Compilers Nikos Papaspyrou Kostis Sagonas nickie@softlab.ntua.gr - - PowerPoint PPT Presentation
Compilers Nikos Papaspyrou Kostis Sagonas nickie@softlab.ntua.gr kostis@cs.ntua.gr National Technical University of Athens School of Electrical and Computer Engineering Software Engineering Laboratory Polytechnioupoli, 15780 Zografou,
nickie@softlab.ntua.gr kostis@cs.ntua.gr
National Technical University of Athens School of Electrical and Computer Engineering Software Engineering Laboratory Polytechnioupoli, 15780 Zografou, Athens, Greece.
March 2017
Nikos Papaspyrou, Kostis Sagonas Compilers March 2017 1 / 216
Nikos Papaspyrou, Kostis Sagonas Compilers March 2017 55 / 216
▶ Parse tree ▶ It can be built in two ways:
▶ Top-down
i.e., starting from the root and moving towards the leaves
▶ Bottom-up
i.e., starting from the leaves and moving towards the root
Nikos Papaspyrou, Kostis Sagonas Compilers March 2017 56 / 216
S a A B c a b A c a B S ε
απόπάνωπροςτακάτω (topdown)
Compilers March 2017 57 / 216
S a A B c a b A c a B S ε
απόπάνωπροςτακάτω (topdown)
a A B c a b A c a B S ε
απόκάτωπροςταπάνω (bottomup)
Compilers March 2017 57 / 216
▶ FIRST sets
▶ Let 𝛽 ∈ (𝑈 ∪ 𝑂)∗ be a string ▶ The set FIRST(𝛽) ⊆ 𝑈 ∪ { 𝜗 } contains the terminal
symbols from which all strings that are produced by 𝛽 are bound to start
▶ If 𝛽 ⇒∗ a𝛾
then a ∈ FIRST(𝛽)
▶ If 𝛽 ⇒∗ 𝜗
then 𝜗 ∈ FIRST(𝛽)
Nikos Papaspyrou, Kostis Sagonas Compilers March 2017 58 / 216
▶ FIRST sets
▶ Let 𝛽 ∈ (𝑈 ∪ 𝑂)∗ be a string ▶ The set FIRST(𝛽) ⊆ 𝑈 ∪ { 𝜗 } contains the terminal
symbols from which all strings that are produced by 𝛽 are bound to start
▶ If 𝛽 ⇒∗ a𝛾
then a ∈ FIRST(𝛽)
▶ If 𝛽 ⇒∗ 𝜗
then 𝜗 ∈ FIRST(𝛽)
Nikos Papaspyrou, Kostis Sagonas Compilers March 2017 58 / 216
▶ FOLLOW sets
▶ Let 𝐵 be a non-terminal symbol ▶ The set FOLLOW(𝐵) ⊆ 𝑈 ∪ { EOF } contains all terminal
symbol that may follow 𝐵 during a production from the start symbol 𝑇
▶ If 𝐵 can be the last symbol in a production, then
EOF ∈ FOLLOW(𝐵)
▶ If 𝑇 ⇒∗ 𝛽𝐵a𝛾
then a ∈ FOLLOW(𝐵)
▶ If 𝑇 ⇒∗ 𝛽𝐵
then EOF ∈ FOLLOW(𝐵)
Nikos Papaspyrou, Kostis Sagonas Compilers March 2017 59 / 216
▶ FOLLOW sets
▶ Let 𝐵 be a non-terminal symbol ▶ The set FOLLOW(𝐵) ⊆ 𝑈 ∪ { EOF } contains all terminal
symbol that may follow 𝐵 during a production from the start symbol 𝑇
▶ If 𝐵 can be the last symbol in a production, then
EOF ∈ FOLLOW(𝐵)
▶ If 𝑇 ⇒∗ 𝛽𝐵a𝛾
then a ∈ FOLLOW(𝐵)
▶ If 𝑇 ⇒∗ 𝛽𝐵
then EOF ∈ FOLLOW(𝐵)
Nikos Papaspyrou, Kostis Sagonas Compilers March 2017 59 / 216
▶ FIRST(𝜗) = { 𝜗 } ▶ FIRST(a𝛾) = { a } ▶ if
𝜗 ∉ FIRST(𝐵) then FIRST(𝐵𝛾) = FIRST(𝐵)
▶ if
𝜗 ∈ FIRST(𝐵) then FIRST(𝐵𝛾) = (FIRST(𝐵) − { 𝜗 }) ∪ FIRST(𝛾)
▶ for each rule 𝐵 → 𝛽, it must be FIRST(𝛽) ⊆ FIRST(𝐵)
Nikos Papaspyrou, Kostis Sagonas Compilers March 2017 60 / 216
▶ Example
FIRST(E) = { id, ( } FIRST(T) = { id, ( } FIRST(F) = { id, ( } FIRST(E’) = { +, 𝜗 } FIRST(T’) = { *, 𝜗 }
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E → T E’ E’ → 𝜗 E’ → + T E’ T → F T’ T’ → 𝜗 T’ → * F T’ F → ( E ) F → id
▶ Example
FIRST(E) = { id, ( } FIRST(T) = { id, ( } FIRST(F) = { id, ( } FIRST(E’) = { +, 𝜗 } FIRST(T’) = { *, 𝜗 }
Nikos Papaspyrou, Kostis Sagonas Compilers March 2017 61 / 216
E → T E’ E’ → 𝜗 E’ → + T E’ T → F T’ T’ → 𝜗 T’ → * F T’ F → ( E ) F → id
▶ Example
FIRST(E) = { id, ( } FIRST(T) = { id, ( } FIRST(F) = { id, ( } FIRST(E’) = { +, 𝜗 } FIRST(T’) = { *, 𝜗 }
Nikos Papaspyrou, Kostis Sagonas Compilers March 2017 61 / 216
E → T E’ E’ → 𝜗 E’ → + T E’ T → F T’ T’ → 𝜗 T’ → * F T’ F → ( E ) F → id
▶ Example
FIRST(E) = { id, ( } FIRST(T) = { id, ( } FIRST(F) = { id, ( } FIRST(E’) = { +, 𝜗 } FIRST(T’) = { *, 𝜗 }
Nikos Papaspyrou, Kostis Sagonas Compilers March 2017 61 / 216
