Professor Emeritus, Madras Christian CollegeAdjunct Professor, Chennai Mathematical Institute

Chennai, Tamil Nadu, Indiaranisiro@gmail.com

PICTURE LANGUAGESPICTURE LANGUAGES

Kolam is a traditional art practised extensively

in the southern part of India,

for decorating courtyards of dwellings.

Picture LanguagesPicture 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

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.

CommandsF:Move forward by a step while drawing a linef: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.

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

Example A 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)

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.

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

Derivation of 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.

•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.

DNA ComputingDNA Computing

DNA ComputingDNA Computing

Leonard Adleman ( 1994 ) solved

An Instance of the Directed Hamiltonian Path Problem(HPP)Directed Hamiltonian Path Problem(HPP)

Solely by manipulating DNA ( deoxyribonucleic acid ) strings

For a mathematical problemmathematical problem

The tools of biologytools of biology are used

DNA strings used to encode informationencode information

enzymes employed to simulate computations

Biological Notions: SBiological Notions: Summaryummary DNADNA: Storage medium for genetic information{A, T, C, G} Bases of nucleotides

Dligonucleotide ( Oligo )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 pairsWaston-Crick pairsA and T, and C and G complementary

AnnealingAnnealing ( base pairing ): 2 complementary single stranded sequences, with opposite polarity, join to form double helix

Reverse ProcessReverse Process: Melting

Biological Notions: SBiological Notions: Summary ummary (Continued)(Continued)

SynthesizeSynthesize: a required polynomial length strand

MixingMixing: Pour contents of test tube to form union

AmplifyingAmplifying: ( copying ) by PCR – Polymerase Chain Reaction

SeparatingSeparating: the strands by length using gel electrophoresis

ExtractingExtracting: strands containing a given pattern as a substring using affinity purification

CuttingCutting: DNA Double strands at specific sites by Restriction Enzymes

LigatingLigating: Pasting DNA strands with compatible stiky ends using DNA Ligases.

Early work Early work Adleman’s molecular algorithmAdleman’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.

The Directed Hamiltonian Path ProblemThe 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 The following (non deterministic) algorithm solves the problemproblem

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.

Adleman implemented the algorithm at a molecular levelAdleman implemented the algorithm at a molecular level

Step 1: Vertex i encoded by random 20-mer DNA sequenceEdges ij formedLigation enables formation of DNA moleculesencoding random paths through the graphs

Step 2: Product of Step 1 amplified by PCRUsing 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

Adleman's Algorithm: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 amplify( T,p1,pn )

3. T ( T,n )

4. For i=1 to n do begin

T +( T,pi )

end

5. Output( detect( T ) )

1. 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.

Lipton's molecular algorithm for SATLipton'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 )

Rani Siromoney, Bireswar Das

DNA Algorithm for Breaking a Propositional Logic Based Cryptosystem

Bulletin of the EATCS, Number 79, February 2003, pp.170-176

Professor Emeritus, Madras Christian CollegeAdjunct Professor, Chennai Mathematical Institute

Chennai, Tamil Nadu, India

The Fifth International Conference on Bio-Inspired Computing: Theories and Applications (BIC-TA 2010)

Liverpool Hope University, UKSep 8 - 10, 2010