Study Material 2IT70 Finite Automata and Process...

Study Material

2IT70 Finite Automata and Process Theory

Technische Universiteit Eindhoven

June 8, 2016

Study material

Updated reader: Chapters 3 and 5

Hopcroft, Motwani & Ullman (2001)

Background reading: Chapters 5-7More specific: 5.1, 5.2; 6.1, 6.2, 6.3; 7.1, 7.2, 7.4

Exercises

Proofs

Lm 3.15bc: combining and splitting productions of a CFGLm 3.28 & Th 3.29: eliminating non-generating variablesLm 3.31 & Th 3.32: eliminating non-reachable variablesTh 3.39 & 3.42: produced by CFG vs. yield of a parse treeTh 3.44 & 3.48: from CFG to PDA, from PDA to ePDATh 3.52: intersection of context-free and regularTh 3.54: Pumping Lemma for CFL

2 IT70 (2016) Study Material 2 / 34

Push-Down Automata & Context-Free Languages



June 8, 2016

Formal definition

push-down automaton P = (Q, Σ, ∆, ∅, →, q0, F )

Q states

Σ input alphabet, τ ∉ Σ

∆ stack alphabet, ∅ ∉∆ empty stack symbol

→ ⊆ Q ×Στ ×∆∅ ×∆∗ ×Q transition relation

with Στ = Σ ∪ {τ} and ∆∅ =∆ ∪ {∅}

q0 ∈ Q initial state

F ⊆ Q final states

accepted language

L(P) = {w ∈ Σ∗ ∣ ∃q ∈ F ∃x ∈∆∗∶ (q0,w , ε) ⊢∗P(q, ε, x) }

2 IT70 (2016) Push-Down Automata & Context-Free Languages 4 / 34

Invariant table

q0 q1 q2 q3a[∅/1] b[1/ε] τ[∅/ε]

a[1/11] b[1/ε]

L(P) = { anbn ∣ n ⩾ 1 }


Invariant table

q0 q1 q2 q3a[∅/1] b[1/ε] τ[∅/ε]

a[1/11] b[1/ε]

L(P) = { anbn ∣ n ⩾ 1 }

state q input w stack x

q0 ε ε

q1 an 1n 1 ⩽ n

q2 anbm 1n−m 1 ⩽ m ⩽ n

q3 anbn ε 1 ⩽ n

if (q,w , x) in invariant table then (q0,w , ε) ⊢∗P(q, ε, x)


Language of a CFG

context-free grammar G = (V , T , R , S )

V variables and T terminals

R ⊆ V ×(V ∪T )∗ production rules A→ α

S ∈ V start symbol

productions ⇒G ⊆ (V∪T ) × (V∪T )

γ ⇒G γ′ if γ = β1Aβ2, A→ α rule of G , γ′ = β1αβ2

production sequences

γ0 ⇒G γ1 ⇒G ⋯⇒G γn

language of a variable

LG(A) = {w ∈ T∗ ∣ A ⇒∗

Gw }

language of the grammar

L(G) = LG(S)


Avoiding the inductive proofs

lemma CFGs G1 = (V1, T1, R1, S1 ) and G2 = (V2, T2, R2, S2 )

moreover V1 and V2 disjoint

define CFG G = ({S} ∪V1 ∪ V2, T1 ∪T2, R , S )

if R = {S → S1 ∣ S2} ∪R1 ∪ R2 then L(G) = L(G1) ∪L(G2)

if R = {S → S1S2} ∪R1 ∪ R2 then L(G) = L(G1) ⋅L(G2)

if R = {S → ε ∣ S1S} ∪ R1 then L(G) = L(G1)∗


Avoiding the inductive proofs (cont.)

CFG G with production rules

S → S1 ∣ S2S1 → aB B → ε ∣ bBS2 → bA A→ ε ∣ aA

then L(G) = { abn, bam ∣ n,m ⩾ 0 }

proof use the lemma

LG(A) = { am ∣ m ⩾ 0 } and LG(B) = { b

n ∣ n ⩾ 0 }

LG(S1) = {a} ⋅ { bn ∣ n ⩾ 0 } and LG(S2) = {b} ⋅ { a

m ∣ m ⩾ 0 }

L(G) = { abn ∣ n ⩾ 0 } ∪ { bam ∣ m ⩾ 0 }


Context-free languages

language L is context-free if L = L(G) for CFG G

{ anbn ∣ n ⩾ 0 } and {wwR ∣ w ∈ {0,1}∗ } are context-free

but not regular


Yield of a parse tree

CFG G = (V , T , R , S )