E → T E’ E’ → 𝜗 E’ → + T E’ T → F T’ T’ → 𝜗 T’ → * F T’ F → ( E ) F → id
▶ Example
FIRST(E) = { id, ( } FIRST(T) = { id, ( } FIRST(F) = { id, ( } FIRST(E’) = { +, 𝜗 } FIRST(T’) = { *, 𝜗 }
Nikos Papaspyrou, Kostis Sagonas Compilers March 2017 61 / 216
E → T E’ E’ → 𝜗 E’ → + T E’ T → F T’ T’ → 𝜗 T’ → * F T’ F → ( E ) F → id
▶ Example
FIRST(E) = { id, ( } FIRST(T) = { id, ( } FIRST(F) = { id, ( } FIRST(E’) = { +, 𝜗 } FIRST(T’) = { *, 𝜗 }
Nikos Papaspyrou, Kostis Sagonas Compilers March 2017 61 / 216
E → T E’ E’ → 𝜗 E’ → + T E’ T → F T’ T’ → 𝜗 T’ → * F T’ F → ( E ) F → id
▶ EOF ∈ FOLLOW(𝑇) ▶ for each rule 𝐵 → 𝛽𝐶𝛾
▶ (FIRST(𝛾) − { 𝜗 }) ⊆ FOLLOW(𝐶) ▶ if
𝜗 ∈ FIRST(𝛾) then FOLLOW(𝐵) ⊆ FOLLOW(𝐶)
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▶ Example
FOLLOW(E) = {
EOF, ) }
FOLLOW(T) = { +, EOF, ) } FOLLOW(F) = { *, +, EOF, ) } FOLLOW(E’) = {
EOF, ) }
FOLLOW(T’) = { +, EOF, ) } FIRST(E’) = { +, 𝜗 } FIRST(T’) = { *, 𝜗 }
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E → T E’ E’ → 𝜗 E’ → + T E’ T → F T’ T’ → 𝜗 T’ → * F T’ F → ( E ) F → id
▶ Example
FOLLOW(E) = { EOF , ) } FOLLOW(T) = { +, EOF, ) } FOLLOW(F) = { *, +, EOF, ) } FOLLOW(E’) = {
EOF, ) }
FOLLOW(T’) = { +, EOF, ) } FIRST(E’) = { +, 𝜗 } FIRST(T’) = { *, 𝜗 }
Nikos Papaspyrou, Kostis Sagonas Compilers March 2017 63 / 216
E → T E’ E’ → 𝜗 E’ → + T E’ T → F T’ T’ → 𝜗 T’ → * F T’ F → ( E ) F → id
▶ Example
FOLLOW(E) = { EOF, ) } FOLLOW(T) = { +, EOF, ) } FOLLOW(F) = { *, +, EOF, ) } FOLLOW(E’) = {
EOF, ) }
FOLLOW(T’) = { +, EOF, ) } FIRST(E’) = { +, 𝜗 } FIRST(T’) = { *, 𝜗 }
Nikos Papaspyrou, Kostis Sagonas Compilers March 2017 63 / 216
E → T E’ E’ → 𝜗 E’ → + T E’ T → F T’ T’ → 𝜗 T’ → * F T’ F → ( E ) F → id
▶ Example
FOLLOW(E) = { EOF, ) } FOLLOW(T) = { +, EOF, ) } FOLLOW(F) = { *, +, EOF, ) } FOLLOW(E’) = {
EOF, ) }
FOLLOW(T’) = { +, EOF, ) } FIRST(E’) = { +, 𝜗 } FIRST(T’) = { *, 𝜗 }
Nikos Papaspyrou, Kostis Sagonas Compilers March 2017 63 / 216
E → T E’ E’ → 𝜗 E’ → + T E’ T → F T’ T’ → 𝜗 T’ → * F T’ F → ( E ) F → id
▶ Example
FOLLOW(E) = { EOF, ) } FOLLOW(T) = { + , EOF, ) } FOLLOW(F) = { *, +, EOF, ) } FOLLOW(E’) = {
EOF, ) }
FOLLOW(T’) = { +, EOF, ) } FIRST(E’) = { +, 𝜗 } FIRST(T’) = { *, 𝜗 }
Nikos Papaspyrou, Kostis Sagonas Compilers March 2017 63 / 216
E → T E’ E’ → 𝜗 E’ → + T E’ T → F T’ T’ → 𝜗 T’ → * F T’ F → ( E ) F → id
▶ Example
FOLLOW(E) = { EOF, ) } FOLLOW(T) = { +, EOF, ) } FOLLOW(F) = { *, +, EOF, ) } FOLLOW(E’) = {
EOF, ) }
FOLLOW(T’) = { +, EOF, ) } FIRST(E’) = { +, 𝜗 } FIRST(T’) = { *, 𝜗 }
Nikos Papaspyrou, Kostis Sagonas Compilers March 2017 63 / 216
E → T E’ E’ → 𝜗 E’ → + T E’ T → F T’ T’ → 𝜗 T’ → * F T’ F → ( E ) F → id
▶ Example
FOLLOW(E) = { EOF, ) } FOLLOW(T) = { +, EOF, ) } FOLLOW(F) = { * , +, EOF, ) } FOLLOW(E’) = {
EOF, ) }
FOLLOW(T’) = { +, EOF, ) } FIRST(E’) = { +, 𝜗 } FIRST(T’) = { *, 𝜗 }
Nikos Papaspyrou, Kostis Sagonas Compilers March 2017 63 / 216
E → T E’ E’ → 𝜗 E’ → + T E’ T → F T’ T’ → 𝜗 T’ → * F T’ F → ( E ) F → id
▶ Example
FOLLOW(E) = { EOF, ) } FOLLOW(T) = { +, EOF, ) } FOLLOW(F) = { *, +, EOF, ) } FOLLOW(E’) = {
EOF, ) }
FOLLOW(T’) = { +, EOF, ) } FIRST(E’) = { +, 𝜗 } FIRST(T’) = { *, 𝜗 }
Nikos Papaspyrou, Kostis Sagonas Compilers March 2017 63 / 216
E → T E’ E’ → 𝜗 E’ → + T E’ T → F T’ T’ → 𝜗 T’ → * F T’ F → ( E ) F → id
▶ Example
FOLLOW(E) = { EOF, ) } FOLLOW(T) = { +, EOF, ) } FOLLOW(F) = { *, +, EOF, ) } FOLLOW(E’) = { EOF, ) } FOLLOW(T’) = { +, EOF, ) } FIRST(E’) = { +, 𝜗 } FIRST(T’) = { *, 𝜗 }
Nikos Papaspyrou, Kostis Sagonas Compilers March 2017 63 / 216
E → T E’ E’ → 𝜗 E’ → + T E’ T → F T’ T’ → 𝜗 T’ → * F T’ F → ( E ) F → id
▶ Example
