context sensitive grammar - indian statistical instituteansuman/flat2018/csl-lba.pdf · de nition...
TRANSCRIPT
Context Sensitive Grammar
Aparna S Vijayan
Department of Computer Science and Automation
December 2, 2011
Aparna S Vijayan (CSA) CSG December 2, 2011 1 / 12
Contents
1 Definition of CSG
2 Context Sensitive Language(CSL)
3 Example of CSL
4 Chomsky Hierachy
Aparna S Vijayan (CSA) CSG December 2, 2011 2 / 12
Contents
1 Definition of CSG
2 Context Sensitive Language(CSL)
3 Example of CSL
4 Chomsky Hierachy
Aparna S Vijayan (CSA) CSG December 2, 2011 2 / 12
Contents
1 Definition of CSG
2 Context Sensitive Language(CSL)
3 Example of CSL
4 Chomsky Hierachy
Aparna S Vijayan (CSA) CSG December 2, 2011 2 / 12
Contents
1 Definition of CSG
2 Context Sensitive Language(CSL)
3 Example of CSL
4 Chomsky Hierachy
Aparna S Vijayan (CSA) CSG December 2, 2011 2 / 12
Contents
1 Definition of CSG
2 Context Sensitive Language(CSL)
3 Example of CSL
4 Chomsky Hierachy
Aparna S Vijayan (CSA) CSG December 2, 2011 2 / 12
Definition
Context Sensitive Grammar(Type1 Grammar)
A context-sensitive grammar (CSG) is an unrestricted grammar inwhich every production has the formα→ β with |β| ≥ |α| (where α and β are strings of nonterminals andterminals).
The concept of context-sensitive grammar was introduced by NoamChomsky in the 1950.
In every derivation the length of the string never decreases.
The term ”context-sensitive” comes from a normal form for thesegrammars,where each production is of the form α1Aα2→α1βα2,withβ 6= ε.
They permit replacement of variable A by string β only in the”context ” α1 - α2.
Aparna S Vijayan (CSA) CSG December 2, 2011 3 / 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.All rules in P are of the form α1Aα2→α1βα2.
A ∈ N ( A is a single nonterminal)α1, α2, β ∈ (N ∪ Σ)+.
Aparna S Vijayan (CSA) CSG December 2, 2011 4 / 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.All rules in P are of the form α1Aα2→α1βα2.
A ∈ N ( A is a single nonterminal)α1, α2, β ∈ (N ∪ Σ)+.
Aparna S Vijayan (CSA) CSG December 2, 2011 4 / 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.All rules in P are of the form α1Aα2→α1βα2.
A ∈ N ( A is a single nonterminal)α1, α2, β ∈ (N ∪ Σ)+.
Aparna S Vijayan (CSA) CSG December 2, 2011 4 / 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.All rules in P are of the form α1Aα2→α1βα2.
A ∈ N ( A is a single nonterminal)α1, α2, β ∈ (N ∪ Σ)+.
Aparna S Vijayan (CSA) CSG December 2, 2011 4 / 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.
All rules in P are of the form α1Aα2→α1βα2.
A ∈ N ( A is a single nonterminal)α1, α2, β ∈ (N ∪ Σ)+.
Aparna S Vijayan (CSA) CSG December 2, 2011 4 / 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.All rules in P are of the form α1Aα2→α1βα2.
A ∈ N ( A is a single nonterminal)α1, α2, β ∈ (N ∪ Σ)+.
Aparna S Vijayan (CSA) CSG December 2, 2011 4 / 12
Context Sensitive Language
The language generated by the Context Sensitive Grammar is calledcontext sensitive language.
If G is a Context Sensitive Grammar thenL(G)=
{w∣∣(w ∈ Σ∗) ∧
(S ⇒+
G w)}
.
Eg 1 of a context sensitive grammarG = {{S ,A,B,C , a, b, c} , {a, b, c} ,P,S}where P is the set of rules.
S→ aSBCS → aBCCB → BCaB → abbB → bbbC → bccC →cc
The language generated by this grammar is {anbncn|n ≥ 1} .