set PTG of all parse trees of G

[X ] single node tree, X ∈ V ∪T

[A→ ε] two node tree, root A, leaf ε for rule A→ ε ∈ R

[A→ PT 1,PT 2, . . . ,PT k]

rule A→ X1⋯Xk ∈ Rparse trees PTi with root Xi

yield function yield ∶ PTG → (V ∪T )∗

yield([X ]) = X yield([A → ε]) = ε

yield([A → PT 1, . . . ,PT k]) = yield(PT 1) ⋅ . . . ⋅ yield(PT k)

parse tree PT is complete if yield(PT ) ∈ T ∗


A parse tree with yield ()(())

S

S S

( S )

ε

( S )

( S )

ε


From CFG to parse trees

theorem CFG G = (V , T , R , S )

A ⇒∗G

w implies w = yield(PT )

for parse tree PT with root A

proof by induction on n :

A ⇒n

Gw implies ∃PT ∈ PTG(A)∶w = yield(PT )

for all A ∈ V and w ∈ T ∗

thus L(G)={w ∈ T ∗ ∣ S ⇒∗G

w }

⊆{ yield(PT ) ∣ PT complete parse tree of G , root S }


From parse tree to CFG

theorem CFG G

for parse tree PT , root A and yield w : Aℓ⇒∗

Gw

proof induction on the height of the parse tree PT

thus { yield(PT ) ∣ PT complete parse tree of G , root S }⊆

{w ∈ T ∗ ∣ S ⇒∗G

w }=L(G)


Useless symbols

CFG G = (V ,T ,R ,S)

symbol X ∈ V ∪T generating if X ⇒∗G

w ∈ T ∗

symbol X ∈ V ∪T reachable if ∃α,β∶S ⇒∗G

αXβ

symbol X ∈ V ∪T is useful if both generating and reachable


Finding of generating variables

CFG G = (V ,T ,R ,S) with L(G) ≠ ∅

symbol X ∈ V ∪T is generating if X ⇒∗G

w ∈ T ∗ andGen(G) = { X ∣ X generating }

lemma put

Gen0 = T

Geni+1 = Geni ∪ { A ∣ A→G α, α ∈ Gen∗i}

Gen = ⋃∞i=0 Geni

then Gen(G) = Gen

theorem CFG G ′ = (V ′,T ,R ′,S) with V ′ = V ∩Gen(G) and

R ′ = { A→ α ∈ R ∣ A ∈ Gen, α ∈ Gen∗ }

then L(G) = L(G ′) and all symbols of G ′ generating


Finding of reachable variables

CFG G = (V ,T ,R ,S) with L(G) ≠ ∅

symbol X ∈ V ∪T is reachable if S ⇒∗G

αXβ andReach(G) = { X ∣ X reachable }

lemma put

Reach0 = {S}

Reachi+1 = Reach i ∪ {X ∣ A→G γ, A ∈ Reach i , γαXβ }

Reach = ⋃∞i=0 Reach i

then Reach(G) = Reach

theorem CFG G ′ = (V ′,T ,R ′,S) with V ′ = V ∩Reach(G) and

R ′ = { A→ α ∈ R ∣ A ∈ Reach }

then L(G) = L(G ′) and all variables of G ′ reachable


From CFG to PDA

theorem CFG G = (V , T , R , S )

a PDA P exists such that L(P) = L(G)

proof P = ({q0,q1,q2}, T , V ∪T , ∅,→,q0,{q2}) with

(1) q0τ[∅/S]ÐÐÐÐ→ q1

(2) q1a[a/ε]ÐÐÐ→ q1 for all a ∈ T (matching step)

(3) q1τ[A/α]ÐÐÐÐ→ q1 for all A → α ∈ R (production step)

(3) q1τ[∅/ε]ÐÐÐÐ→ q2

claim γℓ⇒∗

Gw iff (q1,w , γ) ⊢∗

P(q1, ε, ε)


PDA accepting on empty stack

PDA P = (Q, Σ, ∆, D, →, q0 ) accepting on empty stack

Q finite set of states

Σ input alphabet

∆ data alphabet or stack alphabet

D ∈∆ stack bottom symbol,

→ ⊆ Q ×Στ ×∆ ×∆∗×Q transitions where Στ = Σ ∪ {τ}

q0 ∈ Q initial state.

