1. Implementation of Haskell Modules for Automata and Sticker Systems
　　　　　　　　Kissani Perera＊, Yoshihiro Mizoguchi＊＊
　　　　　　　　＊Graduate School of Mathematics
　　　　　　　　＊＊Faculty of Mathematics
　　　　　　　　　　　　Kyushu university.

2. Outline
Motivations
Objectives
Basic notations
DNA sequences and Dominos
Sticker system
Sticker vs Automata
Grammar Module
Sticker vs Grammar
Advantages of Haskell
Related works
Conclusion and future works

3. 1.Motivations
DNA computing uses DNA instead of traditional computer technologies.
Sticker system is a computing system based on the sticker operations, which in turn, is a model of the techniques used by Adleman[1994] in his DNA experiment.
Paun & Rozenberg[1998] introduced a one concrete method to transform the automata into sticker system.
We have improved an insufficient parts in their results and introduce more simple efficient transformation using Haskell module functions.
Our Haskell module functions can be used as self study materials to learn automata, grammar and language theories.

4. 2.Objectives
Realize the operations used in Sticker system using Haskell module functions.
Discuss the modifications done in our module which gave insufficient results in Paun & Rozenberg[1998].
Discuss the advantages of Haskell module functions.

5. 3.Basic notations: Finite Automata :alphabet, natural number
Q: set of states
: alphabet
q0: initial state
FM: set of final states
: transition function

6. 4.DNA sequences and Dominos
DNA sequences are double stranded sequences composed of four nucleotides A(adenine), C(cytosine),G(guanine), T(thymine). Paired A-T, C-G are called Watson-Crick complementarity.
5’ –AAACTGGAG -3’ + 3’ –TTTGACCTC -5’ “glued” 5’ –AAACTGGAG -3’ ’ –TTTGACCTC -5’ .
Double stranded DNA　sequence

7. y x
y x
x y
x y
ｙ x z
y x z
y x z
y x z
We used incomplete double stranded sequences and anneal them to form the double stranded sequences with the same length.
x y
y x

8. We modify the definition of domino (D)( defined in Paun & Rozenberg) from string of pairs of alphabet to a triple of two strings l, r and integer x.
Here is the definition for our domino.
Domino
be symmetric relation. An element
is a domino over If
Let
be a set of alphabet and Z be a set of integers and

9. Represent domino in our system.
ATGC
C TA -1
ATGC TA
CGT
ATGC

10. Sticker operation is defined as
5.Sticker system
Sticker system is a four tuple
Starting from A and using pairs (r1,r2) in R we can obtain a set of double stranded sequences using sticker operations.

11. Example 1: Sticker operation
Let
（（ATG,TA,0）,（GA,CCT,-1))=(ATGGA,TACCT,0)
ATG TA GA
CCT
ATGGA
TACCT ＋

12. 6.Sticker system vs Automaton
For a finite automaton sticker system
defined in our module as follows.
(*)

13. Language generated by Automata M
For a sticker system , we define a relation as follows
Then

14. Theorem (Paun & Rozenberg[1998])
Example 2：
Consider the following stickers with length 5 generated by 　　　　which cannot be generated by Paun & Rozenberg` definitions.　　　　
["aaaab","abbab","babab","bbaab","aabaa","abaaa","baaaa","bbbaa","aaaba","abbba","babba","bbaba","aabbb","ababb",
"baabb","bbbbb]

15. Corrections for Paun & Rozenberg’s definitionsab bb a
aaa
bba
aba
baa
F =
Our definition b ab
bab a
aaa
bba
aba
baa aa
aaaa
bbaa bb ba
aab
abbb
bbbb
aabb
baaa
abaa
babb
F =

16. Example 3 : Stickers “abbab” and “bbbbb” in example2 can be generated by our module.
aab a b ab ba
baa
aba
bbb
aaa aa
abb
bab
bba bb
abbab
abb ab +
bab A F
bbbbb
bbb b bb
bbbb + A F

17. 7.Grammar module
Grammar is a four tuple G=(T,N,R,S) where
T: termnial symbols
N: non terminal symbols
R: transformation rules
S: start symbol
Language generated by G

18. 8.Sticker system vs Grammar module
For a linear grammar sticker system
is defined in our module as follows:

20. Theorem (Paun &Rozenberg[1998]
Example 4: Consider the grammar
Stickers generated by G
["","ab","aabb","aaabbb","aaaabbbb","aaaaabbbbb","aaaaaabbbbbb","aaaaaaabbbbbbb",
"aaaaaaaabbbbbbbb","aaaaaaaaabbbbbbbbb"]

21. Consider the following sticker generated by
which cannot be generated by the definitions in Paun & Rozenberg[1998]
aaaaabbbbb

22. Corrections for Paun & Rozenberg`s definition
Paun & Rozenberg’s definition for
Our definition for
Similarly for R2 ,R4 ,R5

23. Our module Paun & Rozenberg[1998]
Output for R
Our module　　　　　　　　　　　　 Paun & Rozenberg[1998]

24. Now we can generate the sticker “aaaaabbbbb” in our module as:
Output for A ab
aabb b
abb
aab a A1 A2 A3
Now we can generate the sticker “aaaaabbbbb” in our module as: R6 a aa
aabb
aab bb b bb
aaaaabbbbb A3 R4

25. 9.Advantages of Haskell
Haskell has a description for infinite set (Example 5,6).
We can define the dominos using set theoretical notations in Haskell (Example 7).
10. Related works
HaLex is a Haskell library enables us to model and manipulate regular languages.
HaLex also provide the facilities for defining deterministic and non deterministic finite Automata, Regular expressions etc.
Our module use a description for infinite set, which is not used in HaLex.

26. Example 5:
Let Then produce the infinite set as follows
infinite set {a,b}* is denoted by finite length of expression (sstar ['a','b']).
Using (take) function we can view contents of an infinite set
*AutomatonEx> take 10 $ sstar['a','b']
["","a","b","aa","ba","ab","bb","aaa","baa","aba"]

27. *AutomatonEx> Automaton.language m1
Example 6
Produce infinite set. Therefore use take function to retrieve finite set.
*AutomatonEx> Automaton.language m1
*AutomatonEx> take 10 $ Automaton.language m1
["bb","abb","bbb","aabb","babb","abbb","bbbb","aaabb","baabb","ababb"]

28. Example 7 : Consider the following definition for domino
We can use Haskell set theoretical notation as follows

29. 11.Conclusion and future works
Based on the facts in Paun & Rozenberg [1998] we have modified an insufficient results in there, and introduced a more simple technique to transform Automata into Sticker module using Haskell module functions.
We suppose to publish an online tutorial, which can be used as a study material to learn Automata and language theories.