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CS2303-THEORY OF COMPUTATION

•Chapter: RECENT DEVOLOPMENTS IN LANGUAGE THEORY

1

Mathematics

Mechanics,(Fluid Dynamics, etc)Algebra

Pure applied

Analysis Statistics

Discrete Mathematics

Graph TheoryFormal Languages

& AutomataLogic Fuzzy Sets

Formal Languages & Automata

Cornerstone of Theoretical Computer Science

Origin & Motivation from different sources

New Applications:

1. Cryptography

2. Graphics and Artificial Intelligence

3. Pattern Recognition

4. DNA computing.

Formal Languages

An abstraction of general characteristics of programming languages.

Automaton

A construct which possesses all the indispensablefeatures of a digital computer.

Language

• A system suitable for the expression of certain ideas, facts or concepts, including a set of symbols and rules for their manipulation.

• To study language mathematically we need a mechanism to describe them.

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X – Finite Alphabet

X* - Set of finite words

ExampleX = {a, b}

X* = {λ, a, b, aa, ab, bb, . . . }

Preliminaries

7

Let L X* – language

a a . . . a (n times) = an;

If X = {a, b} thenL = {anbn : n ≥ 1} is a language

Characterizing L by finite set of rules, called Grammars, is Formal Languages.

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• DNA – Deoxyribonucleotide• DNA – molecules are made up of

Adenine CytocineThymine Guanine

nitrgen bounded bases• DNA exists in double stranded linear or circular

form.• DNA molecules are strings (words) over

A T C GX

T A G C

DNA Sequence

A T C G A T G C G C A G TT A G C T A C G C G T C A

• Protein – String over twenty amino acids

• RNA – String over four ribonucleotide

Different variants of words

• Word

• Circular words

• Infinite words, Bi-infinite words

• Two dimensional words (arrays)

• Infinite arrays

• Chain code pictures

Circular words

- Motivated by the structure of DNA sequences

A circular word c is

We denote it by a1a2 … an.

ie… a1a2 … an is one linearlized form of C.

3

2

1

. .

. .

.

n

a

a a

a

Example:

= aabbaa = bbaaaa = baaaab = …..

In other words circular word c is an equivalence class in X* under the relation defined as x y if y is a cyclic permutation of x.

Collection of all circular words is denoted as X .

aa

ba

ba

Infinite words

Infinite word is a function u : N

We represent infinite word as u = u1u2 …

(a sequence of elements of )

Example: = {a, b}u = abaabb…

- Set of all infinite words over - Set of all finite and infinite words over

L - -language

Bi-infinte words are functions from integers Z to .

Array over is of the form

** - collection of all arrays over

If = { o, x}, then

Collection of arrays is called picture languages.

11 12 1

21 22 2

1 2

n

n

m m mn

a a a

a a a

a a a

**

x x x

o x o

o x o

x x x

Kolam is a traditional art practiced extensively

in the southern part of India,

for decorating courtyards of dwellings.

Chain code picture languages

Position Vector or generalized Parikh Vector

u ; = (a1, a2, . . . , an)

P(u) = (p1, p2, . . . , pn) where

where Ai gives all the positions of the letter ai in the word u.

Example: = {a, b}u = abaa

u = ababab… = (ab) ,

1

2i

i ij A

p

3 4 2

1 1 1 1( ) ,

2 2 2 2p u

3 2 4

1 1 1 1( ) ,

2 2 2 2

2 1,3 3

p u

Results

• If u and if p(u) = (p1, p2, . . . pn) then

(i) p1 + p2 + . . . + pn = 1 and

If u * (ii) p1 + p2 + . . . + pn = 1 – (1 / 2n )

.

a

b2

b

a2 a

b

abba

DefinitionL - line language if there exists a line l in R2 such that L = { x : p(x) l }.l – language line of L.

Examples1. L = (anbn) (anbn) line language lies on

y = x/(2n)2. L = (a* b) a - line language lies on x + 2y = 1

Grammar for English language tells us whether a particular sentence is well formed or not.

Typical rule of English grammar<sentence> <noun phrase> <verb pharse> < noun phrase> <article > <noun>< verb phrase > < verb > < article > A / The < noun > boy / dog <verb> runs

The boy runs27

<sentences>

< noun phrase > < verb phrase >

< article > < noun > <verb>

The boy runs

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S aSb, S ab

Replace S by aSb or abS – variable; a, b – constants

S abS aSb aabb = a2b2

S aSb aaSbb aaabbb = a3b3

L = {anbn : n ≥ 1}

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L-Systems

DNA Computing

Membrane Computing

L-Systems

32V R Dare, MCC

33V R Dare, MCC

Aristid Lindenmayer (1925-1989) was a biologist who studied the growth patterns of various types of Algae.