FOLLOW(E) = { EOF, ) } FOLLOW(T) = { +, EOF, ) } FOLLOW(F) = { *, +, EOF, ) } FOLLOW(E’) = { EOF, ) } FOLLOW(T’) = { +, EOF, ) } FIRST(E’) = { +, 𝜗 } FIRST(T’) = { *, 𝜗 }
Nikos Papaspyrou, Kostis Sagonas Compilers March 2017 63 / 216
E → T E’ E’ → 𝜗 E’ → + T E’ T → F T’ T’ → 𝜗 T’ → * F T’ F → ( E ) F → id
▶ Example
FOLLOW(E) = { EOF, ) } FOLLOW(T) = { +, EOF, ) } FOLLOW(F) = { *, +, EOF, ) } FOLLOW(E’) = { EOF, ) } FOLLOW(T’) = { +, EOF, ) } FIRST(E’) = { +, 𝜗 } FIRST(T’) = { *, 𝜗 }
Nikos Papaspyrou, Kostis Sagonas Compilers March 2017 63 / 216
E → T E’ E’ → 𝜗 E’ → + T E’ T → F T’ T’ → 𝜗 T’ → * F T’ F → ( E ) F → id
▶ Example
FOLLOW(E) = { EOF, ) } FOLLOW(T) = { +, EOF, ) } FOLLOW(F) = { *, +, EOF, ) } FOLLOW(E’) = { EOF, ) } FOLLOW(T’) = { +, EOF, ) } FIRST(E’) = { +, 𝜗 } FIRST(T’) = { *, 𝜗 }
Nikos Papaspyrou, Kostis Sagonas Compilers March 2017 63 / 216
E → T E’ E’ → 𝜗 E’ → + T E’ T → F T’ T’ → 𝜗 T’ → * F T’ F → ( E ) F → id
▶ Example
FOLLOW(E) = { EOF, ) } FOLLOW(T) = { +, EOF, ) } FOLLOW(F) = { *, +, EOF, ) } FOLLOW(E’) = { EOF, ) } FOLLOW(T’) = { +, EOF, ) } FIRST(E’) = { +, 𝜗 } FIRST(T’) = { *, 𝜗 }
Nikos Papaspyrou, Kostis Sagonas Compilers March 2017 63 / 216
E → T E’ E’ → 𝜗 E’ → + T E’ T → F T’ T’ → 𝜗 T’ → * F T’ F → ( E ) F → id
▶ Example
FOLLOW(E) = { EOF, ) } FOLLOW(T) = { +, EOF, ) } FOLLOW(F) = { *, +, EOF, ) } FOLLOW(E’) = { EOF, ) } FOLLOW(T’) = { +, EOF, ) } FIRST(E’) = { +, 𝜗 } FIRST(T’) = { *, 𝜗 }
Nikos Papaspyrou, Kostis Sagonas Compilers March 2017 63 / 216
E → T E’ E’ → 𝜗 E’ → + T E’ T → F T’ T’ → 𝜗 T’ → * F T’ F → ( E ) F → id
▶ Parsing starts from the root ▶ Every time, we seek:
▶ the non-terminal symbol to replace
⇒ usually we pick the left-most
▶ the production rule to apply
⇒ based on the next 𝑙 tokens of the input string: LL(𝑙)
▶ We repeat until the non-terminal symbols have been
exhausted
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▶ Necessary conditions:
▶ No left recursion (direct or indirect) ▶ No common prefix in alternative rules
▶ Sometimes it is possible to transform a grammar into an
equivalent LL(1)
⇒ left recursion elimination ⇒ left factorization
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▶ Substitution
A → 𝛽1 ∣ … ∣ 𝛽𝑜 B → 𝛾1A𝛾2 ⇒ A → 𝛽1 ∣ … ∣ 𝛽𝑜 B → 𝛾1𝛽1𝛾2 ∣ … ∣ 𝛾1𝛽𝑜𝛾2
▶ Left factorization
A → 𝛽𝛾1 ∣ … ∣ 𝛽𝛾𝑜 ⇒ A → 𝛽B B → 𝛾1 ∣ … ∣ 𝛾𝑜
▶ Direct left recursion elimination
A → A𝛽1 ∣ … ∣ A𝛽𝑜 ∣ 𝛾1 ∣ … ∣ 𝛾𝑛 ⇒ A → 𝛾1B ∣ … ∣ 𝛾𝑛B B → 𝛽1B ∣ … ∣ 𝛽𝑜B ∣ 𝜗
Nikos Papaspyrou, Kostis Sagonas Compilers March 2017 66 / 216
A → 𝛽1 ∣ … ∣ 𝛽𝑜 is transformed into a procedure whose body is of the form:
if token ∈ FIRST(𝛽1) then code to recognize 𝛽1 … else if token ∈ FIRST(𝛽𝑜) then code to recognize 𝛽𝑜 else if 𝜗 ∉ FIRST(𝛽1) ∪ … ∪ FIRST(𝛽𝑜) then syntax error else if token ∉ FOLLOW(𝐵) then syntax error end if
Nikos Papaspyrou, Kostis Sagonas Compilers March 2017 67 / 216
▶ They use a stack where they put grammar symbols —
initially just 𝑇
▶ Every time, they examine the top of the stack:
▶ If it is a terminal symbol and it’s the same as the next input
symbol, then both are removed
▶ If it is a non-terminal symbol, then they apply a rule
depending on the next symbol of the input
▶ Success: both the stack and the input string are empty
άδειες
Nikos Papaspyrou, Kostis Sagonas Compilers March 2017 68 / 216
The algorithm for generating table 𝑁 defines the class of LL(1) languages
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ΠίνακαςM συντακτικήςανάλυσης LL(1) ΣυντακτικόςΑναλυτήςLL(1) συβολοσειράεισόδου στοίβα
σύβολο κορυφή
E → T E’ E’ → + T E’ ∣ 𝜗 T → F T’ T’ → * F T’ ∣ 𝜗 F → ( E ) ∣ id FIRST(E) = FIRST(T) = FIRST(F) = { id, ( } FIRST(E’) = { +, 𝜗 } FIRST(T’) = { *, 𝜗 } FOLLOW(E) = FOLLOW(E’) = { ), EOF } FOLLOW(T) = FOLLOW(T’) = { +, ), EOF } FOLLOW(F) = { *, +, ), EOF } id + * ( )
