1. Table-driven parsing Parsing performed by a finite state machine.
Parsing algorithm is language-independent.
FSM driven by table (s) generated automatically from grammar.
Language generator tables
Input
parser
stack
tables

2. Pushdown Automata
A context-free grammar can be recognized by a finite state machine with a stack: a PDA.
The PDA is defined by set of internal states and a transition table.
The PDA can read the input and read/write on the stack.
The actions of the PDA are determined by its current state, the current top of the stack, and the current input symbol.
There are three distinguished states:
start state: nothing seen
accept state: sentence complete
error state: current symbol doesn’t belong.

3. Top-down parsing
Parse tree is synthesized from the root (sentence symbol).
Stack contains symbols of rhs of current production, and pending non-terminals.
Automaton is trivial (no need for explicit states)
Transition table indexed by grammar symbol G and input symbol a. Entries in table are terminals or productions: P ABC…

4. Top-down parsing Actions:
initially, stack contains sentence symbol
At each step, let S be symbol on top of stack, and a be the next token on input.
if T (S, a) is terminal a, read token, pop symbol from stack
if T (S, a) is production P ABC…., remove S from stack, push the symbols A, B, C on the stack (A on top).
If S is the sentence symbol and a is the end of file, accept.
If T (S, a) is undefined, signal error.
Semantic action: when starting a production, build tree node for non-terminal, attach to parent.

5. Table-driven parsing and recursive descent parsing
Recursive descent: every production is a procedure. Call stack holds active procedures corresponding to pending non-terminals.
Table-driven parser: recursion simulated with explicit stack.

6. Building the parse table
Define two functions on the symbols of the grammar: FIRST and FOLLOW.
For a non-terminal N, FIRST (N) is the set of terminal symbols that can start any derivation from N.
First (If_Statement) = {if}
First (Expr) = {id, ( }
FOLLOW (N) is the set of non-terminals that can appear after a string derived from N:
Follow (Expr) = {+, ), $ }

7. Computing FIRST (N) If N e First (N) includes e
if N aABC First (N) includes a
if N X1X2 First (N) includes First (X1)
if N X1X2… and X e,
First (N) includes First (X2)
Obvious generalization to First (a) where a is X1X2...

8. Computing First (N) Grammar for expressions, without left-recursion:
E TE’ | T
E’ TE’ | e
T FT’ | F
T’ *FT’ | e
F id | (E)
First (F) = { id, ( }
First (T’) = { *, e} First (T) = { id, ( }
First (E’) = { +, e} First (E) = { id, ( }

9. Computing Follow (N)
Follow (N) is computed from productions in which N appears on the rhs
For the sentence symbol S, Follow (S) includes $
if A a N b, Follow (N) includes First (b)
because an expansion of N will be followed by an expansion from b
if A a N, Follow (N) includes Follow (A)
because N will be expanded in the context in which A is expanded
if A a N B , B e, Follow (N) includes Follow (A)

10. Computing Follow (N) Follow (E) = { ), $ } Follow (E’) = { ), $ }
E TE’ | T
E’ TE’ | e
T FT’ | F
T’ *FT’ | e
F id | (E)
Follow (E) = { ), $ } Follow (E’) = { ), $ }
Follow (T) = First (E’ ) + Follow (E’) = { +, ), $ }
Follow (T’) = Follow (T) = { +, ), $ }
Follow (F) = First (T’) + Follow (T’) = { *, +, ), $ }

11. Building parse tables for each production P: A a loop
for each terminal a in First (a) loop
T (A, a) := P;
end loop;
if e in First (a), then
for each terminal b in Follow (a) loop T (A, b) := P; end loop;
end if;
All other entries are errors.
If two assignments conflict, parse table cannot be built.

12. LL (1) grammars
If table construction is successful, grammar is LL (1): left-to right, leftmost derivation with one-token lookahead.
If construction fails, can conceive of LL (2), etc.
Ambiguous grammars are never LL (k)
If a terminal is in First for two different productions of A, the grammar cannot be LL (1).
Grammars with left-recursion are never LL (k)
Some useful constructs are not LL (k)

13. Bottom-up parsing Synthesize tree from fragments
Automaton performs two actions:
shift: push next symbol on stack
reduce: replace symbols on stack
Automaton synthesizes (reduces) when end of a production is recognized
States of automaton encode synthesis so far, and expectation of pending non-terminals
Automaton has potentially large set of states
Technique more general than LL (k)

14. LR (k) parsing
Left-to-right, rightmost derivation with k-token lookahead.
Most general parsing technique for deterministic grammars.
In general, not practical: tables too large (10^6 states for C++, Ada).
Common subsets: SLR, LALR (1).

15. The states of the LR(0) automaton
An item is a point within a production, indicating that part of the production has been recognized:
A a . B b ,
seen the expansion of a, expect to see expansion of B
A state is a set of items
Transition within states are determined by terminals and non-terminals
Parsing tables are built from automaton:
action: shift / reduce depending on next symbol
goto: change state depending on synthesized non-terminal

16. Building LR (0) states If a state includes: A a . B b
it also includes every state that is the start of B:
B X Y Z
Informally: if I expect to see B next, I expect to start anything that B can start with:
X G H I
States are built by closure from individual items.

17. A grammar of expressions: initial state
E’ E
E E + T | T; left-recursion ok here.
T T * F | F;
F id | (E)
S0 = { E’ .E, E .E + T, E T,
T T * F, T F,
F id, F ( E ) }

18. Adding states If a state has item A a .a b,
and the next symbol in the input is a, we shift a on the stack and enter a state with item
A a a.b
and everything else brought in by closure
if a state has as item A a. , this indicates the end of a production: reduce action.
If a state has an item A a .N b, then after a reduction that find an N, go to a state with A a N. b

19. The LR (0) states S1 = { E’ E., E E. + T } S2 = { E T., T T. * F }
S3 = { T F. }
S4 = { F (. E), } + S0 (by closure)
S5 = { F id. }
S6 = { E E +. T, T .T * F, T .F, F .id, F .(E)}
S7 = { T T *. F, F .id, F .(E)}
S8 = { F (E.), E E.+ T}
S9 = { E E + T., T T.* F}
S10 = { T T * F.}, S11 = {F (E).}

20. Building SLR tables
An arc between two states labelled with a terminal is a shift action.
An arc between two states labelled with a non-terminal is a goto action.
if a state contains an item A a. , (a reduce item)
the action is to reduce by this production, for all terminals in Follow (A).
If there are shift-reduce conflicts or reduce-reduce conflicts, more elaborate techniques are needed.