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Formal Grammars

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Eelco Visser
Eelco Visser

The document discusses formal grammars and their applications in language specification and parsing. It introduces key concepts such as formal grammars, derivation, terminal and non-terminal symbols, and different types of grammars including context-sensitive, context-free and regular grammars. It also discusses applications of formal grammars in parsing and how they relate to theoretical computer science concepts like decidability and complexity of the word problem for different grammar types.

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IN4303 2016-2017 Compiler Construction Formal Grammars language specification Guido Wachsmuth, Eelco Visser
Formal Grammars 2 3 * +7 21 3 * +7 21 Exp ExpExp Exp Exp parser
Formal Grammars 3 syntax definition errors parse generate check parse table generic parser
Formal Grammars Stratego DynSem NaBL2 SDF3 ESV editor SPT tests 4 syntax definition concrete syntax abstract syntax static semantics name binding type system dynamic semantics translation interpretation
Formal Grammars Stratego DynSem NaBL2 SDF3 ESV editor SPT tests 5 syntax definition concrete syntax abstract syntax static semantics name binding type system dynamic semantics translation interpretation
Formal Grammars Stratego DynSem NaBL2 SDF3 ESV editor SPT tests 6 syntax definition concrete syntax abstract syntax static semantics name binding type system dynamic semantics translation interpretation
Formal Grammars 7 P A R E N T A L ADVISORY THEORETICAL CONTENT
Formal Grammars 8 formal languages
Formal Grammars 9 infinite productivity
Formal Grammars 10 finite models
Formal Grammars 11 Philosophy Linguistics lexicology grammar morphology syntax phonology semantics Interdisciplinary Computer Science syntax semantics
Formal Grammars 12 Philosophy Linguistics lexicology grammar morphology syntax phonology semantics Interdisciplinary Computer Science syntax semantics
Formal Grammars 13
Formal Grammars 13 vocabulary Σ finite, nonempty set of elements (words, letters) alphabet
Formal Grammars 13 vocabulary Σ finite, nonempty set of elements (words, letters) alphabet string over Σ finite sequence of elements chosen from Σ word, sentence, utterance
Formal Grammars 13 vocabulary Σ finite, nonempty set of elements (words, letters) alphabet string over Σ finite sequence of elements chosen from Σ word, sentence, utterance formal language λ set of strings over a vocabulary Σ λ ⊆ Σ*
Formal Grammars 14 formal grammars
Formal Grammars 15 formal grammar G derivation relation G formal language L(G) ⊆ Σ* L(G) = {w∈Σ* | S G * w}
Formal Grammars 16 G = (N, Σ, P, S) Num → Digit Num Num → Digit Digit → “0” Digit → “1” Digit → “2” Digit → “3” Digit → “4” Digit → “5” Digit → “6” Digit → “7” Digit → “8” Digit → “9” decimal numbers morphology
Formal Grammars 17 G = (N, Σ, P, S) Num → Digit Num Num → Digit Digit → “0” Digit → “1” Digit → “2” Digit → “3” Digit → “4” Digit → “5” Digit → “6” Digit → “7” Digit → “8” Digit → “9” decimal numbers morphology Σ: finite set of terminal symbols
Formal Grammars 18 G = (N, Σ, P, S) Num → Digit Num Num → Digit Digit → “0” Digit → “1” Digit → “2” Digit → “3” Digit → “4” Digit → “5” Digit → “6” Digit → “7” Digit → “8” Digit → “9” decimal numbers morphology Σ: finite set of terminal symbols N: finite set of non-terminal symbols
