Recursive descent for grammars with contexts
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1 39th International Conference on Current Trends in Theory and Practice of Computer Science Špindleruv Mlýn, Czech Republic Recursive descent parsing for grammars with contexts Ph.D. student, Department of Mathematics and Statistics, University of Turku Turku Centre for Computer Science (TUCS) Finland January 26 31, 2013
2 What a recursive descent is? subclass of grammars allowing recursive descent: LL(k)-grammars k look-ahead symbols
3 What a recursive descent is? subclass of grammars allowing recursive descent: LL(k)-grammars k look-ahead symbols LL(1) context-free grammar generating { a n b n n 0 }: S asb ε S() { a() { if (look-ahead is a ) { if (current symbol is a ) a(); S(); b(); advance position by 1 } else error } }
4 What a recursive descent is? subclass of grammars allowing recursive descent: LL(k)-grammars k look-ahead symbols LL(1) context-free grammar generating { a n b n n 0 }: S asb ε S() { a() { if (look-ahead is a ) { if (current symbol is a ) a(); S(); b(); advance position by 1 } else error } } First compilers for Pascal are recursive descent parsers. Implemented by hand (N. Wirth, 1970). Program code can be easily generated automatically.
5 Extension of CFGs with Boolean operations Conjunctive grammars (Okhotin, 2001) A α 1 &... & α k, α i (Σ N)
6 Extension of CFGs with Boolean operations Conjunctive grammars (Okhotin, 2001) A α 1 &... & α k, α i (Σ N) a string w has property A w has all the properties α 1,..., α k
7 Extension of CFGs with Boolean operations Conjunctive grammars (Okhotin, 2001) A α 1 &... & α k, α i (Σ N) a string w has property A w has all the properties α 1,..., α k { a n b n c n n 0 } S AB & DC A aa ε B bbc ε C cc ε D adb ε
8 Extension of CFGs with Boolean operations Conjunctive grammars (Okhotin, 2001) A α 1 &... & α k, α i (Σ N) a string w has property A w has all the properties α 1,..., α k { a n b n c n n 0 } S AB & DC A aa ε B bbc ε C cc ε D adb ε non-context-free languages generated complexity of basic parsing algorithms preserved nontrivial properties of subclasses
9 Grammars with contexts (B, Okhotin, 2012) A α 1 &... & α k
10 Grammars with contexts (B, Okhotin, 2012) A α 1 &... & α k & β 1 &... & β m
11 Grammars with contexts (B, Okhotin, 2012) A α 1 &... & α k & β 1 &... & β m & γ 1 &... & γ n
12 Grammars with contexts (B, Okhotin, 2012) A α 1 &... & α k & β 1 &... & β m & γ 1 &... & γ n Given a string x = u w v: each α i defines w
13 Grammars with contexts (B, Okhotin, 2012) A α 1 &... & α k & β 1 &... & β m & γ 1 &... & γ n Given a string x = u w v: each α i defines w each β i defines v (the right context of w)
14 Grammars with contexts (B, Okhotin, 2012) A α 1 &... & α k & β 1 &... & β m & γ 1 &... & γ n Given a string x = u w v: each α i defines w each β i defines v (the right context of w) each γ i defines wv (the extended right context of w)
15 Grammars with contexts (B, Okhotin, 2012) A α 1 &... & α k & β 1 &... & β m & γ 1 &... & γ n Given a string x = u w v: each α i defines w each β i defines v (the right context of w) each γ i defines wv (the extended right context of w) Semantics by logical deduction of elementary propositions [X, w v] : a string w written in right context v has the property X
16 Grammars with contexts (B, Okhotin, 2012) A α 1 &... & α k & β 1 &... & β m & γ 1 &... & γ n Given a string x = u w v: each α i defines w each β i defines v (the right context of w) each γ i defines wv (the extended right context of w) Semantics by logical deduction of elementary propositions [X, w v] : a string w written in right context v has the property X {a n b n c} {a n b 2n d}
17 Grammars with contexts (B, Okhotin, 2012) A α 1 &... & α k & β 1 &... & β m & γ 1 &... & γ n Given a string x = u w v: each α i defines w each β i defines v (the right context of w) each γ i defines wv (the extended right context of w) Semantics by logical deduction of elementary propositions [X, w v] : a string w written in right context v has the property X {a n b n c} {a n b 2n d} S Ac Bd A aab ε L(A) = {a n b n } B abbb ε L(B) = {a n b 2n }
18 Grammars with contexts (B, Okhotin, 2012) A α 1 &... & α k & β 1 &... & β m & γ 1 &... & γ n Given a string x = u w v: each α i defines w each β i defines v (the right context of w) each γ i defines wv (the extended right context of w) Semantics by logical deduction of elementary propositions [X, w v] : a string w written in right context v has the property X {a n b n c} {a n b 2n d} S Ac Bd A aab ε L(A) = {a n b n } B abbb ε L(B) = {a n b 2n } the language is not standard LL(k) for any k
