Context Sensitive Grammar

Transcription

1 Context Sensitive Grammar Aparna S Vijayan Department of Computer Science and Automation December 2, 2011 Aparna S Vijayan (CSA) CSG December 2, / 12

2 Contents Aparna S Vijayan (CSA) CSG December 2, / 12

3 Contents 1 Definition of CSG Aparna S Vijayan (CSA) CSG December 2, / 12

4 Contents 1 Definition of CSG 2 Context Sensitive Language(CSL) Aparna S Vijayan (CSA) CSG December 2, / 12

5 Contents 1 Definition of CSG 2 Context Sensitive Language(CSL) 3 Example of CSL Aparna S Vijayan (CSA) CSG December 2, / 12

6 Contents 1 Definition of CSG 2 Context Sensitive Language(CSL) 3 Example of CSL 4 Chomsky Hierachy Aparna S Vijayan (CSA) CSG December 2, / 12

7 Definition Context Sensitive Grammar(Type1 Grammar) A context-sensitive grammar (CSG) is an unrestricted grammar in which every production has the form α β with β α (where α and β are strings of nonterminals and terminals). The concept of context-sensitive grammar was introduced by Noam Chomsky in the In every derivation the length of the string never decreases. The term context-sensitive comes from a normal form for these grammars,where each production is of the form α 1 Aα 2 α 1 βα 2,with β ɛ. They permit replacement of variable A by string β only in the context α 1 - α 2. Aparna S Vijayan (CSA) CSG December 2, / 12

8 Formal Definition A Context Sensitive Grammar is a 4-tuple, G = (N,Σ, P, S) where: Aparna S Vijayan (CSA) CSG December 2, / 12

9 Formal Definition A Context Sensitive Grammar is a 4-tuple, G = (N,Σ, P, S) where: N=Set of non terminal symbols. Aparna S Vijayan (CSA) CSG December 2, / 12

10 Formal Definition A Context Sensitive Grammar is a 4-tuple, G = (N,Σ, P, S) where: N=Set of non terminal symbols. Σ=Set of terminal symbols. Aparna S Vijayan (CSA) CSG December 2, / 12

11 Formal Definition A Context Sensitive Grammar is a 4-tuple, G = (N,Σ, P, S) where: N=Set of non terminal symbols. Σ=Set of terminal symbols. S=Start symbol of the production. Aparna S Vijayan (CSA) CSG December 2, / 12

12 Formal Definition A Context Sensitive Grammar is a 4-tuple, G = (N,Σ, P, S) where: N=Set of non terminal symbols. Σ=Set of terminal symbols. S=Start symbol of the production. P=Finite set of productions. Aparna S Vijayan (CSA) CSG December 2, / 12

13 Formal Definition A Context Sensitive Grammar is a 4-tuple, G = (N,Σ, P, S) where: N=Set of non terminal symbols. Σ=Set of terminal symbols. S=Start symbol of the production. P=Finite set of productions. All rules in P are of the form α 1 Aα 2 α 1 βα 2. A N ( A is a single nonterminal) α 1, α 2, β (N Σ) +. Aparna S Vijayan (CSA) CSG December 2, / 12

14 Context Sensitive Language The language generated by the Context Sensitive Grammar is called context sensitive language. Aparna S Vijayan (CSA) CSG December 2, / 12

15 Context Sensitive Language The language generated by the Context Sensitive Grammar is called context sensitive language. If G is a Context Sensitive Grammar then L(G)= { w (w Σ ) ( S + G w)}. Aparna S Vijayan (CSA) CSG December 2, / 12

16 Context Sensitive Language The language generated by the Context Sensitive Grammar is called context sensitive language. If G is a Context Sensitive Grammar then L(G)= { w (w Σ ) ( S + G w)}. Eg 1 of a context sensitive grammar G = {{S, A, B, C, a, b, c}, {a, b, c}, P, S}where P is the set of rules. S asbc S abc CB BC ab ab bb bb bc bc cc cc The language generated by this grammar is {a n b n c n n 1}. Aparna S Vijayan (CSA) CSG December 2, / 12

17 Context Sensitive Language The derivation for the string aabbcc is Aparna S Vijayan (CSA) CSG December 2, / 12

18 Context Sensitive Language The derivation for the string aabbcc is S asbc Aparna S Vijayan (CSA) CSG December 2, / 12

19 Context Sensitive Language The derivation for the string aabbcc is S asbc aabcbc Aparna S Vijayan (CSA) CSG December 2, / 12

20 Context Sensitive Language The derivation for the string aabbcc is S asbc aabcbc aabcbc Aparna S Vijayan (CSA) CSG December 2, / 12

21 Context Sensitive Language The derivation for the string aabbcc is S asbc aabcbc aabcbc aabbcc Aparna S Vijayan (CSA) CSG December 2, / 12

22 Context Sensitive Language The derivation for the string aabbcc is S asbc aabcbc aabcbc aabbcc aabbcc Aparna S Vijayan (CSA) CSG December 2, / 12

