Transcript CS355 - Theory of Computation

CS355 - Theory of
Computation
Context-Free Languages
Last Time
• Is L = {0ix | i ≥ 0, x {0,1}* and |x| ≤ i}
regular?
• Is L = {0i | i is prime} regular?
Dr. A. Mooney, Dept. of Computer Science, NUI Maynooth
Context-Free Languages
• Context-Free Languages (CFL) are
described using Context-Free Grammars
(CFG).
• A CFG is a simple recursive method of
specifying grammar rules which can
generate strings in a language – these
languages are the CFL’s.
Dr. A. Mooney, Dept. of Computer Science, NUI Maynooth
Context-Free Grammars
• The following is an example of a CFG, call
it G1:
– A → 1A0
–A→B
–B→#
(A and B = variables)
(0, 1 and # = terminals)
• A grammar consists of a collection of
substitution rules (projections).
• A is start variable in this case – usually
occurs on left hand side of topmost rule.
Dr. A. Mooney, Dept. of Computer Science, NUI Maynooth
Context-Free Grammars
• Use grammar to describe a language by
generating each string of language:
– Write down start variable.
– Find a variable and a rule which starts with
that variable. Replace written variable with the
right hand side of this rule.
– Repeat the second step until no variables
remain.
Dr. A. Mooney, Dept. of Computer Science, NUI Maynooth
Context-Free Grammars
• G1 generates the string 1111#0000 using
the following sequences:
• A → 1A0 → 11A00 → 111A000 →
1111A0000 → 1111B0000 → 1111#0000
Dr. A. Mooney, Dept. of Computer Science, NUI Maynooth
Context-Free Grammars
• All strings generated in this manner
constitute the language of the grammar.
• L(G1) = language of grammar G1.
• Can show that L(G1) is {1n#0n | n ≥ 0}.
• Any language that can be generated by
some context-free grammar is called a
context-free language.
Dr. A. Mooney, Dept. of Computer Science, NUI Maynooth
Context-Free Grammars
• Note:
– For convenience if a variable has several
rules they are often abbreviated:
– A → 1A0 and A → B may be represented as:
A → 1A0 | B, where “|” represents “or”.
Dr. A. Mooney, Dept. of Computer Science, NUI Maynooth
Definition of a CFG
• A context-free grammar is a 4-tuple (V, Σ , R, S),
where
–
–
–
–
V are the variables (finite set)
Σ are the terminal states (finite set)
R is the set of rules
S is the start variable, SV
Dr. A. Mooney, Dept. of Computer Science, NUI Maynooth
Context-Free Grammars
• In grammar G1, V = {A, B}, Σ = {0, 1, #},
S = A, and R is the collections of the rules:
– A → 1A0
–A→B
–B→#
Dr. A. Mooney, Dept. of Computer Science, NUI Maynooth
Context-Free Grammars
• Consider G3 = ({S}, {a,b}, R, S). The set of
rules R, is
• S → aSb | SS | ε
• This grammar generates strings such as
ab, abab, aababb and aaabbb.
• Note that L(G3) is the language of all
strings of properly nested parnetheses.
Dr. A. Mooney, Dept. of Computer Science, NUI Maynooth
Context-Free Grammars
• Consider the language of palindromes:
– L = {w {a,b}* | w = wr}
• This is not regular, but the language can be
generated by the following rules:
– S → aSa S → bSb
S→ a S→b
S→ε
• V = {S}, S = S, Σ = {a,b} and R are rules above.
• Check to ensure if produces palindromes.
• What about a CFG for the non-palindromes?
Dr. A. Mooney, Dept. of Computer Science, NUI Maynooth
Designing CFG’s
• Many CFL’s are the union of simpler ones.
• Construct the smaller simpler CFG’s and
then construct them to give the larger CFG
for the CFL.
Dr. A. Mooney, Dept. of Computer Science, NUI Maynooth
Designing CFG’s
• Construct a grammar for the language:
{0n1n | n ≥ 0}  {1n0n | n ≥ 0}.
• Firstly construct the grammar
• S1 → 0 S11 | ε for the language {0n1n | n ≥ 0}, and the grammar
• S2 → 1 S20 | ε for the language {1n0n | n ≥ 0}, and then add the
rule S → S1| S2 to give the grammar:
– S → S1| S2
– S1 → 0 S11 | ε
– S2 → 1 S20 | ε
Dr. A. Mooney, Dept. of Computer Science, NUI Maynooth
Designing CFG’s
• We can construct a CFG for a regular language
by first constructing a DFA for the language.