Formal Grammars 19 G = (N, Σ, P, S) Num → Digit Num Num → Digit Digit → “0” Digit → “1” Digit → “2” Digit → “3” Digit → “4” Digit → “5” Digit → “6” Digit → “7” Digit → “8” Digit → “9” decimal numbers morphology Σ: finite set of terminal symbols N: finite set of non-terminal symbols S∈N: start symbol
Formal Grammars 20 G = (N, Σ, P, S) Num → Digit Num Num → Digit Digit → “0” Digit → “1” Digit → “2” Digit → “3” Digit → “4” Digit → “5” Digit → “6” Digit → “7” Digit → “8” Digit → “9” decimal numbers morphology Σ: finite set of terminal symbols N: finite set of non-terminal symbols S∈N: start symbol P⊆N×(N∪Σ)* : set of production rules
Formal Grammars 21 Num Digit Num 4 Num 4 Digit Num 4 3 Num 4 3 Digit Num 4 3 0 Num 4 3 0 Digit 4 3 0 3
 Num → Digit Num Digit → “4” Num → Digit Num Digit → “3” Num → Digit Num Digit → “0” Num → Digit Digit → “3” decimal numbers production
Formal Grammars 21 Num Digit Num 4 Num 4 Digit Num 4 3 Num 4 3 Digit Num 4 3 0 Num 4 3 0 Digit 4 3 0 3
 Num → Digit Num Digit → “4” Num → Digit Num Digit → “3” Num → Digit Num Digit → “0” Num → Digit Digit → “3” decimal numbers production leftmost derivation
Formal Grammars 22 Num Digit Num Digit Digit Num Digit Digit Digit Num Digit Digit Digit Digit Digit Digit Digit 3 Digit Digit 0 3 Digit 3 0 3 4 3 0 3
 Num → Digit Num Num → Digit Num Num → Digit Num Num → Digit Digit → “3” Digit → “0” Digit → “3” Digit → “4” decimal numbers production
Formal Grammars 22 Num Digit Num Digit Digit Num Digit Digit Digit Num Digit Digit Digit Digit Digit Digit Digit 3 Digit Digit 0 3 Digit 3 0 3 4 3 0 3
 Num → Digit Num Num → Digit Num Num → Digit Num Num → Digit Digit → “3” Digit → “0” Digit → “3” Digit → “4” decimal numbers production rightmost derivation
Formal Grammars 23 G = (N, Σ, P, S) Exp → Num Exp → Exp “+” Exp Exp → Exp “−” Exp Exp → Exp “*” Exp Exp → Exp “/” Exp Exp → “(” Exp “)” binary expressions syntax
Formal Grammars 24 G = (N, Σ, P, S) Exp → Num Exp → Exp “+” Exp Exp → Exp “−” Exp Exp → Exp “*” Exp Exp → Exp “/” Exp Exp → “(” Exp “)” binary expressions syntax Σ: finite set of terminal symbols
Formal Grammars 25 G = (N, Σ, P, S) Exp → Num Exp → Exp “+” Exp Exp → Exp “−” Exp Exp → Exp “*” Exp Exp → Exp “/” Exp Exp → “(” Exp “)” binary expressions syntax Σ: finite set of terminal symbols N: finite set of non-terminal symbols
Formal Grammars 26 G = (N, Σ, P, S) Exp → Num Exp → Exp “+” Exp Exp → Exp “−” Exp Exp → Exp “*” Exp Exp → Exp “/” Exp Exp → “(” Exp “)” binary expressions syntax Σ: finite set of terminal symbols N: finite set of non-terminal symbols S∈N: start symbol
Formal Grammars 27 G = (N, Σ, P, S) Exp → Num Exp → Exp “+” Exp Exp → Exp “−” Exp Exp → Exp “*” Exp Exp → Exp “/” Exp Exp → “(” Exp “)” binary expressions syntax Σ: finite set of terminal symbols N: finite set of non-terminal symbols S∈N: start symbol P⊆N×(N∪Σ)* : set of production rules
Formal Grammars 28 Exp Exp + Exp Exp + Exp * Exp 3 + Exp * Exp 3 + 4 * Exp 3 + 4 * 5
 Exp → Exp “+” Exp Exp → Exp “*” Exp Exp → Num Exp → Num Exp → Num binary expressions production
Formal Grammars 29 formal grammar G = (N, Σ, P, S) nonterminal symbols N terminal symbols Σ production rules P ⊆ (N∪Σ)* N (N∪Σ)* × (N∪Σ)* start symbol S∈N
Formal Grammars 29 formal grammar G = (N, Σ, P, S) nonterminal symbols N terminal symbols Σ production rules P ⊆ (N∪Σ)* N (N∪Σ)* × (N∪Σ)* start symbol S∈N nonterminal symbol
Formal Grammars 29 formal grammar G = (N, Σ, P, S) nonterminal symbols N terminal symbols Σ production rules P ⊆ (N∪Σ)* N (N∪Σ)* × (N∪Σ)* start symbol S∈N context