23 Context Sensitive Language The derivation for the string aabbcc is S asbc aabcbc aabcbc aabbcc aabbcc aabbcc Aparna S Vijayan (CSA) CSG December 2, / 12

24 Context Sensitive Language The derivation for the string aabbcc is S asbc aabcbc aabcbc aabbcc aabbcc aabbcc aabbcc Aparna S Vijayan (CSA) CSG December 2, / 12

25 Example2 CSG L = (#a = #b = #c) G2 =({S, A, B, C, a, b, c}, {a, b, c}, P, S) S ABC S ABCS AB BA AC CA BC CB BA AB CA AC CB BC A a B b C c Aparna S Vijayan (CSA) CSG December 2, / 12

26 Hierarchy of Formal Languages Aparna S Vijayan (CSA) CSG December 2, / 12

27 Relation between Formal Languages The CFL s not containing ɛ are properly contained in the context sensitive languages Not all Context Sensitive Languages are Context Free. Every Context Sensitive Language is recusive. Aparna S Vijayan (CSA) CSG December 2, / 12

28 Chomsky Hierachy Described by Chomsky in Four classes of language Type 3,Type2,Type1,Type 0 from most restrictive to most general(unestricted). Each level of hierarchy can be characterized by a class of grammar. Aparna S Vijayan (CSA) CSG December 2, / 12

29 Chomsky Hierarchy Type Language productions Device 3 Regular A ab, A a Finite Automaton 2 CFL A α Pushdown Automaton 1 CSL α β, β α Linear Bounded Automaton 0 RE α β Turing Machine A,B Nonterminals α,β string of terminals and nonterminals a terminal symbol Aparna S Vijayan (CSA) CSG December 2, / 12

30 References Introduction to Automata Theory Languages and Computation by JOHN E HOPCROFT, JEFFERY D.ULLMAN. Introduction to Languages and the Theory of Computation JOHN MARTIN grammar Aparna S Vijayan (CSA) CSG December 2, / 12

31 Linear Bounded Automata Indu John Department of Computer Science and Automation Indian Institute of Science, Bangalore December 1, 2011 Indu John (Department of CSA) Linear Bounded Automata December 1, / 14

32 Overview Indu John (Department of CSA) Linear Bounded Automata December 1, / 14

33 Overview Definition Indu John (Department of CSA) Linear Bounded Automata December 1, / 14

34 Overview Definition Results about LBAs Indu John (Department of CSA) Linear Bounded Automata December 1, / 14

35 Overview Definition Results about LBAs CSLs and LBAs Indu John (Department of CSA) Linear Bounded Automata December 1, / 14

36 Definition A Turing machine that uses only the tape space occupied by the input is called a linear-bounded automaton (LBA). A linear bounded automaton is a nondeterministic Turing machine M = (Q, Σ, Γ, δ, s, t, r) such that: There are two special tape symbols < and >(the left end marker and right end marker). The TM begins in the configuration (s, < x >, 0). The TM cannot replace < or > with anything else, nor move the tape head left of < or right of >. Indu John (Department of CSA) Linear Bounded Automata December 1, / 14

37 LBA An equivalent definition of an LBA is that it uses only k times the amount of space occupied by the input string, where k is a constant fixed for the particular machine. Indu John (Department of CSA) Linear Bounded Automata December 1, / 14

38 LBA An equivalent definition of an LBA is that it uses only k times the amount of space occupied by the input string, where k is a constant fixed for the particular machine. Possible to simulate k tape cells with a single tape cell, by increasing the size of the tape alphabet Indu John (Department of CSA) Linear Bounded Automata December 1, / 14

39 LBA An equivalent definition of an LBA is that it uses only k times the amount of space occupied by the input string, where k is a constant fixed for the particular machine. Possible to simulate k tape cells with a single tape cell, by increasing the size of the tape alphabet Examples: {a n b n c n n 0}; counting number of a s Indu John (Department of CSA) Linear Bounded Automata December 1, / 14

40 LBA An equivalent definition of an LBA is that it uses only k times the amount of space occupied by the input string, where k is a constant fixed for the particular machine. Possible to simulate k tape cells with a single tape cell, by increasing the size of the tape alphabet Examples: {a n b n c n n 0}; counting number of a s This limitation makes the LBA a somewhat more accurate model of computers that actually exist than a Turing machine, whose definition assumes unlimited tape. Indu John (Department of CSA) Linear Bounded Automata December 1, / 14

41 History Indu John (Department of CSA) Linear Bounded Automata December 1, / 14

42 History In 1960, Myhill introduced an automaton model today known as deterministic linear bounded automaton. Indu John (Department of CSA) Linear Bounded Automata December 1, / 14

43 History In 1960, Myhill introduced an automaton model today known as deterministic linear bounded automaton. Shortly thereafter, Landweber proved that the languages accepted by a deterministic LBA are always context-sensitive. Indu John (Department of CSA) Linear Bounded Automata December 1, / 14