EOF
E E → TE’ E → TE’ E’ E’ → +TE’ E’ → 𝜗 E’ → 𝜗 T T → FT’ T → FT’ T’ T’ → 𝜗 T’ → *FT’ T’ → 𝜗 T’ → 𝜗 F F → id F → (E)
Nikos Papaspyrou, Kostis Sagonas Compilers March 2017 70 / 216
E → T E’ E’ → + T E’ ∣ 𝜗 T → F T’ T’ → * F T’ ∣ 𝜗 F → ( E ) ∣ id FIRST(E) = FIRST(T) = FIRST(F) = { id, ( } FIRST(E’) = { +, 𝜗 } FIRST(T’) = { *, 𝜗 } FOLLOW(E) = FOLLOW(E’) = { ), EOF } FOLLOW(T) = FOLLOW(T’) = { +, ), EOF } FOLLOW(F) = { *, +, ), EOF } id + * ( )
EOF
E E → TE’ E → TE’ E’ E’ → +TE’ E’ → 𝜗 E’ → 𝜗 T T → FT’ T → FT’ T’ T’ → 𝜗 T’ → *FT’ T’ → 𝜗 T’ → 𝜗 F F → id F → (E)
Nikos Papaspyrou, Kostis Sagonas Compilers March 2017 70 / 216
E → T E’ E’ → + T E’ ∣ 𝜗 T → F T’ T’ → * F T’ ∣ 𝜗 F → ( E ) ∣ id FIRST(E) = FIRST(T) = FIRST(F) = { id, ( } FIRST(E’) = { +, 𝜗 } FIRST(T’) = { *, 𝜗 } FOLLOW(E) = FOLLOW(E’) = { ), EOF } FOLLOW(T) = FOLLOW(T’) = { +, ), EOF } FOLLOW(F) = { *, +, ), EOF } id + * ( )
EOF
E E → TE’ E → TE’ E’ E’ → +TE’ E’ → 𝜗 E’ → 𝜗 T T → FT’ T → FT’ T’ T’ → 𝜗 T’ → *FT’ T’ → 𝜗 T’ → 𝜗 F F → id F → (E)
Nikos Papaspyrou, Kostis Sagonas Compilers March 2017 70 / 216
E → T E’ E’ → + T E’ ∣ 𝜗 T → F T’ T’ → * F T’ ∣ 𝜗 F → ( E ) ∣ id FIRST(E) = FIRST(T) = FIRST(F) = { id, ( } FIRST(E’) = { +, 𝜗 } FIRST(T’) = { *, 𝜗 } FOLLOW(E) = FOLLOW(E’) = { ), EOF } FOLLOW(T) = FOLLOW(T’) = { +, ), EOF } FOLLOW(F) = { *, +, ), EOF } id + * ( )
EOF
E E → TE’ E → TE’ E’ E’ → +TE’ E’ → 𝜗 E’ → 𝜗 T T → FT’ T → FT’ T’ T’ → 𝜗 T’ → *FT’ T’ → 𝜗 T’ → 𝜗 F F → id F → (E)
Nikos Papaspyrou, Kostis Sagonas Compilers March 2017 70 / 216
E → T E’ E’ → + T E’ ∣ 𝜗 T → F T’ T’ → * F T’ ∣ 𝜗 F → ( E ) ∣ id FIRST(E) = FIRST(T) = FIRST(F) = { id, ( } FIRST(E’) = { +, 𝜗 } FIRST(T’) = { *, 𝜗 } FOLLOW(E) = FOLLOW(E’) = { ), EOF } FOLLOW(T) = FOLLOW(T’) = { +, ), EOF } FOLLOW(F) = { *, +, ), EOF } id + * ( )
EOF
E E → TE’ E → TE’ E’ E’ → +TE’ E’ → 𝜗 E’ → 𝜗 T T → FT’ T → FT’ T’ T’ → 𝜗 T’ → *FT’ T’ → 𝜗 T’ → 𝜗 F F → id F → (E)
Nikos Papaspyrou, Kostis Sagonas Compilers March 2017 70 / 216
E → T E’ E’ → + T E’ ∣ 𝜗 T → F T’ T’ → * F T’ ∣ 𝜗 F → ( E ) ∣ id FIRST(E) = FIRST(T) = FIRST(F) = { id, ( } FIRST(E’) = { +, 𝜗 } FIRST(T’) = { *, 𝜗 } FOLLOW(E) = FOLLOW(E’) = { ), EOF } FOLLOW(T) = FOLLOW(T’) = { +, ), EOF } FOLLOW(F) = { *, +, ), EOF } id + * ( )
EOF
E E → TE’ E → TE’ E’ E’ → +TE’ E’ → 𝜗 E’ → 𝜗 T T → FT’ T → FT’ T’ T’ → 𝜗 T’ → *FT’ T’ → 𝜗 T’ → 𝜗 F F → id F → (E)
Nikos Papaspyrou, Kostis Sagonas Compilers March 2017 70 / 216
E → T E’ E’ → + T E’ ∣ 𝜗 T → F T’ T’ → * F T’ ∣ 𝜗 F → ( E ) ∣ id FIRST(E) = FIRST(T) = FIRST(F) = { id, ( } FIRST(E’) = { +, 𝜗 } FIRST(T’) = { *, 𝜗 } FOLLOW(E) = FOLLOW(E’) = { ), EOF } FOLLOW(T) = FOLLOW(T’) = { +, ), EOF } FOLLOW(F) = { *, +, ), EOF } id + * ( )
EOF
E E → TE’ E → TE’ E’ E’ → +TE’ E’ → 𝜗 E’ → 𝜗 T T → FT’ T → FT’ T’ T’ → 𝜗 T’ → *FT’ T’ → 𝜗 T’ → 𝜗 F F → id F → (E)
Nikos Papaspyrou, Kostis Sagonas Compilers March 2017 70 / 216
E → T E’ E’ → + T E’ ∣ 𝜗 T → F T’ T’ → * F T’ ∣ 𝜗 F → ( E ) ∣ id FIRST(E) = FIRST(T) = FIRST(F) = { id, ( } FIRST(E’) = { +, 𝜗 } FIRST(T’) = { *, 𝜗 } FOLLOW(E) = FOLLOW(E’) = { ), EOF } FOLLOW(T) = FOLLOW(T’) = { +, ), EOF } FOLLOW(F) = { *, +, ), EOF } id + * ( )
EOF
E E → TE’ E → TE’ E’ E’ → +TE’ E’ → 𝜗 E’ → 𝜗 T T → FT’ T → FT’ T’ T’ → 𝜗 T’ → *FT’ T’ → 𝜗 T’ → 𝜗 F F → id F → (E)
Nikos Papaspyrou, Kostis Sagonas Compilers March 2017 70 / 216
E → T E’ E’ → + T E’ ∣ 𝜗 T → F T’ T’ → * F T’ ∣ 𝜗 F → ( E ) ∣ id FIRST(E) = FIRST(T) = FIRST(F) = { id, ( } FIRST(E’) = { +, 𝜗 } FIRST(T’) = { *, 𝜗 } FOLLOW(E) = FOLLOW(E’) = { ), EOF } FOLLOW(T) = FOLLOW(T’) = { +, ), EOF } FOLLOW(F) = { *, +, ), EOF } id + * ( )