Formal Grammars 29 formal grammar G = (N, Σ, P, S) nonterminal symbols N terminal symbols Σ production rules P ⊆ (N∪Σ)* N (N∪Σ)* × (N∪Σ)* start symbol S∈N replacement
Formal Grammars 29 formal grammar G = (N, Σ, P, S) nonterminal symbols N terminal symbols Σ production rules P ⊆ (N∪Σ)* N (N∪Σ)* × (N∪Σ)* start symbol S∈N
Formal Grammars 30 type-0, unrestricted: P ⊆ (N∪Σ)* N (N∪Σ)* × (N∪Σ)* type-1, context-sensitive: (a A c, a b c) type-2, context-free: P ⊆ N × (N∪Σ)* type-3, regular: (A, x) or (A, xB)
Formal Grammars 31 formal grammars context-sensitive context-free regular
Formal Grammars 32 formal grammar G derivation relation G formal language L(G) ⊆ Σ* L(G) = {w∈Σ* | S G * w}
Formal Grammars 33 formal grammar G = (N, Σ, P, S) derivation relation G ⊆ (N∪Σ)* × (N∪Σ)* w G w’ ∃(p, q)∈P: ∃u,v∈(N∪Σ)* : w=u p v ∧ w’=u q v formal language L(G) ⊆ Σ* L(G) = {w∈Σ* | S G * w}
Formal Grammars 34 formal languages context-sensitive context-free regular
Formal Grammars 35 3 * +7 21 3 * +7 21 Exp ExpExp Exp Exp parser Derivation is about productivity. But what about parsing?
Formal Grammars 36 word problem
Formal Grammars 37 word problem χL: Σ* → {0,1} w → 1, if w∈L w → 0, else theoretical computer science decidability & complexity
Formal Grammars 37 word problem χL: Σ* → {0,1} w → 1, if w∈L w → 0, else decidability type-0: semi-decidable type-1, type-2, type-3: decidable theoretical computer science decidability & complexity
Formal Grammars 37 word problem χL: Σ* → {0,1} w → 1, if w∈L w → 0, else decidability type-0: semi-decidable type-1, type-2, type-3: decidable complexity type-1: PSPACE-complete type-2, type-3: P theoretical computer science decidability & complexity
Formal Grammars 37 word problem χL: Σ* → {0,1} w → 1, if w∈L w → 0, else decidability type-0: semi-decidable type-1, type-2, type-3: decidable complexity type-1: PSPACE-complete type-2, type-3: P theoretical computer science decidability & complexity PSPACE⊇NP⊇P
Formal Grammars 38 formal grammars context-sensitive context-free regular theoretical computer science decidability & complexity
Formal Grammars 38 formal grammars context-sensitive context-free regular theoretical computer science decidability & complexity
Formal Grammars 38 formal grammars context-sensitive context-free regular theoretical computer science decidability & complexity
Formal Grammars 39 Exp → Num Exp → Exp “+” Exp Exp → Exp “−” Exp Exp → Exp “*” Exp Exp → Exp “/” Exp Exp → “(” Exp “)” context-free grammars production vs. reduction rules
Formal Grammars 39 Exp → Num Exp → Exp “+” Exp Exp → Exp “−” Exp Exp → Exp “*” Exp Exp → Exp “/” Exp Exp → “(” Exp “)” context-free grammars production vs. reduction rules productive form
Formal Grammars 39 Exp → Num Exp → Exp “+” Exp Exp → Exp “−” Exp Exp → Exp “*” Exp Exp → Exp “/” Exp Exp → “(” Exp “)” Num → Exp Exp “+” Exp → Exp Exp “−” Exp → Exp Exp “*” Exp → Exp Exp “/” Exp → Exp “(” Exp “)” → Exp context-free grammars production vs. reduction rules productive form
Formal Grammars 39 Exp → Num Exp → Exp “+” Exp Exp → Exp “−” Exp Exp → Exp “*” Exp Exp → Exp “/” Exp Exp → “(” Exp “)” Num → Exp Exp “+” Exp → Exp Exp “−” Exp → Exp Exp “*” Exp → Exp Exp “/” Exp → Exp “(” Exp “)” → Exp context-free grammars production vs. reduction rules productive form reductive form
Formal Grammars 40 3 + 4 * 5 Exp + 4 * 5 Exp + Exp * 5 Exp + Exp * Exp Exp + Exp Exp
 Num → Exp Num → Exp Num → Exp Exp “*” Exp → Exp Exp “+” Exp → Exp binary expressions reduction
Formal Grammars 41 3 * +7 21 3 * +7 21 Exp ExpExp Exp Exp parser The word problem is about membership. But what about structure?
