Theory of computing, part 4

Theory of computing, part 4

Course outline 1 Introduction 2
Theoretical background Biochemistry/molecular biology 3 Theoretical background computer science 4 History of the field 5 Splicing systems 6 P systems 7 Hairpins 8 Detection techniques 9 Micro technology introduction 10 Microchips and fluidics 11 Self assembly 12 Regulatory networks 13 Molecular motors 14 DNA nanowires 15 Protein computers 16 DNA computing - summery 17 Presentation of essay and discussion

Turing machines

The language hierarchy
Context-Free Languages Regular Languages

The language hierarchy
Languages accepted by Turing Machines Context-Free Languages Regular Languages

A Turing machine Tape ...... ...... Read-Write head Control Unit

...... ...... The tape No boundaries -- infinite length
Read-Write head The head moves Left or Right

...... ...... The tape Read-Write head The head at each time step:
1. Reads a symbol 2. Writes a symbol 3. Moves Left or Right

Example Time 0 ...... ...... Time 1 ...... ...... 1. Reads 2. Writes
3. Moves left

...... ...... ...... ...... Example Time 1 Time 2 1. Reads 2. Writes
3. Moves right

The input string Input string Blank symbol ...... ...... head
Head starts at the leftmost position of the input string

States and transitions
Write Read Move Left Move Right

Example Time 1 ...... ...... current state

Example Time 1 ...... ...... Time 2 ...... ......

Determinism Turing Machines are deterministic Allowed Not Allowed

...... ...... Example: partial transition function Allowed:
No transition for input symbol

Halting The machine halts if there are no possible transitions to follow

Example ...... ...... No possible transition HALT

Final states Allowed Not Allowed
Final states have no outgoing transitions In a final state the machine halts

Acceptance If machine halts in a final state Accept Input
in a non-final state or If machine enters an infinite loop Reject Input

Turing machine example
A Turing machine that accepts language a*

Time 0

Time 1

Time 2

Time 3

Time 4 Halt & Accept

Rejection example Time 0

Rejection example Time 1 No possible Transition Halt & Reject

Infinite loop example Another Turing machine for language a* and is this one correct???

Infinite loop example Time 0

Infinite loop example Time 2 Time 3 Time 4 Time 5 ... Infinite Loop

Infinite loop example Because of the infinite loop:
The final state cannot be reached The machine never halts The input is not accepted

Another Turing machine example
Turing machine for the language

Time 0

Time 1

Time 2

Time 3

Time 4

Time 5

Time 6

Time 7

Time 8

Time 9

Time 10

Time 11

Time 12

Time 13 Halt & Accept

Observation If we modify the machine for the language
we can easily construct a machine for the language

Formal definitions

Transition function

Turing machine Input alphabet Tape alphabet States Final states
Transition function Initial state blank

Configuration Instantaneous description:

Configuration Time 4 Time 5 A Move:

Configuration Time 4 Time 5 Time 6 Time 7

Configuration Equivalent notation:

Initial configuration
Input string

The accepted language For any Turing Machine Initial state Final state

Standard Turing machine
The machine we described is the standard Deterministic Infinite tape in both directions Tape is the input/output file

Computing functions

Functions A function has: Domain: Result Region:

Functions A function may have many parameters Example:
Addition function

Integer domain Decimal: 5 Binary: 101 Unary: 11111
We prefer unary representation: easier to manipulate with Turing machines

Functions definition A function is computable if
there is a Turing Machine such that: Initial configuration Final configuration initial state final state For all Domain

Functions definition A function is computable if
there is a Turing Machine such that: Initial Configuration Final Configuration For all Domain

Example The function is computable are integers Turing Machine:
Input string: unary Output string: unary

Example Start initial state The 0 is the delimiter that
separates the two numbers

Example Start initial state Finish final state

Example The 0 helps when we use the result for other operations Finish
final state

Turing machine for function

Time 0 Execution Example: (2) (2) Final Result

Time 0

Time 1

Time 2

Time 3

Time 4

Time 5

Time 6

Time 7

Time 8

Time 9

Time 10

Time 11

Time 12 HALT & accept

Another example The function is computable is integer Turing Machine:
Input string: unary Output string: unary

Another example Start initial state Finish final state

Pseudocode Turing Machine Pseudocode for Replace every 1 with $
Repeat: Find rightmost $, replace it with 1 Go to right end, insert 1 Until no more $ remain

Example Turing Machine for

Example Start Finish

Another example The function if if is computable

Another example if Turing Machine for if Input: Output: or

Pseudocode Repeat Match a 1 from with a 1 from
Until all of or is matched If a 1 from is not matched erase tape, write 1 else erase tape, write 0

Combining Turing machines

Combining Turing machines
Block Diagram input Turing Machine output

Example if if Adder Comparer Eraser

Turing’s thesis

Turing’s thesis Question:
Do Turing machines have the same power with a digital computer? Intuitive answer: Yes There is no formal answer!!!

