Theory of computing, part 4

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

Turing machines

Regular Languages Context-Free Languages The language hierarchy

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

Tape Read-Write head Control Unit A Turing machine

Read-Write head No boundaries -- infinite length 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 The tape

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

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

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

Read Write Move Left Move Right States and transitions

Time 1 current state Example

Time Time 2 Example

Time Time 2 Example

Time Time 2 Example

Allowed Not Allowed Turing Machines are deterministic Determinism

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

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

No possible transitionHALT Example

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

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

A Turing machine that accepts language a * Turing machine example

Time 0 Turing machine example

Time 1 Turing machine example

Time 2 Turing machine example

Time 3 Turing machine example

Time 4 Halt & Accept Turing machine example

Time 0 Rejection example

Time 1 No possible Transition Halt & Reject Rejection example

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

Time 0 Infinite loop example

Time 1 Infinite loop example

Time 2 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 Infinite loop example

Turing machine for the language Another Turing machine example

Time 0 Another Turing machine example

Time 1 Another Turing machine example

Time 2 Another Turing machine example

Time 3 Another Turing machine example

Time 4 Another Turing machine example

Time 5 Another Turing machine example

Time 6 Another Turing machine example

Time 7 Another Turing machine example

Time 8 Another Turing machine example

Time 9 Another Turing machine example

Time 10 Another Turing machine example

Time 11 Another Turing machine example

Time 12 Another Turing machine example

Halt & Accept Time 13 Another Turing machine example

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

Formal definitions

Transition function

States Input alphabetTape alphabet Transition function Initial stateblank Final states Turing machine

Instantaneous description: Configuration

Time 4Time 5 A Move: Configuration

Time 4Time 5 Time 6Time 7 Configuration

Equivalent notation: Configuration

Input string Initial configuration

For any Turing Machine Initial state Final state The accepted language

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

Computing functions

A function Domain:Result Region: has: Functions

A function may have many parameters Example: Addition function Functions

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

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

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

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

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

Start Finish final state initial state Example

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

Turing machine for function Turing machine example

Execution Example: Time 0 Final Result (2) Turing machine example

Time 0 Turing machine example

Time 1 Turing machine example

Time 2 Turing machine example

Time 3 Turing machine example

Time 4 Turing machine example

Time 5 Turing machine example

Time 6 Turing machine example

Time 7 Turing machine example

Time 8 Turing machine example

Time 9 Turing machine example

Time 10 Turing machine example

Time 11 Turing machine example

HALT & accept Time 12 Turing machine example

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

Start Finish final state initial state

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 Pseudocode

Turing Machine for Example

Start Finish Example

The function is computable if Another example

Turing Machine for Input: Output: or if Another example

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

Combining Turing machines

Block Diagram Turing Machineinputoutput Combining Turing machines

if Comparer Adder Eraser Example

Turing’s thesis

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

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

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 Computer science law

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

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