRST( γ ) and FIRST( δ ) are disjoint, the next symbol can distinguish between these productions • A simple algorithm computes FIRST http://csiweb.ucd.ie/staff/acater/comp30330.html Compiler Construction

5. FOLLOW sets • Meaningful for the nonterminals of a particular grammar • FOLLOW( A) = set of terminals that can follow A in some sentential form • i.e. the set of d such that S aAdg • includes sentinel $ iff S aA • FOLLOW sets are computed by a simple algorithm http://csiweb.ucd.ie/staff/acater/comp30330.html Compiler Construction

6. Computing FIRST and FOLLOW For grammar symbol X, If X is a terminal, FIRST(X) = {X} If X ::= Y1 Y2 … Yk is a production, k>0 add all of FIRST(Y1) except e into FIRST(X), then if FIRST(Y1) contains e add all of FIRST(Y2) except e, and so on If X ::= e is a production, add e to FIRST(X) For string of symbols X1X2…Xn, Add non-e symbols of FIRST(X1), then non-e symbols of each FIRST(Xk) if e is in all FIRST(Xj) for j<k, then e if e is in all FIRST(Xj), 1<=j<=n Place $ in FOLLOW(S) If A ::= aBb is a production, put everything in FIRST(b) except e into both FOLLOW(B) … and anything that copied (in)directly from FOLLOW(B) If A ::= aB is a production, or A ::= aBb and be then add everything in FOLLOW(B) into FOLLOW(A) etc http://csiweb.ucd.ie/staff/acater/comp30330.html Compiler Construction

7. LL(1) grammar properties • LL(1) grammars are never left-recursive • LL(1) grammars are never ambiguous • Necessary & Sufficient Condition: • For any two productions A ::= a | b • FIRST( a ) and FIRST ( b ) are disjoint • If be then FOLLOW(A) and FIRST( a ) are disjoint • &vv – although it is unnecessary to say so since a and b are any two http://csiweb.ucd.ie/staff/acater/comp30330.html Compiler Construction

8. Predictive Parsing Table • Indexed by (1) Nonterminal and (2) next input token • Empty cells correspond to error • Multiply-filled cells occur when the grammar is ambiguous http://csiweb.ucd.ie/staff/acater/comp30330.html Compiler Construction