Turing’s thesis Any computation carried out by mechanical means can be performed by a Turing Machine (1930)

Computer science law A computation is mechanical if and only if it can be performed by a Turing Machine There is no known model of computation more powerful than Turing Machines

Definition of an algorithm
An algorithm for function is a Turing Machine which computes

Algorithms are Turing machines
When we say: There exists an algorithm We mean: There exists a Turing Machine that executes the algorithm

Molecular application

Literature A nanoscale programmable computing machine with input, output, software and hardware made of biomolecules Nature 414, (2001) Ehud Shapiro, Dept of Computer Science & Applied Math Weizmann Institute, Israel

Finite automaton, an example
An even number of b’s b S0, a  S0 S0, b  S1 S1, a  S1 S1, b  S0 a a S0 S1 b Two-states, two-symbols automaton

Automaton 1 b a b S0 S0, a  S0 S0, b  S1 S1, a  S1 S1, b  S0
An even number of b’s S0, a  S0 S0, b  S1 S1, a  S1 S1, b  S0 S0 b a b

Automaton 1 S0, b  S1 b a b S0 S0, a  S0 S0, b  S1 S1, a  S1
An even number of b’s S0, b  S1 S0, a  S0 S0, b  S1 S1, a  S1 S1, b  S0 S0 b a b

Automaton 1 a b S1 S0, a  S0 S0, b  S1 S1, a  S1 S1, b  S0
An even number of b’s S0, a  S0 S0, b  S1 S1, a  S1 S1, b  S0 S1 a b

Automaton 1 S1, a  S1 a b S1 S0, a  S0 S0, b  S1 S1, a  S1
An even number of b’s S1, a  S1 S0, a  S0 S0, b  S1 S1, a  S1 S1, b  S0 S1 a b

Automaton 1 b S1 S0, a  S0 S0, b  S1 S1, a  S1 S1, b  S0
An even number of b’s S0, a  S0 S0, b  S1 S1, a  S1 S1, b  S0 S1 b

Automaton 1 S1, b  S0 b S1 S0, a  S0 S0, b  S1 S1, a  S1
An even number of b’s S1, b  S0 S0, a  S0 S0, b  S1 S1, a  S1 S1, b  S0 S1 b

Automaton 1 S0 S0, a  S0 S0, b  S1 S1, a  S1 S1, b  S0
An even number of b’s S0, a  S0 S0, b  S1 S1, a  S1 S1, b  S0 S0 The output

Rationale for the molecular design

CTGGCT GACCGA CGCAGC GCGTCG a b

CTGGCT GACCGA CGCAGC GCGTCG GGCT CAGC a b
Rationale for the molecular design CTGGCT GACCGA CGCAGC GCGTCG a b S0, a S0, b GGCT CAGC

CTGGCT GACCGA CGCAGC GCGTCG GGCT CAGC CTGGCT GA CGCAGC CG a b
Rationale for the molecular design CTGGCT GACCGA CGCAGC GCGTCG a b S0, a S0, b GGCT CAGC S1, a S1, b CTGGCT GA CGCAGC CG

CAGCCTGGCTCGCAGCTGTCGC GACCGAGCGTCGACAGCG
Rationale for the molecular design Transitions S0, b CAGCCTGGCTCGCAGCTGTCGC GACCGAGCGTCGACAGCG a b t

Rationale for the molecular design Transitions S0, b CAGCCTGGCTCGCAGCTGTCGC GACCGAGCGTCGACAGCG a b t S0, b  S1

CTGGCTCGCAGCTGTCGC GAGCGTCGACAGCG Rationale for the molecular design
Transitions S1, a CTGGCTCGCAGCTGTCGC GAGCGTCGACAGCG b t S0, b  S1

Transitions S1, a CTGGCTCGCAGCTGTCGC GAGCGTCGACAGCG b t S1, a  S1

CGCAGCTGTCGC CGACAGCG Rationale for the molecular design S1, b t
Transitions S1, b CGCAGCTGTCGC CGACAGCG t S1, a  S1

