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Bottom-Up Parsing Basics

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Example: Rightmost Derivation

S -> ABc A -> Aa | aB -> bF | bF -> f F | f

Input String: aaabffc

S

A B c

A a b F

A a f F

a f

S => ABc => AbFc => AbfFc => Abffc => Aabffc =>* aaabffc

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Example: Symbol Stack

Input String: aa | abffcSymbol Stack: A

S

A B c

A a b F

A a f F

a f

Input String: aaab | ffcSymbol Stack: Ab

Input String: aaabff | cSymbol Stack: AbF

Input String: aaabff | cSymbol Stack: AB

The symbol stack summarizes the consumed string.

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Example: Item (NFA state)

Input String: aa | abffc

NFA State : A -> A.a S

A B c

A a b F

A a f F

a f

Input String: aaab | ffc

NFA State : B -> b.F

Input String: aaabff | c

NFA State : B -> bF. Input String: aaabff | c

NFA State : S -> AB.c

The NFA state shows the part of a rule reachable using the consumed string.

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Example: State transitions

Input String: aaa | bffcaaab|ffc aaabf|fc

S

A B c

A a b F

A a f F

a f

S => A . B c

=> A b . F c

=> A b f . F c

S -> A.BcB -> .bFB -> .b …

B -> b.FB -> b.F -> .f FF -> .f …

b

State Transition

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Symbol stack and State stack

S -> .ABc

A -> .Aa

A -> .a

Input String: $ aaa b f | fc

Symbol Stack: $ A b f

S -> A . Bc

A -> A.aB -> .bF

B -> .b

B -> b.FB -> b. F -> . f FF -> . f

F -> f.FF -> f.

A b f

State stack

reduce shift

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Note 1 : Redundancy

• The state stack can be generated from the sequence of symbols on the symbol stack, by tracing the derivation from the start symbol, using the productions.– The state stack is used for efficiency.– It avoids running the entire DFA

(containing production fragments) to determine the next state each time.

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Why trace the entire symbol stack?

S -> [ A ] | { B }A -> a A | aB -> a B | a

It is not possible to determine the “current” production (state)by merely inspecting a fixed number of stack symbols.

Input 1: $ [ a a a | a ]Symbol Stack: $ [ a a a State:

A -> a . AA -> .a

Input 2: $ { a a a | a }Symbol Stack: $ { a a a State:

B -> a . BB -> .a

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DFA State : Input strings with eqvt. history

: RHS of rules sharing a prefix

Input String1: ab | ffcInput String2: ab | ge

Symbol Stack: $ A b

State

B -> b.FB -> b. F -> . f FF -> . fD -> b.g …

The DFA state captures all possible rule prefixes reachable from the start symbol using the consumed input.

S -> ABc | ADeA -> Aa | aB -> bF | bD -> bg | dBF -> f F | f

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Stacking states is adequate

S => A B c => A b c S => A D e => A d B e => A d b e

The effect of reduction by B -> b depends on the previous state.

S -> A.Bc

B -> .b S -> AB.c

D -> d.BB -> .b

B -> b.

B

b

D -> dB.B

S -> A.De

D -> .dB

D -> .bg

d

b

S -> AD.eD

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Note 2 : Redundancy

• The state stack contains enough information to regenerate the symbol stack. Note that all arcs into a DFA state carry the same symbol label.– As a consequence, the symbol stack can be

avoided by making the “reduce”-entry in the parse table specify the number of stack elements (length of RHS) to be popped and the (LHS) nonterminal to be pushed to make the transition.

– In practice, the (synthesized) semantic attribute associated with a stack symbol is allocated and computed on a parallel stack.

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Relation to Grammars

• The DFA naturally corresponds to a left-linear grammar.

S -> A.Bc

B -> .b

B -> b. b

Grammar Rule: P Q bQ P

• The DFA state is obtained from the NFA states using the standard construction.