Professor Emeritus, Madras Christian College Adjunct Professor, Chennai Mathematical Institute

1. MCC Alumni Get-Together YMCA, London Sep 19, 2010 Rani Siromoney Professor Emeritus, Madras Christian College Adjunct Professor, Chennai Mathematical Institute Chennai, Tamil Nadu, India ranisiro@gmail.com

2. PICTURE LANGUAGES Kolam is a traditional art practised extensively in the southern part of India, for decorating courtyards of dwellings.

4. Picture Languages Kolam figures grouped into families attracted interest of theoretical computer scientists concerned with analysis and description of pictures through the use of picture languages, which use sets of basic units and specific, formal rules for combining the units

5. For Gift Siromoney, Rani Siromoney, Kamala Krithivasan and K.G. Subramanian, Kolam designs became a rich source of figures that served as a stimulus for the creation of new types of picture languages. Other computer scientists in addition to the Madras group have used picture languages to describe Kolam families.

6. Commands F:Move forward by a step while drawing a line f:Move forward by a step without drawing a line+:Turn left (counter clockwise) by an angle of d degrees. -:Turn right (clockwise) by an angle of d degrees.

7. Studies on the traditional Art of Kolam • Working Paper I, May 1985, Gift Siromoney • * Studies to examine methods used by rural folk to memorize complicated patterns • * Concerned here, mainly with Kambi Kolam  (Literally, wire decoration). • * Each kambi ( thread) begins and ends at the same point. i.e. each kambi is an unending line • According to one Kolam Practioner (KP) a “proper” • kambi kolam should consist of a single kambi (closed pattern) • * If a kolam contained more than one kambi, then the greater the number of kambis,easier to memorise the Kolam* To memorize a kolam, the number of closed patterns or kambis identified, and executed one after another

8. ExampleA subject was shown the kolam(figure e)First she plotted the pullis (dots) as a 5 x 5 matrix.(fig a) Next she drew a closed sub pattern.(fig b)She repeated it 3 times using rotational symmetry of the kolam. (fig d)Finally she drew the border design (global pattern)

9. Kolam Moves Gift Siromoney conducted an experiment to find out how simple village women (very often not literate) learn, store complicated patterns in their memory and retrieve them with ease while drawing the kolam. He found that kolam practitioners remember, describe and draw the designs in terms of "moves" such as 'going forward’, 'taking a right turn’, 'taking a U-turn to the right' … , reminiscent of “interpretations" used in computer graphics as sequences of commands which control a "turtle" Treating each kind of a move as a terminal sign, each single kambi kolam represents a picture cycle. Thus kambi kolam designs provide us with illustrative examples of picture cycle languages.

10. To avoid producing angular versions of the kolam figures Gift Siromoney introduced kolam moves to draw smooth curves and loops instead of using linear turtle moves, he defined seven kolam moves based on the women’s descriptions of their actions

13. Derivationof multi-kambi kolam from single kambi kolam According to one KP, a ‘proper’ kambi kolam should consist of a single kambi If a kolam did contain more than one kambi ,then the greater the number of kambis the easier it is to memorize the kolam. -*A single kambi kolam can be converted into a multi-kambi-kolam by applying a cut at a crossing.*A cut and join (delink) operation fuses ends together,two at a time, after cut at a cross which produces four ends.*A cut and join operation at a crossing when used on a single strand can at most increase the number of kambis by one.*In figure, four cuts are introduced and single kambi kolam becomes a five kambi kolamwhich is more easily memorized than the single kambi kolam.

15. A cut and connect operation can link two adjacent corners. • A cut is introduced such that it goes through two adjacent • rounded corners producing four ends. • *These ends are connected either forming a crossing • alternately two new adjacent rounded corners. • -Two kambis when used in a cut and connect operation will • fuse into one same kambi. • - If two adjacent corners belong to the same kambi then a • cut and connect operation can produce two kambis • or just a kambi with an additional crossing.