Aparna S Vijayan (CSA) CSG December 2, 2011 5 / 12
Context Sensitive Language
The language generated by the Context Sensitive Grammar is calledcontext sensitive language.
If G is a Context Sensitive Grammar thenL(G)=
{w∣∣(w ∈ Σ∗) ∧
(S ⇒+
G w)}
.
Eg 1 of a context sensitive grammarG = {{S ,A,B,C , a, b, c} , {a, b, c} ,P,S}where P is the set of rules.
S→ aSBCS → aBCCB → BCaB → abbB → bbbC → bccC →cc
The language generated by this grammar is {anbncn|n ≥ 1} .
Aparna S Vijayan (CSA) CSG December 2, 2011 5 / 12
Context Sensitive Language
The language generated by the Context Sensitive Grammar is calledcontext sensitive language.
If G is a Context Sensitive Grammar thenL(G)=
{w∣∣(w ∈ Σ∗) ∧
(S ⇒+
G w)}
.
Eg 1 of a context sensitive grammarG = {{S ,A,B,C , a, b, c} , {a, b, c} ,P,S}where P is the set of rules.
S→ aSBCS → aBCCB → BCaB → abbB → bbbC → bccC →cc
The language generated by this grammar is {anbncn|n ≥ 1} .Aparna S Vijayan (CSA) CSG December 2, 2011 5 / 12
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, 2011 6 / 12
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, 2011 6 / 12
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, 2011 6 / 12
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, 2011 6 / 12
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, 2011 6 / 12
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, 2011 6 / 12
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, 2011 6 / 12
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, 2011 6 / 12
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 →cAparna S Vijayan (CSA) CSG December 2, 2011 7 / 12
Relation between Formal Languages
The CFL’s not containing ε are properly contained in the contextsensitive languages
Not all Context Sensitive Languages are Context Free.
Every Context Sensitive Language is recusive.
Aparna S Vijayan (CSA) CSG December 2, 2011 9 / 12
Chomsky Hierachy
Described by Chomsky in 1956.
Four classes of language Type 3,Type2,Type1,Type 0 from mostrestrictive to most general(Unestricted).
Each level of hierarchy can be characterized by a class of grammar.
Aparna S Vijayan (CSA) CSG December 2, 2011 10 / 12
Chomsky Hierarchy
Type Language productions Device3 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, 2011 11 / 12
References
Introduction to Automata Theory Languages and Computation by ”JOHN EHOPCROFT”,”JEFFERY D.ULLMAN”.
Introduction to Languages and the Theory of Computation ”JOHN MARTIN”
http://en.wikipedia.org/wiki/Context-sensitive grammar
http://adammikeal.org/courses/compute/Chomsky
Aparna S Vijayan (CSA) CSG December 2, 2011 12 / 12
Linear Bounded Automata
Indu John
Department of Computer Science and AutomationIndian Institute of Science, Bangalore
December 1, 2011
Indu John (Department of CSA) Linear Bounded Automata December 1, 2011 1 / 14
Overview
Definition
Results about LBAs
CSLs and LBAs
Indu John (Department of CSA) Linear Bounded Automata December 1, 2011 2 / 14
Overview
Definition
Results about LBAs
CSLs and LBAs
Indu John (Department of CSA) Linear Bounded Automata December 1, 2011 2 / 14
Overview
Definition
Results about LBAs
CSLs and LBAs
Indu John (Department of CSA) Linear Bounded Automata December 1, 2011 2 / 14
Overview
Definition
Results about LBAs
CSLs and LBAs
Indu John (Department of CSA) Linear Bounded Automata December 1, 2011 2 / 14
Definition
A Turing machine that uses only the tape space occupied by the input iscalled a linear-bounded automaton (LBA).
A linear bounded automaton is a nondeterministic Turing machineM = (Q,Σ, Γ, δ, s, t, r) such that:
There are two special tape symbols < and >(the left end marker andright end marker).The TM begins in the configuration (s, < x >, 0).The TM cannot replace < or > with anything else, nor move the tapehead left of < or right of >.
