CS 3240: Languages and Computation

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CS 3240 Languages and Computation

• General LR(1) and LALR(1) Parsing

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Review

• LR(0)
• What are LR(0) items?
• How is DFA for LR(0) defined?
• Simple LR(1)
• How does SLR(1) extends LR(0)?
• General LR(1)
• What are LR(1) items?
• How is DFA for LR(1) items defined?
• Their corresponding parsing algorithms
• Conflicts of actions

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Finite Automata for LR(1)

• Transition from A??.X?,a to A??X.?,a with
  symbol X
• Note that this does not cause new lookahead
• Given LR(1) item A??.B?,a where B is a
  nonterminal, add ?-transition to items B?.?,b
  for every production rule B?? and every token b
  in First(?a)

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GLR(1) Parsing Algorithm

• If statecontains any item A??.X?,a and current
  input token is X, then
• shift X
• push new state ?(A??.X?,a,X)
• If state contains any item A??.,a and next
  input token is a, then
• remove ? and corresponding states from stack
• If rule is S?S.,, then accept if input token
  is
• backup the DFA to the state s from which
  construction of ? began
• push A and new state ?(s,A) onto stack

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Examples

• Example 1
• S ? ( S ) a
• Parse string ((a))
• Example 2
• S ? id V E V ? id E ? V number

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LALR(1) Parsing

• LR(1) has too high complexity (too many states)
• How to reduce number of states?
• If LR(1) items in two states differ only by
  lookahead variables, then merge the states
• Definition The core of a state of the DFA of
  LR(1) items is the set of LR(0) items consisting
  of the first components of all LR(1) items in the
  state.

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Principles of LALR(1) Parsing

• First principle (observation)
• The core of a state of the LR(1) DFA is a state
  of the LR(0) DFA
• Second principle (observation)
• Given two states s1 and s2 of the LR(1) DFA with
  the same core, if there is a transition
  t1?(s1,X), then there is a transition t2?(s2,X)
  where t1 and t2 have the same core
• Based on these principles, we merge LR(1) states
  with the same cores with a set of lookahead
  symbols in each item

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Examples

• Example 1
• Consider grammar S ? ( S ) a
• Parse string ((a))
• Example 2
• S ? id V E V ? id E ? V number

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Remarks

• One can construct the LALR(1) DFA directly from
  LR(0) DFA by propagating lookaheads
• LALR(1) grammars is a subset of LR(1)
• It is possible (bur rarely) that LALR(1)
  introduces conflicts that are not present in
  LR(1)
• Almost all grammars used by programming languages
  are LALR(1)