LR(1) Parsers Part III Last Parser Lecture. Copyright 2010, Keith D. Cooper & Linda Torczon, all rights reserved.

LR(1) Parser Recap Bottom-up conceptual algorithm LR(1) Skeleton Parser Construct Control DFA Closure and goto functions Today Construct ACTION AND GOTO tables

Example from SheepNoise S 0 : { [Goal SheepNoise, EOF], [SheepNoise SheepNoise baa, EOF], [SheepNoise baa, EOF], [SheepNoise SheepNoise baa, baa], [SheepNoise baa, baa] } S 1 = Goto(S 0, SheepNoise) = { [Goal SheepNoise, EOF], [SheepNoise SheepNoise baa, EOF], [SheepNoise SheepNoise baa, baa] } S 2 = Goto(S 0, baa) = { [SheepNoise baa, EOF], [SheepNoise baa, baa] } S 3 = Goto(S 1, baa) = { [SheepNoise SheepNoise baa, EOF], [SheepNoise SheepNoise baa, baa] } 0 Goal SheepNoise 1 SheepNoise SheepNoise baa 2 baa

Filling in the ACTION and GOTO Tables The algorithm x is the state number set S x CC item i S x if i is [A β aδ,b] and goto(s x,a) = S k, a T then ACTION[x,a] shift k else if i is [S S,EOF] then ACTION[x,EOF] accept else if i is [A β,a] then ACTION[x,a] reduce A β n NT if goto(s x,n) = S k then GOTO[x,n] k

Filling in the ACTION and GOTO Tables The algorithm set S x CC before T shift item i S x if i is [A β aδ,b] and goto(s x,a) = S k, a T then ACTION[x,a] shift k else if i is [S S,EOF] then ACTION[x,EOF] accept else if i is [A β,a] then ACTION[x,a] reduce A β n NT if goto(s x,n) = S k then GOTO[x,n] k

Filling in the ACTION and GOTO Tables The algorithm set S x CC item i S x if i is [A β aδ,b] and goto(s x,a) = S k, a T then ACTION[x,a] shift k else if i is [S S,EOF] then ACTION[x,EOF] accept else if i is [A β,a] then ACTION[x,a] reduce A β n NT if goto(s x,n) = S k then GOTO[x,n] k have Goal accept

Example from SheepNoise S 0 : { [Goal SheepNoise, EOF], [SheepNoise SheepNoise baa, EOF], [SheepNoise baa, EOF], [SheepNoise SheepNoise baa, baa], [SheepNoise baa, baa] } S 1 = Goto(S 0, SheepNoise) = { [Goal SheepNoise, EOF], [SheepNoise SheepNoise baa, EOF], [SheepNoise SheepNoise baa, baa] } S 2 = Goto(S 0, baa) = { [SheepNoise baa, EOF], before T shift k [SheepNoise baa, baa] } S 3 = Goto(S 1, baa) = { [SheepNoise SheepNoise baa, EOF], 0 Goal SheepNoise 1 SheepNoise SheepNoise baa 2 baa if i is [A β aδ,b] and goto(s x,a) = S k, a T [SheepNoise SheepNoise baa, baa] } then ACTION[x,a] shift k

Example from SheepNoise S 0 : { [Goal SheepNoise, EOF], [SheepNoise SheepNoise baa, EOF], [SheepNoise baa, EOF], [SheepNoise SheepNoise baa, baa], [SheepNoise baa, baa] } S 1 = Goto(S 0, SheepNoise) = before T shift k { [Goal SheepNoise, EOF], [SheepNoise SheepNoise baa, EOF], [SheepNoise SheepNoise baa, baa] } S 2 = Goto(S 0, baa) = { [SheepNoise baa, EOF], [SheepNoise baa, baa] } so, ACTION[s 0,baa] is shift S 2 (clause 1) (items define same entry) S 3 = Goto(S 1, baa) = { [SheepNoise SheepNoise baa, EOF], [SheepNoise SheepNoise baa, baa] } 0 Goal SheepNoise 1 SheepNoise SheepNoise baa 2 baa

Example from SheepNoise S 0 : { [Goal SheepNoise, EOF], [SheepNoise SheepNoise baa, EOF], [SheepNoise baa, EOF], [SheepNoise SheepNoise baa, baa], [SheepNoise baa, baa] } S 1 = Goto(S 0, SheepNoise) = { [Goal SheepNoise, EOF], [SheepNoise SheepNoise baa, EOF], [SheepNoise SheepNoise baa, baa] } S 2 = Goto(S 0, baa) = { [SheepNoise baa, EOF], [SheepNoise baa, baa] } so, ACTION[S 1,baa] is shift S 3 (clause 1) S 3 = Goto(S 1, baa) = { [SheepNoise SheepNoise baa, EOF], [SheepNoise SheepNoise baa, baa] } 0 Goal SheepNoise 1 SheepNoise SheepNoise baa 2 baa if i is [A β aδ,b] and goto(s x,a) = S k, a T then ACTION[x,a] shift k