He devised L-systems in 1968 to provide a formal description of such simple multi-cellular organisms.

Later, this system was extended to describe higher plants and complex structures.

• He found some very many interesting properties in the geometry

of a tree, the blood vessels, a cauliflower, the roots of a plant, or

a fern.

• One interesting property is the property of self similarity.

• He also found that several mathematicians like Cantor, Sierpinski,

Koch, Hilbert etc. had already studied this property, and have

defined several interesting mathematical objects. We will discuss

about a few

L-systems are used in:

• generating realistic plant like images like trees, bushes and flowers.

• for producing generative art

• for producing music using algorithmic composition techniques

• to produce fractal images

• to produce antennas and integrated circuits to hold enormous amount of memory

• to build robust buildings with minimum weight and maximum strength

36V R Dare, MCC

• The distinguishing feature of an L System over the formal language is that in an L- system the production rules are applied in parallel

Definition: A DOL system is a triple

G=(S, h,w), where S is an alphabet, h is an

endomorphism (homomorphism to itself)

defined on S* and w, called an axiom, is an

element of S* The word sequence E(G)

generated by G consists of the words h0(w),

h1(w), h2(w)… and the language of G is

L(G) = {hi(w)/ i >=0}

Examples:

G=({a,b},h, ab) where h(a)=a, h(b)=ab

Then, E(G) = ab, aab, aaab, ...,

and the language of G is

L(G) = {an b / n >0}

Cantor Dust

variables : { A, B } start : A {starting character string} rules : (A → ABA), (B → BBB) Let A mean "draw forward" and B mean "move forward".This produces the famous Cantor’s fractal set on a real

Fibonnacci sequenceIf we define the following simple grammar:

variables : { A, B }

rules : (A → B), (B → AB) then this L-system produces the following sequence of strings:

n = 0 : A n = 1 : B n = 2 : AB n = 3 : BAB n = 4 : ABBAB n = 5 : BABABBAB n = 6 : ABBABBABABBAB n = 7 : BABABBABABBABBABABBAB

Sierpinski Trianglevariables : { A, B }

start : A

rules : (A → B−A−B), (B → A+B+A)

angle : 60 �Here, A and B both mean "draw forward", +

means "turn left by angle", and − means "turn

right by angle"

Sierpinski Triangle

Koch Island

Start: F+F+F+F p: F -> F+F-F-FF+F+F-F

F: draw +: move right -: move left

Plant images

Start: F rule: F -> F[-F]F[+F][F]

Other visually appealing shapes produced by L systems

plant-like objects produced by L systems

angle 22.5START -> FF -> FF-[-F+F+F]+[+F-F-F]

DNA Computing

• DNA – Deoxyribonucleotide• DNA – molecules are made up of

Adenine CytocineThymine Guanine

nitrgen bounded bases• DNA exists in double stranded linear or circular

form.• DNA molecules are strings (words) over

A T C GX

T A G C

• Model the action of restriction enzymes on DNA – sequences

• Define an algebraic structure on the set of all DNA-Sequences.

Splicing Operation on DNA sequences occurs in 2-steps

Step – 1: Cutting DNA modules by restriction enzymes.

Step – 2: Ligase enzyme binds them together.

Restriction enzymes added (i) DpnI, (ii) Bal IDpn I GA TCActs on CT AG

A A A G A T C A A A Ax

T T T C T A G T T T T

x1 x2

A A A A A A AG A T C

C T A GxT T T T T T T

Step - 1

Bal I T G G C C Aacts on A C C G G T

C C C C C C C

yG G G G

T G G C C A

A C C G G T G G G

C C C C T G G C C A C C Cy

G G G G A C C G G T G G G

C C C C C

xG G

T C G A

AG GG C T G

A A A A AyT T

C G C

C

G

T TG C G T

T T T T TZ

A A

G C G C

CA AG C G A

Taq I

Sc1nI

Hha I

Restricted Enzymes added are Toq I, Sc1NI, HhaI.

Consider the DNA-sequences

= … …

where N is any arbitrary deoxyribonucleotide.Apply the enzyme EcoRI on

• EcoRI operates at six term sequence• The cut takes place at x.• The cut is a staggered one.