44 History In 1960, Myhill introduced an automaton model today known as deterministic linear bounded automaton. Shortly thereafter, Landweber proved that the languages accepted by a deterministic LBA are always context-sensitive. In 1964, Kuroda introduced the more general model of (nondeterministic) linear bounded automata, and showed that the languages accepted by them are precisely the context-sensitive languages. Indu John (Department of CSA) Linear Bounded Automata December 1, / 14

45 Number of configurations Suppose that a given LBA M has q states, m characters in the tape alphabet, and the input length is n. Then M can be in at most α(n) = Tapecontents {}}{ m n Headposition {}}{ n State {}}{ q configurations. Indu John (Department of CSA) Linear Bounded Automata December 1, / 14

46 Results about LBA Halting Problem The halting problem is solvable for linear bounded automata. Halt LBA = {< M, w > M is an LBA and M halts on w} is decidable. An LBA that stops on input w must stop in at most α( w ) steps Indu John (Department of CSA) Linear Bounded Automata December 1, / 14

47 Results about LBA Halting Problem The halting problem is solvable for linear bounded automata. Halt LBA = {< M, w > M is an LBA and M halts on w} is decidable. An LBA that stops on input w must stop in at most α( w ) steps Membership problem The membership problem is solvable for linear bounded automata. A LBA = {< M, w > M is an LBA and M accepts w} is decidable. Indu John (Department of CSA) Linear Bounded Automata December 1, / 14

48 Results about LBA Emptiness Problem The emptiness problem is unsolvable for linear bounded automata. For every Turing machine there is a linear bounded automaton which accepts the set of strings which are valid halting computations for the Turing machine. Indu John (Department of CSA) Linear Bounded Automata December 1, / 14

49 LBA Problems Indu John (Department of CSA) Linear Bounded Automata December 1, / 14

50 LBA Problems 1 Is the class of languages accepted by LBA equal to the class of languages accepted by deterministic LBA? Indu John (Department of CSA) Linear Bounded Automata December 1, / 14

51 LBA Problems 1 Is the class of languages accepted by LBA equal to the class of languages accepted by deterministic LBA? Open Problem! Indu John (Department of CSA) Linear Bounded Automata December 1, / 14

52 LBA Problems 1 Is the class of languages accepted by LBA equal to the class of languages accepted by deterministic LBA? Open Problem! 2 Is the class of languages accepted by LBA closed under complement? Indu John (Department of CSA) Linear Bounded Automata December 1, / 14

53 LBA Problems 1 Is the class of languages accepted by LBA equal to the class of languages accepted by deterministic LBA? Open Problem! 2 Is the class of languages accepted by LBA closed under complement? Yes. (Immerman Szelepcsenyi Theorem) Indu John (Department of CSA) Linear Bounded Automata December 1, / 14

54 LBAs and CSLs Theorem(Landweber-Kuroda) A language is accepted by an LBA iff it is context sensitive. Indu John (Department of CSA) Linear Bounded Automata December 1, / 14

55 Proof If L is a CSL, then L is accepted by some LBA. Let G = (N, Σ, S, P) be the given grammar such that L(G) = L. Construct LBA M with tape alphabet Σ {N Σ}(2- track machine) First track holds input string w Second track holds a sentential form α of G, initialized to S. Indu John (Department of CSA) Linear Bounded Automata December 1, / 14

56 Proof... If w = ɛ, M halts without accepting. Repeat : 1 Non-deterministically select a position i in α. 2 Non-deterministically select a production β γ of G. 3 If β appears beginning in position i of α, replace β by γ there. If the sentential form is longer than w, LBA halts without accepting. 4 Compare the resulting sentential form with w on track 1. If they match, accept. If not go to step 1. Indu John (Department of CSA) Linear Bounded Automata December 1, / 14

57 Proof... If there is a linear bounded automaton M accepting the language L, then there is a context sensitive grammar generating L {ɛ}. Indu John (Department of CSA) Linear Bounded Automata December 1, / 14

58 Proof... If there is a linear bounded automaton M accepting the language L, then there is a context sensitive grammar generating L {ɛ}. Sketch of proof : Derivation simulates moves of LBA Indu John (Department of CSA) Linear Bounded Automata December 1, / 14

59 Proof... If there is a linear bounded automaton M accepting the language L, then there is a context sensitive grammar generating L {ɛ}. Sketch of proof : Derivation simulates moves of LBA Three types of productions 1 Productions that can generate two copies of a string in Σ, along with some symbols that act as markers to keep the two copies separate. 2 Productions that can simulate a sequence of moves of M. During this portion of a derivation, one of the two copies of the original string is left unchanged; the other, representing the input tape to M, is modified accordingly. 3 Productions that can erase everything but the unmodified copy of the string, provided that the simulated moves of M applied to the other copy cause M to accept. Indu John (Department of CSA) Linear Bounded Automata December 1, / 14

60 References John Martin, Introduction to Languages and the Theory of Computation, Tata McGraw-Hill, Third Edition. John Hopcroft, Jeffery Ullman, Introduction to Automata Theory, Languages and Computation Indu John (Department of CSA) Linear Bounded Automata December 1, / 14