Transitions S1, b CGCAGCTGTCGC CGACAGCG t S1, b  S0

Transitions S0, t TCGC S1, b  S0

Transitions S0, t TCGC Output: S0

Rationale for the molecular design Transition procedure: a concept S0, b CAGCCTGGCTCGCAGCTGTCGC GACCGAGCGTCGACAGCG a b t

Rationale for the molecular design Transition procedure: a concept S0, b CAGCCTGGCTCGCAGCTGTCGC GACCGAGCGTCGACAGCG a b t GTCG 4 nt 8 nt S0, b -> S1

Rationale for the molecular design Transition procedure: a concept GTCG 4 nt 8 nt CAGCCTGGCTCGCAGCTGTCGC GACCGAGCGTCGACAGCG b t S0, b -> S1

Transition procedure: a concept S1, a CTGGCTCGCAGCTGTCGC GAGCGTCGACAGCG b t S0, b -> S1

Transition procedure: a concept S1, a CTGGCTCGCAGCTGTCGC GAGCGTCGACAGCG b t S1, a -> S1

CTGGCTCGCAGCTGTCGC GAGCGTCGACAGCG GACC
Rationale for the molecular design Transition procedure: a concept S1, a CTGGCTCGCAGCTGTCGC GAGCGTCGACAGCG b t GACC 6 nt 10 nt S1, a -> S1

CTGGCTCGCAGCTGTCGC GAGCGTCGACAGCG GACC
Rationale for the molecular design Transition procedure: a concept GACC 6 nt 10 nt CTGGCTCGCAGCTGTCGC GAGCGTCGACAGCG t S1, a -> S1

Transition procedure: a concept S1, b CGCAGCTGTCGC CGACAGCG t S1, a -> S1

CGCAGCTGTCGC CGACAGCG GCGT Rationale for the molecular design S1, b t
Transition procedure: a concept S1, b CGCAGCTGTCGC CGACAGCG t GCGT 8 nt 12 nt S1, b -> S0

CGCAGCTGTCGC CGACAGCG GCGT Rationale for the molecular design
Transition procedure: a concept GCGT 8 nt 12 nt CGCAGCTGTCGC CGACAGCG S1, b -> S0

TCGC Rationale for the molecular design S0, t Output: S0
Transition procedure: a concept S0, t TCGC Output: S0

TCGC AGCG Rationale for the molecular design S0, t Output: S0
In situ detection S0, t Detection molecule for S0 output TCGC AGCG Output: S0

TCGC AGCG Rationale for the molecular design Output: S0
In situ detection TCGC Reporter molecule for S0 output AGCG Output: S0

Inside the transition molecule
GTCG 4 nt 8 nt S0,b -> S1

GGATGACGAC CCTACTGCTG GTCG FokI Inside the transition molecule
4 nt GGATGACGAC CCTACTGCTG GTCG 8 nt S0,b -> S1

9 nt 4 nt GGATGACGAC CCTACTGCTG GTCG 8 nt 13 nt S0,b -> S1

9 nt GGATGACGAC CCTACTGCTG GTCG 13 nt S0,b -> S1

GACC 6 nt 10 nt S1,a -> S1

GGATGACG CCTACTGC GACC FokI Inside the transition molecule
9 nt 6 nt GGATGACG CCTACTGC GACC 10 nt 13 nt S1,a -> S1

GGATGACG CCTACTGC GACC FokI Inside the transition molecule
9 nt GGATGACG CCTACTGC GACC 13 nt S1,a -> S1

8 nt GCGT 12 nt S1,b -> S0

GGATGG CCTACC GCGT FokI Inside the transition molecule S1,b -> S0
9 nt 8 nt GGATGG CCTACC GCGT 12 nt 13 nt S1,b -> S0

GGATGG CCTACC GCGT FokI Inside the transition molecule S1,b -> S0
9 nt GGATGG CCTACC GCGT 13 nt S1,b -> S0

GGATGACGAC CCTACTGCTG GTCG GGATGACG CCTACTGC GACC GGATGG CCTACC GCGT
Inside the transition molecule GGATGACGAC CCTACTGCTG S0 -> S1 GTCG S0 -> S0 GGATGACG CCTACTGC GACC S1 -> S1 GGATGG CCTACC S1 -> S0 GCGT

Transition rules

Automata programmes used

Transition molecules

Input and detection molecules

Experiments on programmes A1-A6

Computation over 6-letter long input

Parallel computation

Identification of the components

Inspection of the intermediates

Estimate of system fidelity

Summary 1012 automata run independently and in parallel
on potentially distinct inputs in 120 ml at room temperature at combined rate of 109 transitions per second with accuracy greater than 99.8% per transition, consuming less than Watt. 12/9/2018

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