Applications of CFGs Beyond CFGs Recap Radboud University Nijmegen Final lecture: Applications, Chomsky hierarchy, and Recap H. Geuvers and J. Rot Institute for Computing and Information Sciences Radboud University Nijmegen Version: 2016-17 J. Rot Version: 2016-17 Formal Languages, Grammars and Automata 1 / 28 Applications of CFGs Beyond CFGs Recap Radboud University Nijmegen Outline Applications of CFGs Beyond CFGs Recap J. Rot Version: 2016-17 Formal Languages, Grammars and Automata 2 / 28 Applications of CFGs Beyond CFGs Recap Radboud University Nijmegen Programming languages Most programming languages are (deterministic) context-free. There are tools to automatically build: • a lexical analyzer (`lexer') from regular expressions. \if x = 2 then P else Q00 if x = 2 then P else Q ; • a parser from a CFG. ifthenelse ; = P Q x 2 J. Rot Version: 2016-17 Formal Languages, Grammars and Automata 4 / 28 Applications of CFGs Beyond CFGs Recap Radboud University Nijmegen Lindenmayer systems \The development of an organism...may be considered as the execution of a `developmental program' present in the fertilized egg... A central task of developmental biology is to discover the underlying algorithm from the course of development." { Lindenmayer & Rozenberg (1976) Example: Start with A, expand once per iteration: A ! AB 0 A B ! A 1 AB 2 ABA 3 ABAAB 4 ABAABABA ::: J. Rot Version: 2016-17 Formal Languages, Grammars and Automata 5 / 28 Applications of CFGs Beyond CFGs Recap Radboud University Nijmegen Lindenmayer systems Drawing a Lindenmayer system: • F: move forward • +: rotate counter clockwise • −: rotate clockwise • [: push location/angle • ]: pop location/angle Example: F ! F + F − −F + F 0 F 1 F+F--F+F 2 F+F--F+F+F+F--F+F--F+F--F+F+F+F--F+F 3 ::: J. Rot Version: 2016-17 Formal Languages, Grammars and Automata 6 / 28 Applications of CFGs Beyond CFGs Recap Radboud University Nijmegen Lindenmayer systems S ! F [+S][−S] S ! F − [[S] + S] + F [+FS] − S F ! FF J. Rot Version: 2016-17 Formal Languages, Grammars and Automata 7 / 28 Applications of CFGs Beyond CFGs Recap Radboud University Nijmegen Lindenmayer systems (cont'd) Example: Penrose Tiling (P3) S ! [N] + +[N] + +[N] + +[N] + +[N] M ! OF + +PF − − − −NF [−OF − − − −MF ] + + N ! +OF − −PF [− − −MF − −NF ]+ O ! −MF + +NF [+ + +OF + +PF ]− P ! − − OF + + + +MF [+PF + + + +NF ] − −NF J. Rot Version: 2016-17 Formal Languages, Grammars and Automata 8 / 28 Applications of CFGs Beyond CFGs Recap Radboud University Nijmegen Natural language S = hsentencei ! hnoun-phraseihverb-phrasei: hsentencei ! hnoun-phraseihverb-phraseihobject-phrasei: hnoun-phrasei ! hnameiharticleihnouni hnamei ! John j Jill hnouni ! bicycle j mango harticlei ! a j the hverb-phrasei ! hverbi j hadverbihverbi hverbi ! eats j rides hadverbi ! slowly j frequently hadjective-listi ! hadjectiveihadjective-listi j λ hadjectivei ! big j juicy j yellow hobject-phrasei ! hadjective-listihnamei hobject-phrasei ! harticleihadjective-listihnouni Jill frequently eats a juicy yellow mango. belongs to this language J. Rot Version: 2016-17 Formal Languages, Grammars and Automata 9 / 28 Applications of CFGs Beyond CFGs Recap Radboud University Nijmegen Natural language • Many sentences can be modelled using CFGs, e.g.: ...because I saw Cecilia feed the hippopotamuses . • But some (particularly crazy ) natural languages have non-context-free features like,cross-serial dependencies: ...omdat ik Cecilia de nijlpaarden zag voeren. • To capture these, one needs more power than CFGs J. Rot Version: 2016-17 Formal Languages, Grammars and Automata 10 / 28 Applications of CFGs Beyond CFGs Recap Radboud University Nijmegen Context-sensitive languages • Whereas context-free grammars have rules like this: X ! w X 2 V ; w 2 (Σ [ V )∗ • ...a context-sensitive grammar has rules like this: αX β ! αwβ with X 2 V , α; β; w 2 (Σ [ V )∗, w 6= λ. • Context-sensitive grammars generate context-sensitive languages. J. Rot Version: 2016-17 Formal Languages, Grammars and Automata 12 / 28 Applications of CFGs Beyond CFGs Recap Radboud University Nijmegen Context-sensitive languages Example: fanbncn j n > 0g S ! aBC j aSBC CB ! XB XB ! XC XC ! BC aB ! ab bB ! bb bC ! bc cC ! cc J. Rot Version: 2016-17 Formal Languages, Grammars and Automata 13 / 28 Applications of CFGs Beyond CFGs Recap Radboud University Nijmegen Turing machines • An unrestricted grammar has rules like: u ! v u; v 2 (Σ [ V )∗ • Recognisable by Turing machines • The stack is replaced by an infinite tape: ··· ··· • Transitions look like this: a=b; a=b; ! p q p q which read a, write b, and move left or right on the tape • We no longer need a separate input. Just use the tape: ··· a b b ··· J. Rot Version: 2016-17 Formal Languages, Grammars and Automata 14 / 28 Applications of CFGs Beyond CFGs Recap Radboud University Nijmegen Enumerable and computable languages • Languages recognisable by Turing