Formal Grammars 42 syntax trees
Formal Grammars 43 Exp Num Exp +Exp Exp Exp −Exp Exp Exp *Exp Exp Exp /Exp Exp Exp Exp( ) context-free grammars tree construction rules
Formal Grammars 44 binary expressions tree construction 3 + *4 5
Formal Grammars 44 binary expressions tree construction Exp 3 + *4 5
Formal Grammars 44 binary expressions tree construction Exp Exp 3 + *4 5
Formal Grammars 44 binary expressions tree construction Exp Exp 3 + *4 5 Exp
Formal Grammars 44 binary expressions tree construction Exp Exp Exp 3 + *4 5 Exp
Formal Grammars 44 binary expressions tree construction Exp Exp Exp 3 + *4 5 Exp Exp
Formal Grammars 45 binary expressions ambiguity Exp Exp Exp 3 + *4 5 Exp Exp Exp Exp Exp 3 + *4 5 Exp Exp
Formal Grammars 46 syntax trees different trees for same sentence derivations different leftmost derivations for same sentence different rightmost derivations for same sentence NOT just different derivations for same sentence context-free grammars ambiguity
Formal Grammars 47 parse trees parent node: nonterminal symbol child nodes: terminal symbols abstract syntax trees (ASTs) abstract over terminal symbols convey information at parent nodes abstract over injective production rules syntax trees parse trees & abstract syntax trees
Formal Grammars 48 binary expressions parse tree & abstract syntax tree Exp Exp Exp Exp Exp (3 + *4 5 ) Exp
Formal Grammars 48 binary expressions parse tree & abstract syntax tree Const Mul Const 3 4 5 Const Add Exp Exp Exp Exp Exp (3 + *4 5 ) Exp
Formal Grammars 49 Except where otherwise noted, this work is licensed under
Formal Grammars 50 attribution slide title author license 1 The Pine, Saint Tropez Paul Signac public domain 2, 3, 35, 41 PICOL icons Melih Bilgil CC BY 3.0 9 Writing Caitlin Regan CC BY 2.0 10 Latin Grammar Anthony Nelzin 13, 15, 29-34, 38 Noam Chomsky Fellowsisters CC BY-NC-SA 2.0

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Formal Grammars

  • 1. IN4303 2016-2017 Compiler Construction Formal Grammars language specification Guido Wachsmuth, Eelco Visser
  • 2. Formal Grammars 2 3 * +7 21 3 * +7 21 Exp ExpExp Exp Exp parser
  • 3. Formal Grammars 3 syntax definition errors parse generate check parse table generic parser
  • 4. Formal Grammars Stratego DynSem NaBL2 SDF3 ESV editor SPT tests 4 syntax definition concrete syntax abstract syntax static semantics name binding type system dynamic semantics translation interpretation
  • 5. Formal Grammars Stratego DynSem NaBL2 SDF3 ESV editor SPT tests 5 syntax definition concrete syntax abstract syntax static semantics name binding type system dynamic semantics translation interpretation
  • 6. Formal Grammars Stratego DynSem NaBL2 SDF3 ESV editor SPT tests 6 syntax definition concrete syntax abstract syntax static semantics name binding type system dynamic semantics translation interpretation
  • 7. Formal Grammars 7 P A R E N T A L ADVISORY THEORETICAL CONTENT
  • 14. Formal Grammars 13 vocabulary Σ finite, nonempty set of elements (words, letters) alphabet