• A DFA may be converted into an equivalent CFG
as follows:
• Make a variable Ri for each state qi of the DFA.
• Add the rule Ri → aRj to the CFG if δ(qi, a) = qj is a transition
in the DFA.
• Add the rule Ri → ε if qi is an accept state of the DFA.
• Make R0 the start state of the grammar where q0 is the start
state of the machine.
• Verify that it works!!
Dr. A. Mooney, Dept. of Computer Science, NUI Maynooth
Union, concatenation and closure
of CFG’s
• Theorem: If L1 and L2 are CFL’s then L1  L2,
L1L2 and L*1 are also CFL’s.
• That is, the context-free languages are closed
under union, concatenation and Kleene-closure.
• The proofs are constructive.
• Begin with two grammars: G1 = (V1, Σ , R1, S1)
and G2 = (V2, Σ , R2, S2), generating CFL’s L1
and L2 respectively.
Dr. A. Mooney, Dept. of Computer Science, NUI Maynooth
Union of CFG’s
• The new CFG Gx is made as:
–
–
–
–
Σ remains the same
Sx is the new start variable
Vx = V1  V2 {Sx}
Rx = R1 R2 {Sx → S1|S2}
• Explanation: All we have done is augment the
variable set with a new start state and then
allowed the new start state to map to either of
the two grammars. So, we’ll generate strings
from either L1 or L2, i.e. L1 L2
Dr. A. Mooney, Dept. of Computer Science, NUI Maynooth
Concatenation of CFG’s
• The new CFG Gy is made as:
–
–
–
–
Σ remains the same
Sy is the new start variable
Vy = V1
V2
{Sy}
Ry = R1 R2
{Sx → S1S2}
 
 
• Explanation: Again, all we have done is to augment the
variable set with a new start state, and then allowed the
new start state to map to the concatenation of the two
original start symbols. So, we will generate strings that
begin with strings from L1 and end with strings from L2,
i.e. L1L2
Dr. A. Mooney, Dept. of Computer Science, NUI Maynooth
Kleene-Closure of CFG’s
• The new CFG Gz is made as:
–
–
–
–
Σ remains the same
Sz is the new start variable
Vz = V1  {Sz}
Rz = R1  {Sz → S1Sz | ε}
• Explanation: Again we have augmented the
variable set with a new start state, and then
allowed the new start state to map to either
S1Sz or ε. This means we can generate strings
with zero or more strings made from expanding
the variable S1, i.e. L*1
Dr. A. Mooney, Dept. of Computer Science, NUI Maynooth
Pushdown Automata (PDA)
• Pushdown Automata are similar to
nondeterministic finite automata but have
an extra element – stack.
• This stack provided extra memory space.
• Also allows pushdown automata to
recognise some nonregular languages.
Dr. A. Mooney, Dept. of Computer Science, NUI Maynooth
PDA
Finite Automata
Pushdown Automata
state
state
control
control
input
……
stack
input
Dr. A. Mooney, Dept. of Computer Science, NUI Maynooth
PDA
• Why is a stack useful?
– It can hold an unlimited amount of information.
• Remember that a FA was unable to recognise
the language {0n1n | n ≥ 0} because it can’t store
large numbers.
• However, a PDA does not have this problem,
due to the presence of a stack: it can use the
stack to store how many 0’s it has seen.
Dr. A. Mooney, Dept. of Computer Science, NUI Maynooth
PDA
• A PDA can write symbols on the stack and
read them back later.
• Writing a symbol “pushes down” all the
other symbols on the stack.
• Only the top symbol in the stack can ever
be read – once read it is removed.
• Writing a symbol is known as “pushing”
and reading a symbol known as “popping”.
Dr. A. Mooney, Dept. of Computer Science, NUI Maynooth
PDA
• The PDA has no way of checking for an
empty stack.
• Gets around this by placing a special
character, $, on the stack initially.
• Then if it ever sees the $ again it knows
that the stack is effectively empty.
Dr. A. Mooney, Dept. of Computer Science, NUI Maynooth
PDA
• Important:
• Deterministic and non-deterministic PDA’s
are not equivalent in power.
• Non deterministic PDA’s recognise certain
languages which no deterministic
pushdown automata can recognise.
Dr. A. Mooney, Dept. of Computer Science, NUI Maynooth
Formal Definition of PDA’s
• The formal definition of a PDA is similar to
that of a FA, except for the stack.
• The stack contains symbols drawn from
some alphabet.