16. DNA Computing

17. DNA Computing

18. Leonard Adleman ( 1994 ) solved • An Instance of the Directed Hamiltonian Path Problem(HPP) • Solely by manipulating DNA ( deoxyribonucleic acid ) strings • For a mathematical problem • The tools of biology are used • DNA strings used to encode information • enzymesemployed to simulate computations

19. Biological Notions: Summary DNA: Storage medium for genetic information {A, T, C, G} Bases of nucleotides Dligonucleotide ( Oligo ) -Short single-stranded poly-neucleotide chain, usually less than 30 bp long DNA sequences have polarity Two distinct ends 5’ and 3’ Waston-Crick pairs A and T, and C and G complementary Annealing( base pairing ): 2 complementary single stranded sequences, with opposite polarity, join to form double helix Reverse Process: Melting

20. Biological Notions: Summary (Continued) Synthesize: a required polynomial length strand Mixing: Pour contents of test tube to form union Amplifying: ( copying ) by PCR – Polymerase Chain Reaction Separating: the strands by length using gel electrophoresis Extracting: strands containing a given pattern as a substring using affinity purification Cutting: DNA Double strands at specific sites by Restriction Enzymes Ligating: Pasting DNA strands with compatible stiky ends using DNA Ligases.

21. Early work • Adleman’s molecular algorithm • Tools of molecular biology used • To solve an instance of the directed Hamiltonian Path Problem( HPP ) • known to be NP-hard • Graph encoded in molecules of DNA • “operations” of the computation performed with standard protocols and enzymes • Demonstrated the feasibility at the molecular level, solutions to hard problems.

22. The Directed Hamiltonian Path Problem • A directed graph G with designated vertices vin and vout is said to have a Hamiltonian path if and only if there exists a sequence of compatible “one-way” edges e1,e2,…,en that begins at vin ends at vout, and enters every other vertex exactly once. • The following (non deterministic) algorithm solves the problem • Step 1: Generate random paths through graph • Step 2: Keep those that begin with Vin and end with Vout • Step 3: If graph has n vertices, keep paths that • enter exactly n vertices • Step 4: Keep those that enter all vertices at least once • Step 5: If any paths remain, say YES; otherwise NO.

23. Adleman implemented the algorithm at a molecular level Step 1: Vertex i encoded by random 20-mer DNA sequence Edges ij formed Ligation enables formation of DNA molecules encoding random paths through the graphs Step 2: Product of Step 1 amplified by PCR Using primers Ovin and Ovout i.e., only paths beginning with vin and vout are amplified Step 3: Product of Step 2 run on agarose gel Only double standard DNA encoding paths entering exactly seven vertices remain Step 4: Affinity purified Step 5: Amplified by PCR and run on a gel

24. Adleman's Algorithm: Initial Test tube ( multi set ) strings encoding possible paths in Graph G = ( V,E ) where V = {v0,…..vn –1} 1. Input( T ) 2. T famplify( T,p1,pn ) 3. T fj( T,n ) 4. For i=1 to n do begin T f +( T,pi ) end 5. Output( detect( T ) )

25. generates random paths through graph • 2. copies only those strings encoding paths • that begin with vin and end with vout • 3. discards strings encoding paths length  n • 4. Each vertex appears in remaining strings • 5. Detects whether or not a string • encoding a Hamiltonian path is found.

26. Lipton's molecular algorithm for SAT • Lipton extended Adleman's method • To solve the satisfiability problem ( SAT ) • for Boolean formulae, known to be NP-complete • To find values for variables that make • Boolean formula in ( CNF ) true • T a test-tube • ( a multiset of strings from {A, T,C, G} ) . • The set of DNA corresponds to the simple graph Gn Fig 1. The graph Gn which encodes n-bit numbers ( binary )

27. Rani Siromoney, Bireswar Das DNA Algorithm for Breaking a Propositional Logic Based Cryptosystem Bulletin of the EATCS, Number 79, February 2003, pp.170-176

28. Contextual-Insertion to Solve #P Version of the Square-Tiling Problem The Fifth International Conference on Bio-Inspired Computing: Theories and Applications (BIC-TA 2010) Liverpool Hope University, UK Sep 8 - 10, 2010 Rani Siromoney Professor Emeritus, Madras Christian College Adjunct Professor, Chennai Mathematical Institute Chennai, Tamil Nadu, India

29. T Y H A N K O U