  • 15. Formal Grammars 13 vocabulary Σ finite, nonempty set of elements (words, letters) alphabet string over Σ finite sequence of elements chosen from Σ word, sentence, utterance
  • 16. Formal Grammars 13 vocabulary Σ finite, nonempty set of elements (words, letters) alphabet string over Σ finite sequence of elements chosen from Σ word, sentence, utterance formal language λ set of strings over a vocabulary Σ λ ⊆ Σ*
  • 18. Formal Grammars 15 formal grammar G derivation relation G formal language L(G) ⊆ Σ* L(G) = {w∈Σ* | S G * w}
  • 19. Formal Grammars 16 G = (N, Σ, P, S) Num → Digit Num Num → Digit Digit → “0” Digit → “1” Digit → “2” Digit → “3” Digit → “4” Digit → “5” Digit → “6” Digit → “7” Digit → “8” Digit → “9” decimal numbers morphology
  • 20. Formal Grammars 17 G = (N, Σ, P, S) Num → Digit Num Num → Digit Digit → “0” Digit → “1” Digit → “2” Digit → “3” Digit → “4” Digit → “5” Digit → “6” Digit → “7” Digit → “8” Digit → “9” decimal numbers morphology Σ: finite set of terminal symbols
  • 21. Formal Grammars 18 G = (N, Σ, P, S) Num → Digit Num Num → Digit Digit → “0” Digit → “1” Digit → “2” Digit → “3” Digit → “4” Digit → “5” Digit → “6” Digit → “7” Digit → “8” Digit → “9” decimal numbers morphology Σ: finite set of terminal symbols N: finite set of non-terminal symbols
  • 22. Formal Grammars 19 G = (N, Σ, P, S) Num → Digit Num Num → Digit Digit → “0” Digit → “1” Digit → “2” Digit → “3” Digit → “4” Digit → “5” Digit → “6” Digit → “7” Digit → “8” Digit → “9” decimal numbers morphology Σ: finite set of terminal symbols N: finite set of non-terminal symbols S∈N: start symbol
  • 23. Formal Grammars 20 G = (N, Σ, P, S) Num → Digit Num Num → Digit Digit → “0” Digit → “1” Digit → “2” Digit → “3” Digit → “4” Digit → “5” Digit → “6” Digit → “7” Digit → “8” Digit → “9” decimal numbers morphology Σ: finite set of terminal symbols N: finite set of non-terminal symbols S∈N: start symbol P⊆N×(N∪Σ)* : set of production rules
  • 24. Formal Grammars 21 Num Digit Num 4 Num 4 Digit Num 4 3 Num 4 3 Digit Num 4 3 0 Num 4 3 0 Digit 4 3 0 3
 Num → Digit Num Digit → “4” Num → Digit Num Digit → “3” Num → Digit Num Digit → “0” Num → Digit Digit → “3” decimal numbers production
  • 25. Formal Grammars 21 Num Digit Num 4 Num 4 Digit Num 4 3 Num 4 3 Digit Num 4 3 0 Num 4 3 0 Digit 4 3 0 3
 Num → Digit Num Digit → “4” Num → Digit Num Digit → “3” Num → Digit Num Digit → “0” Num → Digit Digit → “3” decimal numbers production leftmost derivation
  • 26. Formal Grammars 22 Num Digit Num Digit Digit Num Digit Digit Digit Num Digit Digit Digit Digit Digit Digit Digit 3 Digit Digit 0 3 Digit 3 0 3 4 3 0 3
 Num → Digit Num Num → Digit Num Num → Digit Num Num → Digit Digit → “3” Digit → “0” Digit → “3” Digit → “4” decimal numbers production
  • 27. Formal Grammars 22 Num Digit Num Digit Digit Num Digit Digit Digit Num Digit Digit Digit Digit Digit Digit Digit 3 Digit Digit 0 3 Digit 3 0 3 4 3 0 3
 Num → Digit Num Num → Digit Num Num → Digit Num Num → Digit Digit → “3” Digit → “0” Digit → “3” Digit → “4” decimal numbers production rightmost derivation