EOF
E E → TE’ E → TE’ E’ E’ → +TE’ E’ → 𝜗 E’ → 𝜗 T T → FT’ T → FT’ T’ T’ → 𝜗 T’ → *FT’ T’ → 𝜗 T’ → 𝜗 F F → id F → (E)
Nikos Papaspyrou, Kostis Sagonas Compilers March 2017 70 / 216
E id + id * id EOF E → T E’ 1 E’ T id + id * id EOF T → F T’ 2 E’ T’ F id + id * id EOF F → id 3 E’ T’ id id + id * id EOF 4 E’ T’ + id * id EOF T’ → 𝜗 5 E’ + id * id EOF E’ → + T E’ 6 E’ T + + id * id EOF 7 E’ T id * id EOF T → F T’ 8 E’ T’ F id * id EOF F → id 9 E’ T’ id id * id EOF 10 E’ T’ * id EOF T’ → * F T’ 11 E’ T’ F * * id EOF 12 E’ T’ F id EOF F → id 13 E’ T’ id id EOF 14 E’ T’
EOF
T’ → 𝜗 15 E’
EOF
E’ → 𝜗 16 𝜗
EOF
success
Nikos Papaspyrou, Kostis Sagonas Compilers March 2017 71 / 216
E id + id * id EOF E → T E’ 1 E’ T id + id * id EOF T → F T’ 2 E’ T’ F id + id * id EOF F → id 3 E’ T’ id id + id * id EOF 4 E’ T’ + id * id EOF T’ → 𝜗 5 E’ + id * id EOF E’ → + T E’ 6 E’ T + + id * id EOF 7 E’ T id * id EOF T → F T’ 8 E’ T’ F id * id EOF F → id 9 E’ T’ id id * id EOF 10 E’ T’ * id EOF T’ → * F T’ 11 E’ T’ F * * id EOF 12 E’ T’ F id EOF F → id 13 E’ T’ id id EOF 14 E’ T’
EOF
T’ → 𝜗 15 E’
EOF
E’ → 𝜗 16 𝜗
EOF
success
Nikos Papaspyrou, Kostis Sagonas Compilers March 2017 71 / 216
E id + id * id EOF E → T E’ 1 E’ T id + id * id EOF T → F T’ 2 E’ T’ F id + id * id EOF F → id 3 E’ T’ id id + id * id EOF 4 E’ T’ + id * id EOF T’ → 𝜗 5 E’ + id * id EOF E’ → + T E’ 6 E’ T + + id * id EOF 7 E’ T id * id EOF T → F T’ 8 E’ T’ F id * id EOF F → id 9 E’ T’ id id * id EOF 10 E’ T’ * id EOF T’ → * F T’ 11 E’ T’ F * * id EOF 12 E’ T’ F id EOF F → id 13 E’ T’ id id EOF 14 E’ T’
EOF
T’ → 𝜗 15 E’
EOF
E’ → 𝜗 16 𝜗
EOF
success
Nikos Papaspyrou, Kostis Sagonas Compilers March 2017 71 / 216
E id + id * id EOF E → T E’ 1 E’ T id + id * id EOF T → F T’ 2 E’ T’ F id + id * id EOF F → id 3 E’ T’ id id + id * id EOF 4 E’ T’ + id * id EOF T’ → 𝜗 5 E’ + id * id EOF E’ → + T E’ 6 E’ T + + id * id EOF 7 E’ T id * id EOF T → F T’ 8 E’ T’ F id * id EOF F → id 9 E’ T’ id id * id EOF 10 E’ T’ * id EOF T’ → * F T’ 11 E’ T’ F * * id EOF 12 E’ T’ F id EOF F → id 13 E’ T’ id id EOF 14 E’ T’
EOF
T’ → 𝜗 15 E’
EOF
E’ → 𝜗 16 𝜗
EOF
success
Nikos Papaspyrou, Kostis Sagonas Compilers March 2017 71 / 216
E id + id * id EOF E → T E’ 1 E’ T id + id * id EOF T → F T’ 2 E’ T’ F id + id * id EOF F → id 3 E’ T’ id id + id * id EOF 4 E’ T’ + id * id EOF T’ → 𝜗 5 E’ + id * id EOF E’ → + T E’ 6 E’ T + + id * id EOF 7 E’ T id * id EOF T → F T’ 8 E’ T’ F id * id EOF F → id 9 E’ T’ id id * id EOF 10 E’ T’ * id EOF T’ → * F T’ 11 E’ T’ F * * id EOF 12 E’ T’ F id EOF F → id 13 E’ T’ id id EOF 14 E’ T’
EOF
T’ → 𝜗 15 E’
EOF
E’ → 𝜗 16 𝜗
EOF
success
Nikos Papaspyrou, Kostis Sagonas Compilers March 2017 71 / 216
E id + id * id EOF E → T E’ 1 E’ T id + id * id EOF T → F T’ 2 E’ T’ F id + id * id EOF F → id 3 E’ T’ id id + id * id EOF 4 E’ T’ + id * id EOF T’ → 𝜗 5 E’ + id * id EOF E’ → + T E’ 6 E’ T + + id * id EOF 7 E’ T id * id EOF T → F T’ 8 E’ T’ F id * id EOF F → id 9 E’ T’ id id * id EOF 10 E’ T’ * id EOF T’ → * F T’ 11 E’ T’ F * * id EOF 12 E’ T’ F id EOF F → id 13 E’ T’ id id EOF 14 E’ T’
EOF
T’ → 𝜗 15 E’
EOF
E’ → 𝜗 16 𝜗
EOF
success
Nikos Papaspyrou, Kostis Sagonas Compilers March 2017 71 / 216
E id + id * id EOF E → T E’ 1 E’ T id + id * id EOF T → F T’ 2 E’ T’ F id + id * id EOF F → id 3 E’ T’ id id + id * id EOF 4 E’ T’ + id * id EOF T’ → 𝜗 5 E’ + id * id EOF E’ → + T E’ 6 E’ T + + id * id EOF 7 E’ T id * id EOF T → F T’ 8 E’ T’ F id * id EOF F → id 9 E’ T’ id id * id EOF 10 E’ T’ * id EOF T’ → * F T’ 11 E’ T’ F * * id EOF 12 E’ T’ F id EOF F → id 13 E’ T’ id id EOF 14 E’ T’
EOF
T’ → 𝜗 15 E’
EOF
E’ → 𝜗 16 𝜗
EOF
success