N N G A A T T C NN

N N C T T A A G NN

N N C T T A A

N N C A A T T G NN

G NN…

…

Step 2:• The staggered strands of DNA reassociate if the ends are in

the neighbourhood of each other by a chemical called “ligase”.

• Consider two DNA sequences, and .

Apply restricted enzyme and it cuts and .

After reassociation original modules are formed.

Two new hybrid molecules are formed by the left half of the first of the original molecule reassociated with the right half of the second and the right half of the first with the left half of the second.

Questions to be analyzed

Suppose we are given a finite set M of DNA sequences and a finite set of

restriction enzymes and ligase enzymes.

What is the nature of the language consisting of all DNA modules that can

arise through the action of restriction enzymes and ligase enzymes.

Splicing rule over an alphabet V isu1 u2 $ u3 u4

where u1, u2, u3, u4 V*, $ , are special symbols not in V.

For the splicing rule r = u1 u2 $ u3 u4, and for x, y, w, z V* ,

(x, y) {w, z} if x = x1u1u2x2, y = y1u3u4y2,

z = x1u1u4y2, w = y1u3u2x2

for x1, x2, y1, y2 V*.

A Splicing scheme – (V, R) V – finite alphabetR – a set of splicing rule over V.

r

For a splicing scheme h = (V, R) and L V*,n (L) = { w, w : (w1, w2) {w, w } for

some w1, w2 L and r R}

A Splicing system or H-system is H = (V, A, R)V – AlphabetA V* - initial languageR – set of splicing rules

Language accepted by H is L(H) = *(A)

r

1( ) ( ) ( )) : 0i i ih h h hL L L i

*

0

( ) ( )ih h

i

L L

An extended H-system H = (V, T, A, R)V – variables, T – terminal alphabetA V* - initial language R – set of splicing rules

L(H) = *(A) T

H – system is finite if A, R are finite

Results Every finite H system generates only regular languages.

The inclusion is strict. A language is regular iff it is generated by a extended

finite H- system.

Generation of words

uaxbv, wcxdz I(a, x, b), (c, x, d) – patterns

The new words formed are u a x d z and w c x b vu a x b v

w c x d zThe language generated by H is the minimal subset of A* which contains I and closed under the operation of splicing.L(H) – languages generated by the splicing system H.

Recognizable languages are also morphic images of languages generated by splicing system H.

MEMBRANE COMPUTING

66

Cell has a complex structure with several compartments called Organelles.

Eg: Nucleus, Golgi apparatus, Ribosomes ,...

Cell and these compartments are covered by cell membrane called Plasma Membrane.

Membrane is a phospholipid bilayer (2 layers of phospholipid molecules) in which protein,

cholestrol, steriods and other molecules are embedded

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68

69

Plasma membrane is only partially Plasma membrane is only partially permeablepermeable

– some non changed molecules cross the membrane almost freely

– Larger molecules can cross a membrane if they are assisted

– Charged ions pass selectively from a region to

another one

70

Basic functions of a biological membrane is to define compartments and to relate

compartments to their environment.

• Ensures some molecules to stay in the Cell.• Blocks some substance like toxic molecules to

stay out of Cell.• Allows some molecule (waste products) to leave

the Cell.• Allows nutrients to enter the Cell.

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Membrane StructureMembrane Structure

Hierarachical arrangement of membranes - all of them placed in a main membrane called Skin Membrane.

A Membrane without Membrane inside is

called Elementary Membrane.

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ExampleExampleSkin membrane

region

Elementary membrane

1

3

25

7

4

6

8 9

1

2 3 4

5 7

6

98Tree associated with Tree associated with

the membranethe membrane

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Region of a Membrane

Multiset of Objects Evolution rules (molecules inside (transformation of

the membrane) molecules)

74

Each region r contains evolution rulesEach region r contains evolution rules

Rule is of the form

a (bc, in)

or

a (bc, out)

or

a (bc, here)

a (bc, in)

a (bc, here)

a (bc, out)

a

bc

a

a

bc

bc

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Re- writing P- systemRe- writing P- system

= (V, T, μ, M1, M2, R1, R2)

• V = {a, b, c, d, d’ ,e, e’} ; T = {a, b, c}

M1 = {de} ; M2 = Φ =

• R1 = {(d ad’b, here), (e ce’, in), (d ab, here) (e c, out)} ; P1 = Φ

• R2 = { (d’ d, out), (e’ e, here) } ; P2 = Φ

12

77

ComputationsComputations

de abeabc

ad'be

ad'bce' ad'bce

adbceaabbce

a2b2c2

{a{annbbnnccnn : n : n1}1}