  • 28. Formal Grammars 23 G = (N, Σ, P, S) Exp → Num Exp → Exp “+” Exp Exp → Exp “−” Exp Exp → Exp “*” Exp Exp → Exp “/” Exp Exp → “(” Exp “)” binary expressions syntax
  • 29. Formal Grammars 24 G = (N, Σ, P, S) Exp → Num Exp → Exp “+” Exp Exp → Exp “−” Exp Exp → Exp “*” Exp Exp → Exp “/” Exp Exp → “(” Exp “)” binary expressions syntax Σ: finite set of terminal symbols
  • 30. Formal Grammars 25 G = (N, Σ, P, S) Exp → Num Exp → Exp “+” Exp Exp → Exp “−” Exp Exp → Exp “*” Exp Exp → Exp “/” Exp Exp → “(” Exp “)” binary expressions syntax Σ: finite set of terminal symbols N: finite set of non-terminal symbols
  • 31. Formal Grammars 26 G = (N, Σ, P, S) Exp → Num Exp → Exp “+” Exp Exp → Exp “−” Exp Exp → Exp “*” Exp Exp → Exp “/” Exp Exp → “(” Exp “)” binary expressions syntax Σ: finite set of terminal symbols N: finite set of non-terminal symbols S∈N: start symbol
  • 32. Formal Grammars 27 G = (N, Σ, P, S) Exp → Num Exp → Exp “+” Exp Exp → Exp “−” Exp Exp → Exp “*” Exp Exp → Exp “/” Exp Exp → “(” Exp “)” binary expressions syntax Σ: finite set of terminal symbols N: finite set of non-terminal symbols S∈N: start symbol P⊆N×(N∪Σ)* : set of production rules
  • 33. Formal Grammars 28 Exp Exp + Exp Exp + Exp * Exp 3 + Exp * Exp 3 + 4 * Exp 3 + 4 * 5
 Exp → Exp “+” Exp Exp → Exp “*” Exp Exp → Num Exp → Num Exp → Num binary expressions production
  • 34. Formal Grammars 29 formal grammar G = (N, Σ, P, S) nonterminal symbols N terminal symbols Σ production rules P ⊆ (N∪Σ)* N (N∪Σ)* × (N∪Σ)* start symbol S∈N
  • 35. Formal Grammars 29 formal grammar G = (N, Σ, P, S) nonterminal symbols N terminal symbols Σ production rules P ⊆ (N∪Σ)* N (N∪Σ)* × (N∪Σ)* start symbol S∈N nonterminal symbol
  • 36. Formal Grammars 29 formal grammar G = (N, Σ, P, S) nonterminal symbols N terminal symbols Σ production rules P ⊆ (N∪Σ)* N (N∪Σ)* × (N∪Σ)* start symbol S∈N context
  • 37. Formal Grammars 29 formal grammar G = (N, Σ, P, S) nonterminal symbols N terminal symbols Σ production rules P ⊆ (N∪Σ)* N (N∪Σ)* × (N∪Σ)* start symbol S∈N replacement
  • 38. Formal Grammars 29 formal grammar G = (N, Σ, P, S) nonterminal symbols N terminal symbols Σ production rules P ⊆ (N∪Σ)* N (N∪Σ)* × (N∪Σ)* start symbol S∈N
  • 39. Formal Grammars 30 type-0, unrestricted: P ⊆ (N∪Σ)* N (N∪Σ)* × (N∪Σ)* type-1, context-sensitive: (a A c, a b c) type-2, context-free: P ⊆ N × (N∪Σ)* type-3, regular: (A, x) or (A, xB)
  • 40. Formal Grammars 31 formal grammars context-sensitive context-free regular
  • 41. Formal Grammars 32 formal grammar G derivation relation G formal language L(G) ⊆ Σ* L(G) = {w∈Σ* | S G * w}
  • 42. Formal Grammars 33 formal grammar G = (N, Σ, P, S) derivation relation G ⊆ (N∪Σ)* × (N∪Σ)* w G w’ ∃(p, q)∈P: ∃u,v∈(N∪Σ)* : w=u p v ∧ w’=u q v formal language L(G) ⊆ Σ* L(G) = {w∈Σ* | S G * w}
  • 43. Formal Grammars 34 formal languages context-sensitive context-free regular
  • 44. Formal Grammars 35 3 * +7 21 3 * +7 21 Exp ExpExp Exp Exp parser Derivation is about productivity. But what about parsing?