19 Grammars with contexts (B, Okhotin, 2012) A α 1 &... & α k & β 1 &... & β m & γ 1 &... & γ n Given a string x = u w v: each α i defines w each β i defines v (the right context of w) each γ i defines wv (the extended right context of w) Semantics by logical deduction of elementary propositions [X, w v] : a string w written in right context v has the property X {a n b n c} {a n b 2n d} S Ac Bd A aab ε L(A) = {a n b n } B abbb ε L(B) = {a n b 2n } the language is not standard LL(k) for any k way out: use right contexts to peek the last symbol of the string
20 Grammars with contexts (B, Okhotin, 2012) A α 1 &... & α k & β 1 &... & β m & γ 1 &... & γ n Given a string x = u w v: each α i defines w each β i defines v (the right context of w) each γ i defines wv (the extended right context of w) Semantics by logical deduction of elementary propositions [X, w v] : a string w written in right context v has the property X {a n b n c} {a n b 2n d} S Ac Bd A aab ε L(A) = {a n b n } B abbb ε L(B) = {a n b 2n } X ax bx ε L(X ) = (a b) the language is not standard LL(k) for any k way out: use right contexts to peek the last symbol of the string
21 Grammars with contexts (B, Okhotin, 2012) A α 1 &... & α k & β 1 &... & β m & γ 1 &... & γ n Given a string x = u w v: each α i defines w each β i defines v (the right context of w) each γ i defines wv (the extended right context of w) Semantics by logical deduction of elementary propositions [X, w v] : a string w written in right context v has the property X {a n b n c} {a n b 2n d} S Xc & Ac Xd & Bd A aab ε L(A) = {a n b n } B abbb ε L(B) = {a n b 2n } X ax bx ε L(X ) = (a b) the language is not standard LL(k) for any k way out: use right contexts to peek the last symbol of the string
22 Results Recursive descent parser:
23 Results Recursive descent parser: conjunction in rules: scan the substring multiple times
24 Results Recursive descent parser: conjunction in rules: scan the substring multiple times right contexts (, ): to choose suitable alternatives of rules
25 Results Recursive descent parser: conjunction in rules: scan the substring multiple times right contexts (, ): to choose suitable alternatives of rules backtracking: to go over the alternatives
26 Results Recursive descent parser: conjunction in rules: scan the substring multiple times right contexts (, ): to choose suitable alternatives of rules backtracking: to go over the alternatives memoization: linear time instead of exponential
27 Results Recursive descent parser: Theory: conjunction in rules: scan the substring multiple times right contexts (, ): to choose suitable alternatives of rules backtracking: to go over the alternatives memoization: linear time instead of exponential mathematically sound definition of right contexts ( look-ahead ) within recursive descent
28 Results Recursive descent parser: Theory: conjunction in rules: scan the substring multiple times right contexts (, ): to choose suitable alternatives of rules backtracking: to go over the alternatives memoization: linear time instead of exponential mathematically sound definition of right contexts ( look-ahead ) within recursive descent higher expressive power, but still linear time
29 Results Recursive descent parser: Theory: conjunction in rules: scan the substring multiple times right contexts (, ): to choose suitable alternatives of rules backtracking: to go over the alternatives memoization: linear time instead of exponential mathematically sound definition of right contexts ( look-ahead ) within recursive descent higher expressive power, but still linear time parsing algorithm proven correct
30 Results Recursive descent parser: Theory: conjunction in rules: scan the substring multiple times right contexts (, ): to choose suitable alternatives of rules backtracking: to go over the alternatives memoization: linear time instead of exponential mathematically sound definition of right contexts ( look-ahead ) within recursive descent Practice: higher expressive power, but still linear time parsing algorithm proven correct implemented as a prototype software
31 Results Recursive descent parser: Theory: conjunction in rules: scan the substring multiple times right contexts (, ): to choose suitable alternatives of rules backtracking: to go over the alternatives memoization: linear time instead of exponential mathematically sound definition of right contexts ( look-ahead ) within recursive descent Practice: higher expressive power, but still linear time parsing algorithm proven correct implemented as a prototype software
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