accepted language N (P) = {w ∈ Σ∗ ∣ ∃q ∈ Q ∶ (q0,w ,D) ⊢∗P(q, ε, ε) }


From ePDA to CFG

ePDA P = (Q,Σ,∆,→,q0,D) gives CFG G = (V , Σ, →, S)

V = {S} ∪ { [qd q′′ ] ∣ q,q′′ ∈ Q, d ∈∆ }

production rules

S → [q0Dq′] for all q′ ∈ Q

[qd q′′ ]→ a[p0d1p1][p1 d2p2]⋯[pk−1dk pk ]

if qa[d/d1⋯dk ]ÐÐÐÐÐÐ→P q′

with p0,p1, . . . ,pk ∈ Q such that p0 = q′ and pk = q

′′

[qd q′′ ]→ [p0d1p1][p1d2p2]⋯[pk−1dk pk ]

if qτ[d/d1⋯dk]ÐÐÐÐÐÐ→P q′

with q′′,p1, . . . ,pk ∈ Q such that p0 = q′ and pk = q

′′

[qd q′′ ]: state q reaches state q′′ when processing stack symbol d


An example construction

q0 q1a[D/1D]

a[1/11] b[1/ε]

τ[D/ε]


An example construction

q0 q1a[D/1D]

a[1/11] b[1/ε]

τ[D/ε]

S → [q0Dq0] ∣ [q0Dq1]

[q0Dq0] → a[q11q0][q0Dq0] ∣ a[q11q1][q1Dq0]

[q0Dq1] → a[q11q0][q0Dq1] ∣ a[q11q1][q1Dq1]

[q11q0] → a[q11q0][q0 1q0] ∣ a[q11q1][q1 1q0]

[q11q1] → a[q11q0][q0 1q1] ∣ a[q11q1][q1 1q1]

[q11q1] → b

[q1Dq1] → ε


Closure properties of context-free languages

language L is context-free if L = L(G) for CFG G

theorem context-free languages L1, L2 and L

(a) the union L1 ∪ L2 is also context-free

(b) the concatenation L1 ⋅ L2 is also context-free

(c) the Kleene closure L∗ is also context-free


Intersection of context-free and regular

theorem context-free language L1, regular language L2then L1 ∩ L2 is context-free

proof PDA P = (Q1, Σ, ∆, ∅, →P , q10 , F ) for L1

DFA D = (Q2, Σ, δ, q20 , F2) for L2

define PDA P ′ = (Q1 ×Q2, Σ, ∆, ∅, →P′ , (q10 ,q

20), F1 × F2) by

(q1,q2)a[d/x]ÐÐÐÐ→P′ (q

′

1,q′

2) iff q1a[d/x]ÐÐÐÐ→P q′1 and δ(q2,a) = q

′

2

(q1,q2)τ[d/x]ÐÐÐÐ→P′ (q

′

1,q2) iff q1τ[d/x]ÐÐÐÐ→P q′1

claim ((q1,q2),w , z) ⊢∗P′((q̄1, q̄2), ε, z

′) iff

(q1,w , z) ⊢∗P(q̄1, ε, z

′) and (q2,w) ⊢∗

D(q̄2, ε)


Pumping lemma for context-free languages

theorem if L ⊆ Σ∗ is a context-free language then

∃m > 0 ∶

∀w ∈ L, ∣w ∣ ⩾ m ∶

∃u, v , x , y , z ∶ w = uv x y z ∧ ∣v x y ∣ ⩽ m ∧ ∣v y ∣ > 0 ∶

∀i ⩾ 0 ∶ uv i x y iz ∈ L


Some languages that are not context-free

Pumping Lemma: if L ⊆ Σ∗ is such that

∀m > 0 ∶

∃w ∈ L, ∣w ∣ ⩾ m ∶

∀u, v , x , y , z ∶ w = uv x y z ∧ ∣v x y ∣ ⩽ m ∧ ∣v y ∣ > 0 ∶

∃i ⩾ 0 ∶ uv i x y iz ∉ L

then L is NOT a context-free language

the language L1 = { anbncn ∣ n ⩾ 0 } is not context-free

for any m > 0, consider ambmcm

the language L2 = {ww ∣ w ∈ {a,b}∗ } is not context-free

for any m > 0, consider ambmambm


Labeled Transition Systems andBisimulation



June 8, 2016

Labeled transition system

labeled transition system S = (Q, Σ, →S , q0 )