  • 46. Formal Grammars 37 word problem χL: Σ* → {0,1} w → 1, if w∈L w → 0, else theoretical computer science decidability & complexity
  • 47. Formal Grammars 37 word problem χL: Σ* → {0,1} w → 1, if w∈L w → 0, else decidability type-0: semi-decidable type-1, type-2, type-3: decidable theoretical computer science decidability & complexity
  • 48. Formal Grammars 37 word problem χL: Σ* → {0,1} w → 1, if w∈L w → 0, else decidability type-0: semi-decidable type-1, type-2, type-3: decidable complexity type-1: PSPACE-complete type-2, type-3: P theoretical computer science decidability & complexity
  • 49. Formal Grammars 37 word problem χL: Σ* → {0,1} w → 1, if w∈L w → 0, else decidability type-0: semi-decidable type-1, type-2, type-3: decidable complexity type-1: PSPACE-complete type-2, type-3: P theoretical computer science decidability & complexity PSPACE⊇NP⊇P
  • 50. Formal Grammars 38 formal grammars context-sensitive context-free regular theoretical computer science decidability & complexity
  • 51. Formal Grammars 38 formal grammars context-sensitive context-free regular theoretical computer science decidability & complexity
  • 52. Formal Grammars 38 formal grammars context-sensitive context-free regular theoretical computer science decidability & complexity
  • 53. Formal Grammars 39 Exp → Num Exp → Exp “+” Exp Exp → Exp “−” Exp Exp → Exp “*” Exp Exp → Exp “/” Exp Exp → “(” Exp “)” context-free grammars production vs. reduction rules
  • 54. Formal Grammars 39 Exp → Num Exp → Exp “+” Exp Exp → Exp “−” Exp Exp → Exp “*” Exp Exp → Exp “/” Exp Exp → “(” Exp “)” context-free grammars production vs. reduction rules productive form
  • 55. Formal Grammars 39 Exp → Num Exp → Exp “+” Exp Exp → Exp “−” Exp Exp → Exp “*” Exp Exp → Exp “/” Exp Exp → “(” Exp “)” Num → Exp Exp “+” Exp → Exp Exp “−” Exp → Exp Exp “*” Exp → Exp Exp “/” Exp → Exp “(” Exp “)” → Exp context-free grammars production vs. reduction rules productive form
  • 56. Formal Grammars 39 Exp → Num Exp → Exp “+” Exp Exp → Exp “−” Exp Exp → Exp “*” Exp Exp → Exp “/” Exp Exp → “(” Exp “)” Num → Exp Exp “+” Exp → Exp Exp “−” Exp → Exp Exp “*” Exp → Exp Exp “/” Exp → Exp “(” Exp “)” → Exp context-free grammars production vs. reduction rules productive form reductive form
  • 57. Formal Grammars 40 3 + 4 * 5 Exp + 4 * 5 Exp + Exp * 5 Exp + Exp * Exp Exp + Exp Exp
 Num → Exp Num → Exp Num → Exp Exp “*” Exp → Exp Exp “+” Exp → Exp binary expressions reduction
  • 58. Formal Grammars 41 3 * +7 21 3 * +7 21 Exp ExpExp Exp Exp parser The word problem is about membership. But what about structure?
  • 60. Formal Grammars 43 Exp Num Exp +Exp Exp Exp −Exp Exp Exp *Exp Exp Exp /Exp Exp Exp Exp( ) context-free grammars tree construction rules
  • 61. Formal Grammars 44 binary expressions tree construction 3 + *4 5
  • 62. Formal Grammars 44 binary expressions tree construction Exp 3 + *4 5
  • 63. Formal Grammars 44 binary expressions tree construction Exp Exp 3 + *4 5
  • 64. Formal Grammars 44 binary expressions tree construction Exp Exp 3 + *4 5 Exp
  • 65. Formal Grammars 44 binary expressions tree construction Exp Exp Exp 3 + *4 5 Exp
  • 66. Formal Grammars 44 binary expressions tree construction Exp Exp Exp 3 + *4 5 Exp Exp
  • 67. Formal Grammars 45 binary expressions ambiguity Exp Exp Exp 3 + *4 5 Exp Exp Exp Exp Exp 3 + *4 5 Exp Exp
  • 68. Formal Grammars 46 syntax trees different trees for same sentence derivations different leftmost derivations for same sentence different rightmost derivations for same sentence NOT just different derivations for same sentence context-free grammars ambiguity
  • 69. Formal Grammars 47 parse trees parent node: nonterminal symbol child nodes: terminal symbols abstract syntax trees (ASTs) abstract over terminal symbols convey information at parent nodes abstract over injective production rules syntax trees parse trees & abstract syntax trees
  • 70. Formal Grammars 48 binary expressions parse tree & abstract syntax tree Exp Exp Exp Exp Exp (3 + *4 5 ) Exp
  • 71. Formal Grammars 48 binary expressions parse tree & abstract syntax tree Const Mul Const 3 4 5 Const Add Exp Exp Exp Exp Exp (3 + *4 5 ) Exp
  • 72. Formal Grammars 49 Except where otherwise noted, this work is licensed under
  • 73. Formal Grammars 50 attribution slide title author license 1 The Pine, Saint Tropez Paul Signac public domain 2, 3, 35, 41 PICOL icons Melih Bilgil CC BY 3.0 9 Writing Caitlin Regan CC BY 2.0 10 Latin Grammar Anthony Nelzin 13, 15, 29-34, 38 Noam Chomsky Fellowsisters CC BY-NC-SA 2.0