Indu John (Department of CSA) Linear Bounded Automata December 1, 2011 3 / 14
LBA
An equivalent definition of an LBA is that it uses only k times theamount of space occupied by the input string, where k is a constantfixed for the particular machine.
Possible to simulate k tape cells with a single tape cell, by increasingthe size of the tape alphabet
Examples: {anbncn|n ≥ 0}; counting number of a’s
This limitation makes the LBA a somewhat more accurate model ofcomputers that actually exist than a Turing machine, whose definitionassumes unlimited tape.
Indu John (Department of CSA) Linear Bounded Automata December 1, 2011 4 / 14
LBA
An equivalent definition of an LBA is that it uses only k times theamount of space occupied by the input string, where k is a constantfixed for the particular machine.
Possible to simulate k tape cells with a single tape cell, by increasingthe size of the tape alphabet
Examples: {anbncn|n ≥ 0}; counting number of a’s
This limitation makes the LBA a somewhat more accurate model ofcomputers that actually exist than a Turing machine, whose definitionassumes unlimited tape.
Indu John (Department of CSA) Linear Bounded Automata December 1, 2011 4 / 14
LBA
An equivalent definition of an LBA is that it uses only k times theamount of space occupied by the input string, where k is a constantfixed for the particular machine.
Possible to simulate k tape cells with a single tape cell, by increasingthe size of the tape alphabet
Examples: {anbncn|n ≥ 0}; counting number of a’s
This limitation makes the LBA a somewhat more accurate model ofcomputers that actually exist than a Turing machine, whose definitionassumes unlimited tape.
Indu John (Department of CSA) Linear Bounded Automata December 1, 2011 4 / 14
LBA
An equivalent definition of an LBA is that it uses only k times theamount of space occupied by the input string, where k is a constantfixed for the particular machine.
Possible to simulate k tape cells with a single tape cell, by increasingthe size of the tape alphabet
Examples: {anbncn|n ≥ 0}; counting number of a’s
This limitation makes the LBA a somewhat more accurate model ofcomputers that actually exist than a Turing machine, whose definitionassumes unlimited tape.
Indu John (Department of CSA) Linear Bounded Automata December 1, 2011 4 / 14
History
In 1960, Myhill introduced an automaton model today known asdeterministic linear bounded automaton.
Shortly thereafter, Landweber proved that the languages accepted bya deterministic LBA are always context-sensitive.
In 1964, Kuroda introduced the more general model of(nondeterministic) linear bounded automata, and showed that thelanguages accepted by them are precisely the context-sensitivelanguages.
Indu John (Department of CSA) Linear Bounded Automata December 1, 2011 5 / 14
History
In 1960, Myhill introduced an automaton model today known asdeterministic linear bounded automaton.
Shortly thereafter, Landweber proved that the languages accepted bya deterministic LBA are always context-sensitive.
In 1964, Kuroda introduced the more general model of(nondeterministic) linear bounded automata, and showed that thelanguages accepted by them are precisely the context-sensitivelanguages.
Indu John (Department of CSA) Linear Bounded Automata December 1, 2011 5 / 14
History
In 1960, Myhill introduced an automaton model today known asdeterministic linear bounded automaton.
Shortly thereafter, Landweber proved that the languages accepted bya deterministic LBA are always context-sensitive.
In 1964, Kuroda introduced the more general model of(nondeterministic) linear bounded automata, and showed that thelanguages accepted by them are precisely the context-sensitivelanguages.
Indu John (Department of CSA) Linear Bounded Automata December 1, 2011 5 / 14
History
In 1960, Myhill introduced an automaton model today known asdeterministic linear bounded automaton.
Shortly thereafter, Landweber proved that the languages accepted bya deterministic LBA are always context-sensitive.
In 1964, Kuroda introduced the more general model of(nondeterministic) linear bounded automata, and showed that thelanguages accepted by them are precisely the context-sensitivelanguages.