machine are called enumerable languages • A languages is computable if both L and L = Σ∗ − L are enumerable. • In other words, there is a Turing machine with terminates telling us w 2 L and a (possibly different) one that terminates telling us w 2= L • Example: fan j n is not primeg. • Church-Turing thesis: \computable" () \computable by Turing machine" • (To be continued: Berekenbaarheid, 2nd year) J. Rot Version: 2016-17 Formal Languages, Grammars and Automata 15 / 28 Applications of CFGs Beyond CFGs Recap Radboud University Nijmegen Chomsky hierarchy fw j w describes a terminating Turing machine g;::: fan j n is primeg;::: fanbncn j n > 0g;::: fanbn j n > 0g;::: fan j n > 0g;::: Regular Context-free Context-sensitive Computable Enumerable J. Rot Version: 2016-17 Formal Languages, Grammars and Automata 16 / 28 Applications of CFGs Beyond CFGs Recap Radboud University Nijmegen Trade-offs Bigger classes of languages: • More languages can be described. • But, you can say less about them. w 2 L? time memory L1 = L2? Regular yes jwj const. yes Deterministic context-free yes jwj jwj no Context-free yes jwj3 jwj2 no Context-sensitive yes 2jwj jwjk no Computable yes 1 1 no Enumerable if w 2 L 1 1 no J. Rot Version: 2016-17 Formal Languages, Grammars and Automata 17 / 28 Applications of CFGs Beyond CFGs Recap Radboud University Nijmegen Exam topic: regular expressions You should know: • the definition of regular expression • how to compute a regular expression from an NFAλ using the elimination-of-states method • how to build an NFAλ from a regular expression using the `toolkit' J. Rot Version: 2016-17 Formal Languages, Grammars and Automata 19 / 28 Applications of CFGs Beyond CFGs Recap Radboud University Nijmegen Exam topic: finite automata You should know: • the definition of DFA, NFA, NFAλ • how to construct a DFA, NFA or NFAλ for a given language • how to construct a DFA from an NFA (the subset construction) • the constructions on DFAs for complement and intersection (product construction) of languages J. Rot Version: 2016-17 Formal Languages, Grammars and Automata 20 / 28 Applications of CFGs Beyond CFGs Recap Radboud University Nijmegen Exam Topic: regular languages You should know: • the closure properties of regular languages (union, intersection, complement), and how to use them to prove that a language is regular (or is not regular) n n • typical non-regular languages (fa b j n 2 Ng and palindromes) • Kleene's theorem • how to apply the pumping lemma to show that a language is not regular J. Rot Version: 2016-17 Formal Languages, Grammars and Automata 21 / 28 Applications of CFGs Beyond CFGs Recap Radboud University Nijmegen Exam topic: grammars You should know: • the definition of context-free grammar (CFG) • how to generate strings in a CFG (with leftmost derivations) • how to find the language generated by a (simple) CFG • how to construct a CFG that generates a given (context-free) language • the definition of regular grammar • how to construct a regular grammar from a NFA • how to construct an NFAλ from a regular grammar J. Rot Version: 2016-17 Formal Languages, Grammars and Automata 22 / 28 Applications of CFGs Beyond CFGs Recap Radboud University Nijmegen Exam topic: pushdown automata and CFLs You should know: • the definition of pushdown automata (PDAs) • how to give a PDA for a given (simple context-free) language • how find the language accepted by a given PDA • how to construct a PDA from a CFG • how to construct a CFG from a PDA • the closure properties of context-free languages J. Rot Version: 2016-17 Formal Languages, Grammars and Automata 23 / 28 Applications of CFGs Beyond CFGs Recap Radboud University Nijmegen Remarks on sets Beware that ; 6= f;g ; 6= λ ; 6= fλg. and ;· L = L ·; = ; and fλg · L = L · fλg = L. Symbols: • w 2 L: \is in". • L1 ⊆ L2: is a subset of, i.e. everything in L1 is also in L2. • L1 [ L2:union, the things in either of the two sets. • L1 \ L2: intersection, only the things in both sets. • L: complement, the words not in L, L = Σ∗ − L. Terminology: Language described using set-notation, examples: ∗ n m fw 2 fa; bg j jwja is eveng; fa b j n < mg; ; J. Rot Version: 2016-17 Formal Languages, Grammars and Automata 24 / 28 Applications of CFGs Beyond CFGs Recap Radboud University Nijmegen Remarks on words and languages Languages • contain words. • can be infinite, but words are finite. The language L∗ • always contains λ. • is not the same as fw n j w 2 L; n ≥ 0g. • is L0 [ L1 [ L2 [··· . J. Rot Version: 2016-17 Formal Languages, Grammars and Automata 25 / 28 Applications of CFGs Beyond CFGs Recap Radboud University Nijmegen Remarks on proofs To prove that L1 = L2, show that • L1 ⊆ L2 (for all w, w 2 L1 implies w 2 L2), and • L1 ⊇ L2 (for all w, w 2 L2 implies w 2 L1). Proof by contradiction • If regular languages have property X , and L1 does not have property X , then L1 is not regular.