• The machine may use different alphabets
for its input and the stack
– We need to specify an input alphabet Σ and a
stack alphabet Γ
Dr. A. Mooney, Dept. of Computer Science, NUI Maynooth
Formal Definition of PDA’s
• In order to formally define a PDA we need to
determine the transition function. Recall:
– Σε = Σ  {ε} and Γε = Γ  {ε}
– The domain of the transition function is Q × Σε × Γε
• Therefore, the current state, next input symbol
read and top state of the stack determine the
next move of the PDA.
• Note that either symbol may be ε meaning that
the machine may move without reading a symbol
from the input or the stack.
Dr. A. Mooney, Dept. of Computer Science, NUI Maynooth
Formal Definition of PDA’s
• What is the range of the transition function?
• The machine may enter some new state and
possibly write to the top of the stack.
• The function δ can represent this by returning a
member of Q along with a member of Γε, i.e. a
member of Q × Γε
• A number of legal next moves may be allowed
– The transition function incorporates this
nondeterminism in the usual way – i.e. returning a set
of members of Q × Γε, that is, a member of P(Q × Γε).
Dr. A. Mooney, Dept. of Computer Science, NUI Maynooth
Formal Definition of PDA’s
• A pushdown automata is a 6-tuple (Q,Σ,Γ,δ,q0,F),
where Q, Σ, Γ, and F are all finite sets, and
– Q is the set of states,
– Σ is the input alphabet,
– Γ is the stack alphabet,
– δ: Q×Σε×Γε → P(Q × Γε) is the transition function,
– q0 Q is the start state, and
– F  Q is the set of accept states.
Dr. A. Mooney, Dept. of Computer Science, NUI Maynooth
PDA
• The computation of a PDA, M, is as follows:
– It accepts input w if w can be written as w = w1w2…wm
where each w1 Σε and sequences of states r0,r1,…,rm
Q and strings s0, s1,…, sm  Γ* exist that satisfy the
following:
• r0 = q0 and s0 = ε: M starts out properly, in start state and
empty stack.
• For i = 0, 1,.., m-1, we have (ri+1, b)  δ(ri, wi+1, a), where si =
at and si+1 = bt for some a,b Γε and t  Γ*: M moves properly
according to the state, stack and next input symbols.
• rm F: Accept states occurs when the input end is reached.
Dr. A. Mooney, Dept. of Computer Science, NUI Maynooth
PDA
• In a state transition we write “a, b → c” to signify that when
machine reads input a it may replace symbol b on top of
the stack with a c.
• Any of a, b, c can be ε.
• If a is ε, the machine may make this transition without
reading any input symbol.
• If b is ε the machine performs transition without reading
and popping any stack symbol.
• If c is ε machine does not write any symbol to stack.
• Can we design a PDA to recognise the language: {0n1n | n
≥ 0} ?
Dr. A. Mooney, Dept. of Computer Science, NUI Maynooth
Context Free
• A language is context free if and only if
some pushdown automata recognises it.
• Every regular language is recognised by a
finite automaton and every finite
automaton is automatically a pushdown
automaton that ignores the stack, we can
note that every regular language is also
a context-free language.
Dr. A. Mooney, Dept. of Computer Science, NUI Maynooth
Regular and Context-Free
Languages
Dr. A. Mooney, Dept. of Computer Science, NUI Maynooth
Non-Context Free Languages
• Recall that we looked at the pumping
lemma previously for showing that certain
languages are not regular.
• We will look at a similar lemma for contextfree languages.
Dr. A. Mooney, Dept. of Computer Science, NUI Maynooth
Pumping Lemma
• The pumping lemma states that every contextfree language has a special value called the
pumping length such that all longer strings in the
language can be “pumped”.
• Pumped, this time, means that the string can be
divided into five parts such that the 2nd and 4th
parts of the string may be repeated together any
number of times and the resulting string still be
part of the language.
Dr. A. Mooney, Dept. of Computer Science, NUI Maynooth
Pumping Lemma
• If A is a context-free language, then there is a
number p (the pumping length) where, if s is a
string in A of length at least p, then s may be
divided into 5 parts, s = uvxyz such that:
– for each i ≥ 0, uvixyiz A
– |vy| > 0, and
– |vxy| ≤ p.
Dr. A. Mooney, Dept. of Computer Science, NUI Maynooth
Pumping Lemma
• Let us look at some examples:
– Use the Pumping Lemma to show that the following
languages are not context-free:
•
•
•
•
B = {anbncn | n ≥ 0}
C = {aibjck | 0 ≤ i ≤ j ≤ k}
D = {ww | w  {0,1}* }
E = {0n1n0n1n | n ≥ 0}
Dr. A. Mooney, Dept. of Computer Science, NUI Maynooth