Nikos Papaspyrou, Kostis Sagonas Compilers March 2017 71 / 216
E id + id * id EOF E → T E’ 1 E’ T id + id * id EOF T → F T’ 2 E’ T’ F id + id * id EOF F → id 3 E’ T’ id id + id * id EOF 4 E’ T’ + id * id EOF T’ → 𝜗 5 E’ + id * id EOF E’ → + T E’ 6 E’ T + + id * id EOF 7 E’ T id * id EOF T → F T’ 8 E’ T’ F id * id EOF F → id 9 E’ T’ id id * id EOF 10 E’ T’ * id EOF T’ → * F T’ 11 E’ T’ F * * id EOF 12 E’ T’ F id EOF F → id 13 E’ T’ id id EOF 14 E’ T’
EOF
T’ → 𝜗 15 E’
EOF
E’ → 𝜗 16 𝜗
EOF
success
Nikos Papaspyrou, Kostis Sagonas Compilers March 2017 71 / 216
E id + id * id EOF E → T E’ 1 E’ T id + id * id EOF T → F T’ 2 E’ T’ F id + id * id EOF F → id 3 E’ T’ id id + id * id EOF 4 E’ T’ + id * id EOF T’ → 𝜗 5 E’ + id * id EOF E’ → + T E’ 6 E’ T + + id * id EOF 7 E’ T id * id EOF T → F T’ 8 E’ T’ F id * id EOF F → id 9 E’ T’ id id * id EOF 10 E’ T’ * id EOF T’ → * F T’ 11 E’ T’ F * * id EOF 12 E’ T’ F id EOF F → id 13 E’ T’ id id EOF 14 E’ T’
EOF
T’ → 𝜗 15 E’
EOF
E’ → 𝜗 16 𝜗
EOF
success
Nikos Papaspyrou, Kostis Sagonas Compilers March 2017 71 / 216
E id + id * id EOF E → T E’ 1 E’ T id + id * id EOF T → F T’ 2 E’ T’ F id + id * id EOF F → id 3 E’ T’ id id + id * id EOF 4 E’ T’ + id * id EOF T’ → 𝜗 5 E’ + id * id EOF E’ → + T E’ 6 E’ T + + id * id EOF 7 E’ T id * id EOF T → F T’ 8 E’ T’ F id * id EOF F → id 9 E’ T’ id id * id EOF 10 E’ T’ * id EOF T’ → * F T’ 11 E’ T’ F * * id EOF 12 E’ T’ F id EOF F → id 13 E’ T’ id id EOF 14 E’ T’
EOF
T’ → 𝜗 15 E’
EOF
E’ → 𝜗 16 𝜗
EOF
success
Nikos Papaspyrou, Kostis Sagonas Compilers March 2017 71 / 216
E id + id * id EOF E → T E’ 1 E’ T id + id * id EOF T → F T’ 2 E’ T’ F id + id * id EOF F → id 3 E’ T’ id id + id * id EOF 4 E’ T’ + id * id EOF T’ → 𝜗 5 E’ + id * id EOF E’ → + T E’ 6 E’ T + + id * id EOF 7 E’ T id * id EOF T → F T’ 8 E’ T’ F id * id EOF F → id 9 E’ T’ id id * id EOF 10 E’ T’ * id EOF T’ → * F T’ 11 E’ T’ F * * id EOF 12 E’ T’ F id EOF F → id 13 E’ T’ id id EOF 14 E’ T’
EOF
T’ → 𝜗 15 E’
EOF
E’ → 𝜗 16 𝜗
EOF
success
Nikos Papaspyrou, Kostis Sagonas Compilers March 2017 71 / 216
E id + id * id EOF E → T E’ 1 E’ T id + id * id EOF T → F T’ 2 E’ T’ F id + id * id EOF F → id 3 E’ T’ id id + id * id EOF 4 E’ T’ + id * id EOF T’ → 𝜗 5 E’ + id * id EOF E’ → + T E’ 6 E’ T + + id * id EOF 7 E’ T id * id EOF T → F T’ 8 E’ T’ F id * id EOF F → id 9 E’ T’ id id * id EOF 10 E’ T’ * id EOF T’ → * F T’ 11 E’ T’ F * * id EOF 12 E’ T’ F id EOF F → id 13 E’ T’ id id EOF 14 E’ T’
EOF
T’ → 𝜗 15 E’
EOF
E’ → 𝜗 16 𝜗
EOF
success
Nikos Papaspyrou, Kostis Sagonas Compilers March 2017 71 / 216
E id + id * id EOF E → T E’ 1 E’ T id + id * id EOF T → F T’ 2 E’ T’ F id + id * id EOF F → id 3 E’ T’ id id + id * id EOF 4 E’ T’ + id * id EOF T’ → 𝜗 5 E’ + id * id EOF E’ → + T E’ 6 E’ T + + id * id EOF 7 E’ T id * id EOF T → F T’ 8 E’ T’ F id * id EOF F → id 9 E’ T’ id id * id EOF 10 E’ T’ * id EOF T’ → * F T’ 11 E’ T’ F * * id EOF 12 E’ T’ F id EOF F → id 13 E’ T’ id id EOF 14 E’ T’
EOF
T’ → 𝜗 15 E’
EOF
E’ → 𝜗 16 𝜗
EOF
success
Nikos Papaspyrou, Kostis Sagonas Compilers March 2017 71 / 216
E id + id * id EOF E → T E’ 1 E’ T id + id * id EOF T → F T’ 2 E’ T’ F id + id * id EOF F → id 3 E’ T’ id id + id * id EOF 4 E’ T’ + id * id EOF T’ → 𝜗 5 E’ + id * id EOF E’ → + T E’ 6 E’ T + + id * id EOF 7 E’ T id * id EOF T → F T’ 8 E’ T’ F id * id EOF F → id 9 E’ T’ id id * id EOF 10 E’ T’ * id EOF T’ → * F T’ 11 E’ T’ F * * id EOF 12 E’ T’ F id EOF F → id 13 E’ T’ id id EOF 14 E’ T’
EOF
T’ → 𝜗 15 E’
EOF
E’ → 𝜗 16 𝜗
EOF
success
Nikos Papaspyrou, Kostis Sagonas Compilers March 2017 71 / 216