Indu John (Department of CSA) Linear Bounded Automata December 1, 2011 5 / 14
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︷︸︸︷mn ∗
Headposition︷︸︸︷n ∗
State︷︸︸︷q
configurations.
Indu John (Department of CSA) Linear Bounded Automata December 1, 2011 6 / 14
Results about LBA
Halting Problem
The halting problem is solvable for linear bounded automata.
HaltLBA = {< 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.
ALBA = {< M,w > |M is an LBA and M accepts w}is decidable.
Indu John (Department of CSA) Linear Bounded Automata December 1, 2011 7 / 14
Results about LBA
Halting Problem
The halting problem is solvable for linear bounded automata.
HaltLBA = {< 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.
ALBA = {< M,w > |M is an LBA and M accepts w}is decidable.
Indu John (Department of CSA) Linear Bounded Automata December 1, 2011 7 / 14
Results about LBA
Emptiness Problem
The emptiness problem is unsolvable for linear bounded automata.
For every Turing machine there is a linear bounded automaton whichaccepts the set of strings which are valid halting computations for theTuring machine.
Indu John (Department of CSA) Linear Bounded Automata December 1, 2011 8 / 14
LBA Problems
1 Is the class of languages accepted by LBA equal to the class oflanguages 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, 2011 9 / 14
LBA Problems
1 Is the class of languages accepted by LBA equal to the class oflanguages 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, 2011 9 / 14
LBA Problems
1 Is the class of languages accepted by LBA equal to the class oflanguages 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, 2011 9 / 14
LBA Problems
1 Is the class of languages accepted by LBA equal to the class oflanguages 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, 2011 9 / 14
LBA Problems
1 Is the class of languages accepted by LBA equal to the class oflanguages 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, 2011 9 / 14
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, 2011 10 / 14
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, 2011 11 / 14
Proof...
If w = ε, 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 theymatch, accept. If not go to step 1.
Indu John (Department of CSA) Linear Bounded Automata December 1, 2011 12 / 14
Proof...
⇒If there is a linear bounded automaton M accepting the language L, thenthere is a context sensitive grammar generating L− {ε}.
Sketch of proof :
Derivation simulates moves of LBA
Three types of productions1 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 isleft unchanged; the other, representing the input tape to M, is modifiedaccordingly.
3 Productions that can erase everything but the unmodified copy of thestring, provided that the simulated moves of M applied to the othercopy cause M to accept.
Indu John (Department of CSA) Linear Bounded Automata December 1, 2011 13 / 14
Proof...
⇒If there is a linear bounded automaton M accepting the language L, thenthere is a context sensitive grammar generating L− {ε}.
Sketch of proof :
Derivation simulates moves of LBA
Three types of productions1 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 isleft unchanged; the other, representing the input tape to M, is modifiedaccordingly.
3 Productions that can erase everything but the unmodified copy of thestring, provided that the simulated moves of M applied to the othercopy cause M to accept.
Indu John (Department of CSA) Linear Bounded Automata December 1, 2011 13 / 14
Proof...
⇒If there is a linear bounded automaton M accepting the language L, thenthere is a context sensitive grammar generating L− {ε}.
Sketch of proof :
Derivation simulates moves of LBA
Three types of productions1 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 isleft unchanged; the other, representing the input tape to M, is modifiedaccordingly.
3 Productions that can erase everything but the unmodified copy of thestring, provided that the simulated moves of M applied to the othercopy cause M to accept.
Indu John (Department of CSA) Linear Bounded Automata December 1, 2011 13 / 14
References
John Martin, ”Introduction to Languages and the Theory of Computation” , TataMcGraw-Hill, Third Edition.
John Hopcroft, Jeffery Ullman, ”Introduction to Automata Theory, Languages andComputation” .
http://www.cs.uky.edu/~lewis/texts/theory/automata/lb-auto.pdf
http://en.wikipedia.org/wiki/Linear_bounded_automaton
http://www.fit.vutbr.cz/~janousek/vyuka/vsl2/newer/class19.pdf
Indu John (Department of CSA) Linear Bounded Automata December 1, 2011 14 / 14