finite/infinite set of states Q

finite/infinite set of actions Σ

transition relation →S ⊆ Q ×Στ ×Q

initial state q0

transitions qαÐ→S q′ for action α ∈ Στ

2 IT70 (2016) Labeled Transition Systems and Bisimulation 26 / 34

Example LTS

a buffer of capacity 2

ε0 1

00 11

10 01

in1 in0

out1 out0

in0

in1

in0

out0

in1out1

out0

out1


An infinite LTS

a counter process

q0 q1 q2 q3 q4

p1 p2 p3 p4

up up up

down down down

downdown

up

down


Branching bisimulation

LTS S = (Q, Σ, →S , q0 )

branching bisimulation relation R ⊆ Q ×Q for S

(i) if R(q,p) and qaÐ→S q′ then exist p̄,p′ ∈ Q such that

pτÐ→∗

S p̄ and p̄aÐ→S p′ with R(q, p̄) and R(q′,p′)

(ii) symmetric condition if R(q,p) and paÐ→S p′


Branching bisimulation

LTS S = (Q, Σ, →S , q0 )

branching bisimulation relation R ⊆ Q ×Q for S

(i) if R(q,p) and qaÐ→S q′ then exist p̄,p′ ∈ Q such that

pτÐ→∗

S p̄ and p̄aÐ→S p′ with R(q, p̄) and R(q′,p′)

(ii) symmetric condition if R(q,p) and paÐ→S p′

(iii) if R(q,p) and qτÐ→S q′ then either exist p̄,p′ ∈ Q such that

pτÐ→∗

S p̄ and p̄τÐ→S p′ with R(q, p̄) and R(q′,p′) or

exists p′ ∈ Q such that pτÐ→∗

S p′ with R(q,p′) and R(q′,p′)

(iv) symmetric condition if R(q,p) and pτÐ→S p′

states q,p ∈ Q branching bisimilar if R(q,p)


Branching bisimulation (cont.)

s1

t1

s2

s ′2

t2

a τ

a

s1

t1

s2

s ′2

t2

τ τ

τ

s1

t1

s2

τ

left-to-right transfer condition for visible actions

left-to-right transfer condition for silent steps


Coloring for branching bisimulation

q0

q4

q1

q5

q2

q6

q3

τa

τ

b

a

b b

τ

0

0

0

0

0

0

0



q0

q1 q2 q3

q4 q5 q6

τa

τ

b

a

b b

τ

1

1 2 2

3 3 3



q0

q1 q2 q3

q4 q5 q6

τa

τ

b

a

b b

τ

4

5 6 6

3 3 3



q0

q1 q2 q3

q4 q5 q6

τa

τ

b

a

b b

τ

7

8 6 6

3 3 3


Composing LTS: an abstract example

a

b

S1

c dS2

processes S1 and S2



a

b

S1

c dS2

a

b

a

b

a

b

c d

c d

c d

S1 ∥∅ S2

process S1 ∥∅ S2



a

b

S1

c dS2

a

b

a

b

a

b

c d

c d

c d

e

f

S1 ∥γ S2

process S1 ∥γ S2 where γ(a, c) = e and γ(b,d) = f



a

b

S1

c dS2

e

f

∂H(S1 ∥γ S2)

process ∂H(S1 ∥γ S2) where γ(a, c) = e, γ(b,d) = f ,H = {a,b, c ,d}



a

b

S1

c dS2

τ

f

τI (∂H(S1 ∥γ S2))

process τI (∂H(S1 ∥γ S2)) where γ(a, c) = e, γ(b,d) = f ,H = {a,b, c ,d} and I = {e}


Study Material



June 8, 2016

Study material

Updated reader: Chapters 3 and 5

Hopcroft, Motwani & Ullman (2001)

Background reading: Chapters 5-7More specific: 5.1, 5.2; 6.1, 6.2, 6.3; 7.1, 7.2, 7.4

Exercises

Proofs

Lm 3.15bc: combining and splitting productions of a CFGLm 3.28 & Th 3.29: eliminating non-generating variablesLm 3.31 & Th 3.32: eliminating non-reachable variablesTh 3.39 & 3.42: produced by CFG vs. yield of a parse treeTh 3.44 & 3.48: from CFG to PDA, from PDA to ePDATh 3.52: intersection of context-free and regularTh 3.54: Pumping Lemma for CFL

2 IT70 (2016) Study Material 34 / 34

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