E id + id * id EOF E → T E’ 1 E’ T id + id * id EOF T → F T’ 2 E’ T’ F id + id * id EOF F → id 3 E’ T’ id id + id * id EOF 4 E’ T’ + id * id EOF T’ → 𝜗 5 E’ + id * id EOF E’ → + T E’ 6 E’ T + + id * id EOF 7 E’ T id * id EOF T → F T’ 8 E’ T’ F id * id EOF F → id 9 E’ T’ id id * id EOF 10 E’ T’ * id EOF T’ → * F T’ 11 E’ T’ F * * id EOF 12 E’ T’ F id EOF F → id 13 E’ T’ id id EOF 14 E’ T’
EOF
T’ → 𝜗 15 E’
EOF
E’ → 𝜗 16 𝜗
EOF
success
Nikos Papaspyrou, Kostis Sagonas Compilers March 2017 71 / 216
E id + id * id EOF E → T E’ 1 E’ T id + id * id EOF T → F T’ 2 E’ T’ F id + id * id EOF F → id 3 E’ T’ id id + id * id EOF 4 E’ T’ + id * id EOF T’ → 𝜗 5 E’ + id * id EOF E’ → + T E’ 6 E’ T + + id * id EOF 7 E’ T id * id EOF T → F T’ 8 E’ T’ F id * id EOF F → id 9 E’ T’ id id * id EOF 10 E’ T’ * id EOF T’ → * F T’ 11 E’ T’ F * * id EOF 12 E’ T’ F id EOF F → id 13 E’ T’ id id EOF 14 E’ T’
EOF
T’ → 𝜗 15 E’
EOF
E’ → 𝜗 16 𝜗
EOF
success
Nikos Papaspyrou, Kostis Sagonas Compilers March 2017 71 / 216
E id + id * id EOF E → T E’ 1 E’ T id + id * id EOF T → F T’ 2 E’ T’ F id + id * id EOF F → id 3 E’ T’ id id + id * id EOF 4 E’ T’ + id * id EOF T’ → 𝜗 5 E’ + id * id EOF E’ → + T E’ 6 E’ T + + id * id EOF 7 E’ T id * id EOF T → F T’ 8 E’ T’ F id * id EOF F → id 9 E’ T’ id id * id EOF 10 E’ T’ * id EOF T’ → * F T’ 11 E’ T’ F * * id EOF 12 E’ T’ F id EOF F → id 13 E’ T’ id id EOF 14 E’ T’
EOF
T’ → 𝜗 15 E’
EOF
E’ → 𝜗 16 𝜗
EOF
success
Nikos Papaspyrou, Kostis Sagonas Compilers March 2017 71 / 216
E id + id * id EOF E → T E’ 1 E’ T id + id * id EOF T → F T’ 2 E’ T’ F id + id * id EOF F → id 3 E’ T’ id id + id * id EOF 4 E’ T’ + id * id EOF T’ → 𝜗 5 E’ + id * id EOF E’ → + T E’ 6 E’ T + + id * id EOF 7 E’ T id * id EOF T → F T’ 8 E’ T’ F id * id EOF F → id 9 E’ T’ id id * id EOF 10 E’ T’ * id EOF T’ → * F T’ 11 E’ T’ F * * id EOF 12 E’ T’ F id EOF F → id 13 E’ T’ id id EOF 14 E’ T’
EOF
T’ → 𝜗 15 E’
EOF
E’ → 𝜗 16 𝜗
EOF
success
Nikos Papaspyrou, Kostis Sagonas Compilers March 2017 71 / 216
▶ Parsing starts from the leaves ▶ Every time, we seek:
▶ the left-most node of the tree ▶ that hasn’t yet been built ▶ but all its children have been built
▶ We repeat until the root is built ▶ Reducing: the selection of the nodes that will become the
children of a new node
Nikos Papaspyrou, Kostis Sagonas Compilers March 2017 72 / 216
Nikos Papaspyrou, Kostis Sagonas Compilers March 2017 73 / 216
B → D ∣ B , D D → a ∣ b
▶ Shift-reduce parsers
▶ They use an initially empty stack where they put symbols
▶ Shift: move the next input symbol to the top of the stack ▶ Reduce: remove from the top of the stack the RHS of some
rule and add the corresponding LHS
▶ Success: the stack contains only 𝑇 and all input symbols
have been exhausted
Nikos Papaspyrou, Kostis Sagonas Compilers March 2017 74 / 216
step stack input action 𝜗 r a , b shift 1 r a , b shift 2 r a , b reduce with D → a 3 r D , b reduce with B → D 4 r B , b shift (not reduce with S → r B) 5 r B , b shift 6 r B , b 𝜗 reduce with D → b 7 r B , D 𝜗 reduce with B → B , D (not reduce with B → D) 8 r B 𝜗 reduce with S → r B 9 S 𝜗 success
Nikos Papaspyrou, Kostis Sagonas Compilers March 2017 75 / 216
S → r B B → D ∣ B , D D → a ∣ b
step stack input action 𝜗 r a , b shift 1 r a , b shift 2 r a , b reduce with D → a 3 r D , b reduce with B → D 4 r B , b shift (not reduce with S → r B) 5 r B , b shift 6 r B , b 𝜗 reduce with D → b 7 r B , D 𝜗 reduce with B → B , D (not reduce with B → D) 8 r B 𝜗 reduce with S → r B 9 S 𝜗 success
Nikos Papaspyrou, Kostis Sagonas Compilers March 2017 75 / 216
S → r B B → D ∣ B , D D → a ∣ b
step stack input action 𝜗 r a , b shift 1 r a , b shift 2 r a , b reduce with D → a 3 r D , b reduce with B → D 4 r B , b shift (not reduce with S → r B) 5 r B , b shift 6 r B , b 𝜗 reduce with D → b 7 r B , D 𝜗 reduce with B → B , D (not reduce with B → D) 8 r B 𝜗 reduce with S → r B 9 S 𝜗 success