Example from SheepNoise S 0 : { [Goal SheepNoise, EOF], [SheepNoise SheepNoise baa, EOF], [SheepNoise baa, EOF], [SheepNoise SheepNoise baa, baa], [SheepNoise baa, baa] } S 1 = Goto(S 0, SheepNoise) = { [Goal SheepNoise, EOF], [SheepNoise SheepNoise baa, EOF], [SheepNoise SheepNoise baa, baa] } so, ACTION[S 1,EOF] is accept (clause 2) S 2 = Goto(S 0, baa) = { [SheepNoise baa, EOF], [SheepNoise baa, baa] } S 3 = Goto(S 1, baa) = { [SheepNoise SheepNoise baa, EOF], 0 Goal SheepNoise 1 SheepNoise SheepNoise baa 2 baa else if i is [S S,EOF] [SheepNoise SheepNoise baa, baa] } then ACTION[x,EOF] accept

Example from SheepNoise S 0 : { [Goal SheepNoise, EOF], [SheepNoise SheepNoise baa, EOF], [SheepNoise baa, EOF], [SheepNoise SheepNoise baa, baa], [SheepNoise baa, baa] } S 1 = Goto(S 0, SheepNoise) = { [Goal SheepNoise, EOF], [SheepNoise SheepNoise baa, EOF], [SheepNoise SheepNoise baa, baa] } S 2 = Goto(S 0, baa) = { [SheepNoise baa, EOF], [SheepNoise baa, baa] } ACTION[S 2,baa] is reduce 2 [SheepNoise SheepNoise baa, baa] } S 3 = Goto(S 1, baa) = { [SheepNoise SheepNoise baa, EOF], 0 Goal SheepNoise 1 SheepNoise SheepNoise baa 2 baa so, ACTION[S 2,EOF] is reduce 2 else if i is [A β,a] then ACTION[x,a] reduce A β

Example from SheepNoise S 0 : { [Goal SheepNoise, EOF], [SheepNoise SheepNoise baa, EOF], [SheepNoise baa, EOF], [SheepNoise SheepNoise baa, baa], [SheepNoise baa, baa] } S 1 = Goto(S 0, SheepNoise) = else if i is [A β,a] then ACTION[x,a] reduce A β [SheepNoise baa, baa] } { [Goal SheepNoise, EOF], [SheepNoise SheepNoise baa, EOF], [SheepNoise SheepNoise baa, baa] } S 2 = Goto(S 0, baa) = { [SheepNoise baa, EOF], S 3 = Goto(S 1, baa) = { [SheepNoise SheepNoise baa, EOF], [SheepNoise SheepNoise baa, baa] } ACTION[S 3,EOF] is reduce 1 0 Goal SheepNoise 1 SheepNoise SheepNoise baa 2 baa ACTION[S 3,baa] is reduce 1, as well

Example from SheepNoise The GOTO Table records Goto transitions on NTs s 0 : { [Goal SheepNoise, EOF], [SheepNoise SheepNoise baa, EOF], [SheepNoise baa, EOF], [SheepNoise SheepNoise baa, baa], [SheepNoise baa, baa] } s 1 = Goto(S 0, SheepNoise) = { [Goal SheepNoise, EOF], [SheepNoise SheepNoise baa, EOF], [SheepNoise SheepNoise baa, baa] } Puts s 1 in GOTO[s 0,SheepNoise] s 2 = Goto(S 0, baa) = { [SheepNoise baa, EOF], [SheepNoise baa, baa] } Based on T, not NT and written into the ACTION table s 3 = Goto(S 1, baa) = { [SheepNoise SheepNoise baa, EOF], [SheepNoise SheepNoise baa, baa] } 0 Goal SheepNoise 1 SheepNoise SheepNoise baa 2 baa Only 1 transition in the entire GOTO table Remember, we recorded these so we don t need to recompute them.

ACTION & GOTO Tables Here are the tables for the SheepNoise grammar The tables ACTION TABLE State EOF baa 0 shift 2 1 accept shift 3 2 reduce 2 reduce 2 3 reduce 1 reduce 1 GOTO TABLE State SheepNoise 0 1 1 0 2 0 3 0 The grammar 0 Goal SheepNoise 1 SheepNoise SheepNoise baa 2 baa

What can go wrong? Shift/reduce error What if set s contains [A β aγ,b] and [B β,a]? First item generates shift, second generates reduce Both set ACTION[s,a] cannot do both actions This is ambiguity, called a shift/reduce error Modify the grammar to eliminate it Shifting will often resolve it correctly (if-then-else)

What can go wrong? Reduce/reduce conflict What is set s contains [A γ, a] and [B γ, a]? Each generates reduce, but with a different production Both set ACTION[s,a] cannot do both reductions This ambiguity is called reduce/reduce conflict Modify the grammar to eliminate it (PL/I s overloading of (...)) In either case, the grammar is not LR(1)

CYK Parser Simple context-free-language parser Worse-case running time is O(n 3 ), space is O(n 2 ) Employs bottom-up parsing and dynamic programming Shunned for many years Even tabular methods [CYK, Earley] should be avoided if the language at hand has a grammar for which more efficient algorithms [LL, LALR] are available. The Theory of Parsing., Aho, Ullman, 1972 But in practice, running time is more like O(n 1.2 ) - Plus computers are now 1,000,000-times faster than in 1972 - And (more importantly) CYK parser is easily parallelizable! Source: Ras Bodik, Slides: Browsing Web 3.0 on 3.0 Watts

CYK Parser (Sequential Version)

The CYK Parser Algorithm (Sequential Version)

Phase IV: Semantic Checking