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Presentation on theme: " && Department of nskinfo-i CS2303-THEORY OF COMPUTATION Chapter: RECENT DEVOLOPMENTS."— Presentation transcript:


2 Mathematics Mechanics, (Fluid Dynamics, etc) Algebra Pure applied AnalysisStatistics Discrete Mathematics Graph Theory Formal Languages & Automata Logic Fuzzy Sets

3 Formal Languages & Automata Cornerstone of Theoretical Computer Science Origin & Motivation from different sources

4 New Applications: 1. Cryptography 2. Graphics and Artificial Intelligence 3. Pattern Recognition 4. DNA computing.

5 Formal Languages An abstraction of general characteristics of programming languages. Automaton A construct which possesses all the indispensable features of a digital computer.

6 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. 6

7 X – Finite Alphabet X* - Set of finite words Example X = {a, b} X* = {λ, a, b, aa, ab, bb,... } Preliminaries 7

8 Let L X* – language a a... a (n times) = a n ; If X = {a, b} then L = {a n b n : n 1} is a language Characterizing L by finite set of rules, called Grammars, is Formal Languages. 8



11 DNA – Deoxyribonucleotide DNA – molecules are made up of AdenineCytocine ThymineGuanine nitrgen bounded bases DNA exists in double stranded linear or circular form. DNA molecules are strings (words) over

12 DNA Sequence A T C G A T G C G C A G T T A G C T A C G C G T C A

13 Protein – String over twenty amino acids RNA – String over four ribonucleotide

14 Different variants of words Word Circular words Infinite words, Bi-infinite words Two dimensional words (arrays) Infinite arrays Chain code pictures

15 Circular words - Motivated by the structure of DNA sequences A circular word c is We denote it by a 1 a 2 … a n. ie… a 1 a 2 … a n is one linearlized form of C.

16 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.

17 Infinite words Infinite word is a function u : N We represent infinite word as u = u 1 u 2 … (a sequence of elements of ) Example: = {a, b} u = abaabb…

18 - 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.

19 Array over is of the form ** - collection of all arrays over If = { o, x}, then Collection of arrays is called picture languages.

20 Kolam is a traditional art practiced extensively in the southern part of India, for decorating courtyards of dwellings.



23 Chain code picture languages

24 Position Vector or generalized Parikh Vector u ; = (a 1, a 2,..., a n ) P(u) = (p 1, p 2,..., p n ) where where A i gives all the positions of the letter a i in the word u. Example: = {a, b} u = abaa u = ababab… = (ab),

25 Results If u and if p(u) = (p 1, p 2,... p n ) then (i) p 1 + p p n = 1 and If u * (ii) p 1 + p p n = 1 – (1 / 2 n ). a b2b2 b a2a2 a b ab ba

26 Definition L - line language if there exists a line l in R 2 such that L = { x : p(x) l }. l – language line of L. Examples 1. L = (a n b n ) (a n b n) line language lies on y = x/(2 n ) 2. L = (a * b) a - line language lies on x + 2y = 1

27 Grammar for English language tells us whether a particular sentence is well formed or not. Typical rule of English grammar A / The boy / dog runs The boy runs 27

28 The boy runs 28

29 S aSb, S ab Replace S by aSb or ab S – variable;a, b – constants S ab S aSb aabb = a 2 b 2 S aSb aaSbb aaabbb = a 3 b 3 L = {a n b n : n 1} 29

30 L-Systems DNA Computing Membrane Computing

31 L-Systems

32 32V R Dare, MCC

33 33V R Dare, MCC

34 Aristid Lindenmayer ( ) 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.

35 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

36 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

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



40 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 Cantors fractal set on a real

41 Fibonnacci sequence If 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



44 Koch Island Start: F+F+F+F p: F -> F+F-F-FF+F+F-F F: draw +: move right -: move left

45 Plant images

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

47 Other visually appealing shapes produced by L systems

48 plant-like objects produced by L systems angle 22.5 START -> F F -> FF-[-F+F+F]+[+F-F-F]

49 DNA Computing


51 DNA – Deoxyribonucleotide DNA – molecules are made up of AdenineCytocine ThymineGuanine nitrgen bounded bases DNA exists in double stranded linear or circular form. DNA molecules are strings (words) over

52 Model the action of restriction enzymes on DNA – sequences Define an algebraic structure on the set of all DNA-Sequences.

53 Splicing Operation on DNA sequences occurs in 2-steps Step – 1: Cutting DNA modules by restriction enzymes. Step – 2: Ligase enzyme binds them together.

54 Restriction enzymes added (i) DpnI,(ii) Bal I Dpn IGA TC Acts on CT AG x1x1 x2x2 Step - 1

55 Bal I T G G C C A acts on A C C G G T

56 Taq I Sc1nI Hha I Restricted Enzymes added are Toq I, Sc1NI, HhaI.

57 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. NNGAATTCNNNNCTTAAGNN NNCTTAA NNCAATTGNN GNN … …

58 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.

59 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.

60 Splicing rule over an alphabet V is u 1 u 2 $ u 3 u 4 where u 1, u 2, u 3, u 4 V *, $, are special symbols not in V. For the splicing rule r = u 1 u 2 $ u 3 u 4, and for x, y, w, z V *, (x, y) {w, z} if x = x 1 u 1 u 2 x 2,y = y 1 u 3 u 4 y 2, z = x 1 u 1 u 4 y 2,w = y 1 u 3 u 2 x 2 for x 1, x 2, y 1, y 2 V *. A Splicing scheme – (V, R) V – finite alphabet R – a set of splicing rule over V. r

61 For a splicing scheme h = (V, R) and L V *, n (L) = { w, w : (w 1, w 2 ) {w, w } for some w 1, w 2 L and r R} A Splicing system or H-system is H = (V, A, R) V – Alphabet A V* - initial language R – set of splicing rules Language accepted by H is L(H) = *(A) r

62 An extended H-system H = (V, T, A, R) V – variables,T – terminal alphabet A 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.

63 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 v u a x b v w c x d z The 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.



66 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

67 67

68 68

69 69 Plasma membrane is only partially permeable – 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 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.

71 71 Membrane 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.

72 72 Example Skin membrane region Elementary membrane Tree associated with the membrane

73 73 Region of a Membrane Multiset of Objects Evolution rules (molecules inside (transformation of the membrane) molecules)

74 74 Each region r contains evolution rules Rule is of the form a (bc, in) or a (bc, out) or a (bc, here)

75 a (bc, in) a (bc, here) a (bc, out) a bc a a

76 76 Re- writing P- system = (V, T, μ, M 1, M 2, R 1, R 2 ) V = {a, b, c, d, d,e, e} ; T = {a, b, c} M 1 = {de} ; M 2 = Φ = R 1 = {(d adb, here), (e ce, in), (d ab, here) (e c, out)} ; P 1 = Φ R 2 = { (d d, out), (e e, here) } ; P 2 = Φ 1 2

77 77 Computations deabe abc ad'be ad ' bce ' ad ' bce adbce aabbce a2b2c2a2b2c2 {a n b n c n : n 1}


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