Nikos Papaspyrou, Kostis Sagonas Compilers March 2017 75 / 216
S → r B B → D ∣ B , D D → a ∣ b
step stack input action 𝜗 r a , b shift 1 r a , b shift 2 r a , b reduce with D → a 3 r D , b reduce with B → D 4 r B , b shift (not reduce with S → r B) 5 r B , b shift 6 r B , b 𝜗 reduce with D → b 7 r B , D 𝜗 reduce with B → B , D (not reduce with B → D) 8 r B 𝜗 reduce with S → r B 9 S 𝜗 success
Nikos Papaspyrou, Kostis Sagonas Compilers March 2017 75 / 216
S → r B B → D ∣ B , D D → a ∣ b
step stack input action 𝜗 r a , b shift 1 r a , b shift 2 r a , b reduce with D → a 3 r D , b reduce with B → D 4 r B , b shift (not reduce with S → r B) 5 r B , b shift 6 r B , b 𝜗 reduce with D → b 7 r B , D 𝜗 reduce with B → B , D (not reduce with B → D) 8 r B 𝜗 reduce with S → r B 9 S 𝜗 success
Nikos Papaspyrou, Kostis Sagonas Compilers March 2017 75 / 216
S → r B B → D ∣ B , D D → a ∣ b
step stack input action 𝜗 r a , b shift 1 r a , b shift 2 r a , b reduce with D → a 3 r D , b reduce with B → D 4 r B , b shift (not reduce with S → r B) 5 r B , b shift 6 r B , b 𝜗 reduce with D → b 7 r B , D 𝜗 reduce with B → B , D (not reduce with B → D) 8 r B 𝜗 reduce with S → r B 9 S 𝜗 success
Nikos Papaspyrou, Kostis Sagonas Compilers March 2017 75 / 216
S → r B B → D ∣ B , D D → a ∣ b
step stack input action 𝜗 r a , b shift 1 r a , b shift 2 r a , b reduce with D → a 3 r D , b reduce with B → D 4 r B , b shift (not reduce with S → r B) 5 r B , b shift 6 r B , b 𝜗 reduce with D → b 7 r B , D 𝜗 reduce with B → B , D (not reduce with B → D) 8 r B 𝜗 reduce with S → r B 9 S 𝜗 success
Nikos Papaspyrou, Kostis Sagonas Compilers March 2017 75 / 216
S → r B B → D ∣ B , D D → a ∣ b
step stack input action 𝜗 r a , b shift 1 r a , b shift 2 r a , b reduce with D → a 3 r D , b reduce with B → D 4 r B , b shift (not reduce with S → r B) 5 r B , b shift 6 r B , b 𝜗 reduce with D → b 7 r B , D 𝜗 reduce with B → B , D (not reduce with B → D) 8 r B 𝜗 reduce with S → r B 9 S 𝜗 success
Nikos Papaspyrou, Kostis Sagonas Compilers March 2017 75 / 216
S → r B B → D ∣ B , D D → a ∣ b
step stack input action 𝜗 r a , b shift 1 r a , b shift 2 r a , b reduce with D → a 3 r D , b reduce with B → D 4 r B , b shift (not reduce with S → r B) 5 r B , b shift 6 r B , b 𝜗 reduce with D → b 7 r B , D 𝜗 reduce with B → B , D (not reduce with B → D) 8 r B 𝜗 reduce with S → r B 9 S 𝜗 success
Nikos Papaspyrou, Kostis Sagonas Compilers March 2017 75 / 216
S → r B B → D ∣ B , D D → a ∣ b
step stack input action 𝜗 r a , b shift 1 r a , b shift 2 r a , b reduce with D → a 3 r D , b reduce with B → D 4 r B , b shift (not reduce with S → r B) 5 r B , b shift 6 r B , b 𝜗 reduce with D → b 7 r B , D 𝜗 reduce with B → B , D (not reduce with B → D) 8 r B 𝜗 reduce with S → r B 9 S 𝜗 success
Nikos Papaspyrou, Kostis Sagonas Compilers March 2017 75 / 216
S → r B B → D ∣ B , D D → a ∣ b
step stack input action 𝜗 r a , b shift 1 r a , b shift 2 r a , b reduce with D → a 3 r D , b reduce with B → D 4 r B , b shift (not reduce with S → r B) 5 r B , b shift 6 r B , b 𝜗 reduce with D → b 7 r B , D 𝜗 reduce with B → B , D (not reduce with B → D) 8 r B 𝜗 reduce with S → r B 9 S 𝜗 success
Nikos Papaspyrou, Kostis Sagonas Compilers March 2017 75 / 216
S → r B B → D ∣ B , D D → a ∣ b
▶ LR(𝑙) ▶ LR(0) ▶ SLR(1) ▶ LALR(1) ▶ LR(1)
Nikos Papaspyrou, Kostis Sagonas Compilers March 2017 76 / 216
Πίνακεςελέγχου συντακτικήςανάλυσης LR(1) ΣυντακτικόςΑναλυτήςLR(1) συβολοσειράεισόδου στοίβα
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